# Traveling fronts for Fisher-KPP lattice equations in almost periodic media

Xing Liang  School of Mathematical Sciences, University of Science and
Technology of China, Hefei, Anhui 230026, China<EMAIL_ADDRESS>,  Hongze
Wang  Chern Institute of Mathematics and LPMC, Nankai University, Tianjin
300071, China<EMAIL_ADDRESS>,  Qi Zhou  Chern Institute of
Mathematics and LPMC, Nankai University, Tianjin 300071, China
<EMAIL_ADDRESS>and  Tao Zhou  School of Mathematical Sciences,
University of Science and Technology of China, Hefei, Anhui 230026, China
<EMAIL_ADDRESS>
###### Abstract.

This paper investigates the existence of almost periodic traveling fronts for
Fisher-KPP lattice equations in one-dimensional almost periodic media. By the
Lyapunov exponent of the linearized operator near the unstable steady state,
we give sufficient condition of the existence of minimal speed of traveling
fronts. Furthermore, it is showed that almost periodic traveling fronts share
the same recurrence property as the structure of the media. As applications,
we give some typical examples which have minimal speed, and the proof of this
depends on dynamical system approach to almost periodic Schrödinger operator.

##  1\. Introduction

###  1.1. Background and main results

Since the pioneer works of [23, 34, 7], the traveling fronts of reaction-
diffusion equations in unbounded domain have become an important branch of the
theory of equations with diffusion. Specially, traveling fronts of the Fisher-
KPP type equation in continuous media

(1.1) $u_{t}-(a(x)u_{x})_{x}=c(x)u(1-u),\quad t\in\mathbb{R},\ x\in\mathbb{R}$

and more general reaction-diffusion equations in heterogeneous media received
intense attention in the last few decades.

As a simplest heterogenous case, traveling fronts in spatially periodic media
were considered widely. First, the definition of spatially periodic traveling
waves was provided by [53, 58] independently, and then [26] proved the
existence of spatially periodic traveling waves of Fisher-KPP equations in the
distributional sense. Then, in the series of works [10, 11], the authors
investigated traveling fronts of Fisher-KPP type equations in high-dimensional
periodic media deeply. The traveling fronts of spatially periodic Fisher-KPP
type equations in discrete lattice

(1.2) $u_{t}(t,n)-u(t,n+1)-u(t,n-1)+2u(t,n)=c(n)u(t,n)(1-u(t,n)),$

were also studied in [24, 39]. Besides above works, a more general framework
was given by [39, 57] to study traveling fronts for Fisher-KPP type equations
and more general diffusion systems.

However, few works on traveling waves of Fisher-KPP equations exist in more
complicated media. Matano [41] first gave a definition of spatially almost
periodic traveling waves and provided some sufficient conditions on the
existence of spatially almost periodic traveling fronts of reaction-diffusion
equations with the bistable nonlinearity. In [38], Liang showed the existence
and uniqueness of the spatially almost periodic traveling front of Fisher-KPP
equations in one-dimensional almost periodic media with free boundary. We also
notice that the propagation problems of (temporally) nonautonomous reaction-
diffusion equations were studied by Shen in a series of works [18, 51, 52].

In this paper, we are concerned with the almost periodic traveling fronts of
the Fisher-KPP equation (1.2). To introduce the definition of the almost
periodic traveling fronts, let we first recall the definition of classical and
periodic traveling fronts. In the case where the media is homogeneous, that
is, $c$ is a constant sequence, the classical traveling fronts are defined by
a solution $u(t,n)=U(n-wt)$ with an invariant profile $U$ and a speed $w$; in
the case where the media is periodic, that is, $c$ is a periodic sequence with
period $N$, the periodic traveling fronts are defined by a solution
$u(t,n)=U(n-wt,n)$ with a periodic profile $U$, $U(\xi,n)=U(\xi,n+N)$ and an
average speed $w$. Then it is natural to consider the recurrence of the
profile depending on the structure of the media if one tries to generalize the
definition of traveling fronts in heterogeneous media. To be exact, in the
almost periodic media, the generalized traveling fronts need to inherit the
almost periodicity of the media.

Before introducing the definition of generalized traveling fronts with almost
periodic recurrence, we give some notions and background. A sequence
$f:\mathbb{Z}\rightarrow\mathbb{R}$ is almost periodic if
${\\{f(\cdot+k)|k\in\mathbb{Z}\\}}$ has a compact closure in
$l^{\infty}(\mathbb{Z})$. Denote by $\mathcal{H}(f)$ the hull of the almost
periodic sequence $f$, i.e.,
$\mathcal{H}(f)=\overline{\\{f(n+k)|k\in\mathbb{Z}\\}}$, the closure in
$l^{\infty}(\mathbb{Z})$. We also denote $g\cdot k=g(\cdot+k),\
g\in\mathcal{H}(f),\ k\in\mathbb{Z}$, and let $l^{\infty}_{loc}(\mathbb{Z})$
be the set of all sequences in $\mathbb{Z}$ with pointwise topology and
$C^{0}(\mathbb{R}\times\mathbb{Z})$ be the set of all continuous functions
with locally uniform topology.

To define the almost periodic traveling fronts, we prefer to consider not only
(1.2), where $c(n)$ is an almost periodic sequence and
$\inf\limits_{\mathbb{Z}}c>0$, but also the family of equations

(1.3) $u_{t}(t,n)-u(t,n+1)-u(t,n-1)+2u(t,n)=g(n)u(t,n)(1-u(t,n)),$

for any $g\in\mathcal{H}(c)$. With these, we have the following precise
definition:

######  Definition 1.1.

Let $v(t,n;g):\mathbb{R}\times\mathbb{Z}\times\mathcal{H}(c)\to\mathbb{R}$. We
say $v(t,n;g)$ is an almost periodic traveling front(with nonzero average
speed) if $v(t,n;g)$ is an entire solution of (1.3) for any
$g\in\mathcal{H}(c)$, and if the following properties hold:

  1. (1)

$\\{v(\cdot,\cdot;g)|g\in\mathcal{H}(c)\\}$ is a one-cover of $\mathcal{H}(c)$
in $C^{0}(\mathbb{R}\times\mathbb{Z})$, i.e. the map $g\in\mathcal{H}(c)\to
v(\cdot,\cdot;g)\in C^{0}({\mathbb{R}}\times{\mathbb{Z}})$ is continuous.

  2. (2)

$v(t,n;g)\to 1$ as $n\to-\infty$, $v(t,n;g)\to 0$ as $n\to\infty$, locally
uniformly in $t$ and uniformly in $g$.

  3. (3)

For any $g\in\mathcal{H}(c)$, there exists
$t(k;g):\mathbb{Z}\times\mathcal{H}(c)\to\mathbb{R}$ such that
$\\{t(\cdot;g)|g\in\mathcal{H}(c)\\}$ is a one-cover of $\mathcal{H}(c)$ in
$l^{\infty}_{loc}(\mathbb{Z})$, and $t(k;g)$ satisfies:

$t(n;g\cdot k)=t(n+k;g)-t(k;g).$

Moreover, $w(g):=\lim\limits_{|n-k|\to\infty}\frac{n-k}{t(n;g)-t(k;g)}$
exists.

  4. (4)

$v(t+t(k;g),n+k;g)=v(t,n;g\cdot k),\ \forall\ t\in\mathbb{R},\
n,k\in\mathbb{Z}.$

The constant $w(g)$ is called the average wave speed of $v(t,n;g)$.

Figure 1.

######  Remark 1.1.

The speed $w(g)$ is nonzero. Indeed,

$t(1;g\cdot k)=t(1+k;g)-t(k;g)$

is bounded with respect to $k$, since $\\{t(k;g)|g\in\mathcal{H}(c)\\}$ is a
one-cover of $\mathcal{H}(c)$ in $l^{\infty}_{loc}(\mathbb{Z})$. Combining
this with the definition of $w(g)$, we deduce that $w(g)$ is nonzero.

######  Remark 1.2.

Actually, Definition 1.1 implies time recurrence of the solution (c.f. Lemma
2.2): For any $\epsilon>0$, there exist relative dense sets
$\\{t_{k}\\}_{k}\subset\mathbb{R}$ and $\\{n_{k}\\}\subset\mathbb{Z}$ such
that

$\sup\limits_{n\in\mathbb{Z}}|v(t_{k},n_{k}+n;g)-v(0,n;g)|\leq\epsilon,$

as shown in Fig 1.

It needs to point out that a more general extension of traveling fronts, so-
called generalized transition front, was presented by Berestycki and Hamel
[8]. For (1.2), the generalized transition front is defined as below.

######  Definition 1.2.

A generalized transition front of (1.2) is an entire solution $u=u(t,n)$ for
which there exists a function $N:\mathbb{R}\rightarrow\mathbb{Z}$ such that

(1.4)
$\lim_{n\rightarrow-\infty}u(t,n+N(t))=1,\lim_{n\rightarrow+\infty}u(t,n+N(t))=0,$

uniformly in $t\in\mathbb{R}$. We say $u$ has an average speed
$w\in\mathbb{R}$, provided

$\lim_{t-s\rightarrow+\infty}\frac{N(t)-N(s)}{t-s}=w.$

######  Remark 1.3.

In the periodic case, the almost periodic traveling front in Definition 1.1 is
exactly the classical periodic traveling front [24]. However, there may exist
a generalized transition front (Definition 1.2) which is not a classical
traveling front [8].

A generalized transition front $u=u(t,x)$ of (1.1) can be defined in the same
way by assuming $N=N(t):\mathbb{R}\rightarrow\mathbb{R}$. Nadin and Rossi [45]
investigated the existence of the generalized transition front of (1.1) when
$a,c$ are almost periodic. Motivated by the works [11, 43, 24, 39], especially
the work of Nadin and Rossi [45], we want to construct the almost periodic
traveling fronts via the eigenvalue problem of the linearized operator of
(1.2) near the equilibrium state $u\equiv 0$:

(1.5) $(\mathcal{L}_{g}u)(n):=u(n+1)+u(n-1)-2u(n)+g(n)u(n)=Eu(n),\
g\in\mathcal{H}(c).$

Here the subcript $g$ is to emphasize the dependence of $g$. We will always
shorten the notation $\mathcal{L}_{c}=\mathcal{L}.$

One novelty of the paper is that we will use method from dynamical systems to
study the operator $\mathcal{L}_{g}$, thus to study (1.3). Note that (1.5) can
be rewritten as

$\begin{pmatrix}u(n+1)\\\ u(n)\end{pmatrix}=A(n)\begin{pmatrix}u(n)\\\
u(n-1)\end{pmatrix},\ $

where $A(n)=\begin{pmatrix}E+2-g(n)&-1\\\ 1&0\end{pmatrix}.$ Let
$A_{n}(g)=A(n-1)\cdots A(0)$ be the transfer matrix. Then the Lyapunov
exponent of $\mathcal{L}_{g}$ at energy $E$ is denoted by $L(E)$ and given by

(1.6)
$L(E):=\lim\limits_{n\rightarrow+\infty}\frac{1}{n}\int_{\mathcal{H}(c)}\ln\|A_{n}(g)\|d\mu\geq
0.$

where $\mu$ is the Haar measure on $\mathcal{H}(c)$. It is known that $L(E)$
is identical for $g\in\mathcal{H}(c)$ (see Proposition 3.4). The Lyapunov
exponent characterizes the decay rate of any solutions of (1.5), and is also a
fundamental topic in smooth dynamical systems.

We always use $\Sigma(\mathcal{L})$ to denote the spectrum of $\mathcal{L}$,
and denote $\lambda_{1}=\max\Sigma(\mathcal{L}).$ Once we have this, we can
state our main results as follows:

######  Theorem 1.1.

Denote

$w^{*}:=\inf_{E>\lambda_{1}}\frac{E}{L(E)},\
\underline{w}:=\lim\limits_{E\searrow\lambda_{1}}\frac{E}{L(E)}.$

Then for (1.3), the following statements hold:

  1. (1)

If $w^{*}<\underline{w}$, then for any $w\in(w^{*},\underline{w})$, there
exists a time-increasing almost periodic traveling front with average wave
speed $w$;

  2. (2)

If $w^{*}<\underline{w}$, then there exists a time-increasing generalized
transition front with average speed $w^{*}$;

  3. (3)

There is no generalized transition fronts with average speed $w<w^{*}.$

Note that the sufficient condition for the existence result in Theorem 1.1(1)
is fulfilled up to a constant perturbation of $g$ (Lemma 4.1), which was first
observed in [45]. Since adding a constant to the potential doesn’t affect the
spectral property of $\mathcal{L}$, in what follows we always assume
$w^{*}<\underline{w}$.

A remarkable fact is Proposition 3.4: For any $E>\lambda_{1}$, there exists a
unique positive solution $\phi_{E}(n)$ of

$\mathcal{L}\phi_{E}=E\phi_{E},\phi_{E}(0)=1,\
\lim\limits_{n\rightarrow\infty}\phi_{E}(n)=0,$

and the limit
$\mu(E):=-\lim\limits_{n\rightarrow\pm\infty}\frac{1}{n}\ln\phi_{E}(n)=L(E)$.
Thus the minimal speed we constructed is the same as that given in [45].

We also should point out that the established almost periodic traveling fronts
share the same recurrence property as the potential $c(n)$. To make it
precisely, if the frequency of the almost periodic sequence $c(n)$ is
$\alpha$, then the frequency of $v(\cdot,\cdot;c)$ is also $\alpha$, as we
will show in Corollary 1.1 and Corollary 1.3.

###  1.2. Applications

Of course, the interesting thing is to give concrete examples in which we can
establish almost periodic traveling fronts for any average wave speed
$w>w^{*}$ (i.e., $\underline{w}=\infty$). Based on the work of [33], Nadin and
Rossi [45] showed that if $a,c$ are finitely differentiable quasi-periodic
function with Diophantine frequency $\alpha$, and $c$ is small enough (the
smallness must depend on $\alpha$), then the operator
$Lu=(a(x)u_{x})_{x}+c(x)u$ has a positive almost periodic function.
Consequently (1.1) has a time-increasing generalized transition front with
average speed $w\in(w^{*},\infty)$. Here we recall that $\alpha$ is
Diophantine (denoted by $\alpha\in\mathrm{DC}_{d}(\gamma,\tau)$), if there
exist $\gamma>0,\tau>d$ such that

$\inf\limits_{j\in\mathbb{Z}}|\langle
k,\alpha\rangle-j|>\frac{\gamma}{|k|^{\tau}},\ 0\neq k\in\mathbb{Z}^{d}.$

Therefore, the natural question is that whether one can remove the arithmetic
condition of $\alpha$, or whether one can remove the smallness of $c$. Now we
answer this question as follows:

######  Corollary 1.1.

Suppose that $c(n)=V(n\alpha)$ where $\alpha\in{\mathbb{T}}^{d}$ is rationally
independent, $V:{\mathbb{T}}^{d}\rightarrow{\mathbb{R}}$ is a positive real
analytic function. Then the following statements hold:

  1. (1)

(1.2) has a time increasing almost periodic traveling front with average wave
speed $w\in(w^{*},\frac{\lambda_{1}}{L(\lambda_{1})})$.

  2. (2)

The traveling front $u(t,n;c)$ can be rewritten as
$u(t,n;c)=U(t+T(n),n\alpha)$, where $U\in
C^{0}(\mathbb{R}\times\mathbb{T}^{d}),\
T\in\ell_{\mathrm{loc}}^{\infty}(\mathbb{Z})$.

Corollary 1.1 (1) reduces $\underline{w}=\infty$ to $L(\lambda_{1})=0$. For
example, based on former results [2, 59], Corollary 5.5 shows that for any
irrational $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, if $V$ is analytic and
close to constant, even the closeness is independent of $\alpha$. Hence
combining Proposition 3.5 and the continuity of the Lyapunov exponent [14,
15], $L(\lambda_{1})=0$. On the other hand, Corollary 1.1 (2) states that if
$c(n)$ is quasi-periodic with frequency $\alpha$, then the resulting traveling
front is also quasi-periodic with frequency $\alpha$.

Corollary 1.1 (1) follows from Theorem 1.1 and the continuity of the Lyapunov
exponent. Thus the assumption that $V$ is analytic is necessary, since the
Lyapunov exponent might be discontinuous in smooth topology [56]. Furthermore,
Corollary 5.4 also states the results for $V$ is just a finitely
differentiable quasi-periodic function as in [45]. To be exact, if $\alpha\in
DC_{d}(\gamma,\tau)$, $V\in C^{s}(\mathbb{T}^{d},\mathbb{R})$ with
$s>6\tau+2$, and $V$ is small enough, then (1.2) has a time increasing almost
periodic traveling front with average wave speed $w\in(w^{*},\infty)$.
However, it is widely believed that if the regularity is worser, then for
generic $V$, the spectrum of (1.5) has no absolutely continuous component [4].
Therefore, most probably $L(\lambda_{1})>0$ by the well-known Kotani’s theory
[35].

As a concrete and typical example, we will take
$V(\theta)=2\kappa\cos\theta+C$ where $C$ is a constant such that
$V(\theta)>0$, then the corresponding linearized operator can be reduced to
the well-known almost Mathieu operator (AMO):

$(\mathcal{L}_{2\kappa\cos-2,\alpha,\theta}u)(n)=u(n+1)+u(n-1)+2\kappa\cos(\theta+n\alpha)u(n).$

The AMO was first introduced by Peierls [48], as a model for an electron on a
2D lattice, acted on by a homogeneous magnetic field [27, 50]. Now, if
$V(\theta)=2\kappa\cos\theta+C$, then we have the following:

######  Corollary 1.2.

Suppose that $c(n)=2\kappa\cos(2\pi n\alpha)+C$ where
$\alpha\in\mathbb{R}\backslash\mathbb{Q}$ and $C$ is a specified constant such
that $c$ has a positive infimum. Then the following statements hold:

  1. (1)

If $|\kappa|\leq 1$, then (1.3) has a time-increasing quasi-periodic traveling
front with average wave speed $w\in(w^{*},\infty).$

  2. (2)

If $|\kappa|>1$, then (1.3) has a time-increasing quasi-periodic traveling
front with average speed $w\in(w^{*},\frac{\lambda_{1}}{|\ln\kappa|}).$

Note that AMO plays the central role in the Thouless et al theory of the
integer quantum Hall effect [54]. This model has been extensively studied not
only because of its importance in physics [46], but also as a fascinating
mathematical object [5, 6, 31, 32]. For us, the example is interesting since
Nadin and Rossi [45] asked is it possible to construct a rigorous example
where $\lim_{E\rightarrow\lambda_{1}}\mu(E)>0$? Then Corollary 1.2 (2) gives
an affirmative answer to their question in the discrete setting. However, it
is still open whether in this case, (1.2) has a time-increasing almost
periodic traveling front with average wave speed $w\in(w^{*},\infty).$

From this aspect, we should mention that compared to [45], we are mainly
concerned with Fisher-KPP lattice equations, since the main results we
mentioned above [2, 3, 14, 15, 31] only work in the discrete case. Whether the
corresponding results are valid for the continuous case are widely open.

We also remark that we are mainly concerned with almost periodic traveling
fronts of (1.2) with average wave speed $w\in(w^{*},\infty)$, while Nadin and
Rossi’s example [45] and Corollary 1.1 only give examples where $c$ are quasi-
periodic. Therefore, it is interesting and important to give concrete examples
of real almost periodic sequence $c(n)$, such that (1.2) has almost periodic
traveling front with average wave speed $w\in(w^{*},\infty).$ To state the
result clearly, let’s show how to construct the desired almost periodic
sequence.

We assume that the frequency $\alpha=(\alpha_{j})_{j\in{\mathbb{N}}}$ belongs
to the infinite dimensional cube $\mathcal{R}_{0}:=[1,2]^{{\mathbb{N}}}$, and
we endow $\mathcal{R}_{0}$ with the probability measure $\mathcal{P}$ induced
by the product measure of the infinite dimensional cube $\mathcal{R}_{0}$. We
now define the set of Diophantine frequencies that was first developed by
Bourgain [13]:

######  Definition 1.3 ([13]).

Given $\gamma\in(0,1),\tau>1$, we denote by $DC_{\infty}(\gamma,\tau)$ the set
of Diophantine frequencies $\alpha\in\mathcal{R}_{0}$ such that

$|\langle
k,\alpha\rangle|\geq\gamma\prod\limits_{j\in\mathbb{N}}\frac{1}{1+|k_{j}|^{\tau}\langle
j\rangle^{\tau}},\ \forall
k\in\mathbb{Z}^{\infty},0<\sum\limits_{j\in\mathbb{N}}|k_{j}|<\infty.$

where $\langle j\rangle:=\max\\{1,|j|\\}$.

As proved in [13, 16], for any $\tau>1$, Diophantine frequencies
$DC_{\infty}(\gamma,\tau)$ are typical in the set $\mathcal{R}_{0}$ in the
sense that there exists a positive constant $C(\tau)$ such that

$\mathcal{P}(\mathcal{R}_{0}\backslash DC_{\infty}(\gamma,\tau))\leq
C(\tau)\gamma.$

Now for any $\alpha\in DC_{\infty}(\gamma,\tau)$, we say that
$c(n):\mathbb{Z}\rightarrow\mathbb{R}$ is an almost periodic sequence with
frequency $\alpha$ and analytic in the strip $r>0$ if we may write it in
totally convergent Fourier series

(1.7)
$c(n)=\sum\limits_{k\in\mathbb{Z}_{*}^{\infty}}\hat{c}(k)\mathrm{e}^{\mathrm{i}\langle
k,\alpha\rangle n}\text{ such that }\hat{c}(k)\in\mathbb{R},\
\sum\limits_{k\in\mathbb{Z}_{*}^{\infty}}|\hat{c}(k)|\mathrm{e}^{r|k|_{1}}<\infty,$

where
$\mathbb{Z}_{*}^{\infty}:=\\{k\in\mathbb{Z}^{\infty}:|k|_{1}:=\sum\limits_{j\in\mathbb{N}}\langle
j\rangle|k_{j}|<\infty\\}$ denotes the set of infinite integer vectors with
finite support. Since $\alpha$ is rationally independent,
$\mathcal{H}(c)={\mathbb{T}}^{\infty}=\prod\limits_{i\in\mathbb{N}}\mathbb{T}^{1}$
with infinite product topology. Once we have these, we can introduce our
precise results as follows:

######  Corollary 1.3.

Let $\gamma\in(0,1),\tau>1$, $r>0$, $\alpha\in DC_{\infty}(\gamma,\tau)$.
Suppose that $c(n)$ is an almost periodic sequence with frequency $\alpha$ and
analytic in the strip $r>0$. Furthermore, suppose that there exists
$\epsilon=\epsilon(\gamma,\tau,r)$ such that

$\sum\limits_{k\in\mathbb{Z}_{*}^{\infty}}|\hat{c}(k)|\mathrm{e}^{r|k|_{1}}<\epsilon(\gamma,\tau,r).$

Then following statements hold:

  1. (1)

(1.2) has a time increasing almost periodic traveling front with average wave
speed $w\in(w^{*},\infty)$

  2. (2)

The front $u(t,n;c)$ can be rewritten as $u(t,n;c)=U(t+T(n),n\alpha)$, where
$U\in C^{0}(\mathbb{R}\times\mathbb{T}^{\infty}),\
T\in\ell_{\mathrm{loc}}^{\infty}(\mathbb{Z})$.

Finally let’s outline the novelty of the proof of these applications. The
proof depends crucially on dynamical approach to almost-periodic Schödinger
operator, i.e., in order to study the spectral property of Schrödinger
operator (1.5), one only needs to study the corresponding Schrödinger cocycle
(c.f. section 2.6). For analytic quasi-periodic potentials (Corollary 1.1 and
Corollary 1.2), the result follows from the continuity of the Lyapunov
exponent for analytic cocycles. For the almost-periodic case, we need to prove
the existence of positive almost-periodic functions. The key observation is
Lemma 5.1, which says that if the Schrödinger cocycle is reduced to a constant
parabolic cocycle, and the conjugacy is close to the identity, then the
corresponding Schrödinger equation has a positive almost-periodic solution.
Here reducibility means the cocycle can be conjugated to a constant cocycle
(c.f. section 5). From this aspect, the powerful method is KAM. For almost-
periodic Hamiltonian systems, KAM was first developed by Pöschel [49]. One can
consult [13, 16, 42] for more study on similar objects. However, it is well-
known traditional KAM method only works for positive measure parameters, and
here we need to fix the energy to be the supremum of the spectrum, thus the
corresponding cocycle is fixed. To solve the difficulty, the method is to make
good use of fibred rotation number which was first developed by Herman [28]
for quasiperiodic cocycles (not necessarily quasiperiodic Schrödinger
cocycle), and it can be extended to the almost periodic setting.

###  1.3. Structure of the paper

In the section 2, we introduce some preliminary knowledge which will be needed
in our proof. In the section 3, we introduce and investigate some properties
of the generalized principal eigenvalue and the Lyapunov exponent which turn
out to be powerful techniques in constructing the almost periodic traveling
front with any average wave speed $w>w^{*}$. Moreover, we also show the reason
why the existence of positive almost periodic solution of (1.5) implies
$\underline{w}=\infty$, c.f. Proposition 3.5.

In the section 4, we prove Theorem 1.1 by the following steps: First we
establish the almost traveling front with any average wave speed $w>w^{*}$ by
constructing super-sub solution, and get the monotonicity of the fronts in
$t$, thus proves (1) in Theorem 1.1. Next we make use of the properties of
spreading speed to deduce that even the generalized transition fronts with the
average speed $w<w^{*}$ can not exist, and then (2) is proved. Lastly, we
construct a generalized transition front with the critical wave speed $w^{*}$
by pulling back the solution of the Cauchy problem associated with the initial
datum Heaviside function, and then we finish the proof.

In the section 5, we use KAM method to get the positive quasi-periodic
(almost-periodic) solution of (1.5) with the positive infimum when
$c=V(\cdot\alpha+\theta)$ is very close to a constant, where $V\in
C^{s}(\mathbb{T}^{d},\mathbb{R})(V\in
C^{\omega}(\mathbb{T}^{\infty},\mathbb{R}))$. This will help us to prove
Corollary 1.3. At last, we finish all the proofs of applications.

##  2\. Preliminaries

###  2.1. Maximum principle, existence and uniqueness for the Cauchy Problem

The maximum principle on the whole space can be stated as follows:

######  Proposition 2.1 (Maximum principle[18]).

Let $v\in\ell^{\infty}(\mathbb{Z})$. Assume that for any bounded interval
$I=[0,t_{0}]\subset[0,\infty)$, $u$ is bounded in $I\times\mathbb{Z}$. If $u$
satisfies

(2.1) $\left\\{\begin{aligned}
u_{t}(n)-u(n+1)-u(n-1)+\big{(}2-v(n)\big{)}u(n)&\geq 0&\text{a.e. in
}I\times{\mathbb{Z}},\\\ u(n)&\geq 0&\text{ in
}\\{0\\}\times{\mathbb{Z}},\end{aligned}\right.$

then $u\geq 0$ in $I\times\mathbb{Z}.$

The following Harnack inequality is a very useful technique when we study the
properties of the solution of (1.2). We will present it here for the reader’s
convenience.

######  Proposition 2.2 (Harnack inequality[40]).

Assume that $u$ is bounded on $(0,\infty)\times\mathbb{Z}$ and solves (2.1).
Then for any $(t,n)\in(0,\infty)\times\mathbb{Z},\ T>0$, there exists a
positive constant $C=C(T)$ such that

$u(t,n)\leq C(T)u(t+T,m),\ m\in\\{n\pm 1,n\\}.$

######  Remark 2.1.

The proof of the Harnack inequality could be found in [40] with the initial
value $u(0,n)$ having a finite support. However, we should notice that the
argument can be applied to (2.1) similarly with minor modification.

Combining the Harnack inequality with the maximum principle, we can deduce the
strong maximum principle as follows:

######  Corollary 2.1 (Strong maximum principle[40]).

Under the assumption of Proposition 2.1, either $u\equiv 0$ or $u>0$ in
$I\times\mathbb{Z}$.

The comparison principle is a consequence of the strong maximum principle, and
it is useful for us to construct the almost periodic traveling front. To state
it, we first give the definition of super-sub solutions:

Let $\bar{u},\underline{u}\in C(\mathbb{R}\times\mathbb{Z})$ be two bounded
functions. We say that $\bar{u}$ is a supersolution of (1.2) if for any given
$n\in\mathbb{Z}$, $\bar{u}$ is absolutely continuous in $t$ and satisfies

$\bar{u}_{t}-\bar{u}(n+1)-\bar{u}(n-1)+2\bar{u}(n)-c\bar{u}(1-\bar{u})\geq
0\text{ for a.e. }t\in(0,\infty),$

and $\underline{u}$ is a subsolution if for any given $n\in\mathbb{Z}$,
$\underline{u}$ is absolutely continuous in $t$ and satisfies

$\underline{u}_{t}-\underline{u}(n+1)-\underline{u}(n-1)+2\underline{u}(n)-c\underline{u}(1-\underline{u})\leq
0\text{ for a.e. }t\in(0,\infty).$

The strong comparison principle is given by

######  Proposition 2.3 (Strong comparison principle).

Let $\overline{u}$ and $\underline{u}$ be a supersolution and a subsolution of
(1.2) repectively. If $\underline{u}(0,n)\leq\overline{u}(0,n)$ in
$\mathbb{Z}$, then $\underline{u}<\overline{u}$ or
$\underline{u}\equiv\overline{u}$ in $(0,\infty)\times\mathbb{Z}$.

Usually, well-behaved Cauchy Problem possesses the property that it admits a
unique global solution. Moreover, existence and uniqueness is vital for us
when we construct the generalized transition front with minimal speed $w^{*}$.

######  Theorem 2.1 ([47]).

For any initial value $\varphi(n)\in\ell^{\infty}(\mathbb{Z})$, there exists a
unique $u\in C^{0}(\mathbb{R}\times\mathbb{Z})$ with
$u(t,\cdot)\in\ell^{\infty}(\mathbb{Z})$ for any $t\in(0,\infty)$ such that

$\left\\{\begin{aligned} &u_{t}(n)-u(n+1)-u(n-1)+2u(n)=c(n)u(1-u)&\text{ in
}(0,\infty)\times\mathbb{Z}\\\ &u(0,n)=\varphi(n).\end{aligned}\right.$

###  2.2. Quadratic form and Critical operator

Denote $l_{c}(\mathbb{Z})$ the space of real valued functions on $\mathbb{Z}$
with compact support. The associated bilinear form $l$ of $-\mathcal{L}$ is
defined on $l_{c}(\mathbb{Z})\times l_{c}(\mathbb{Z})$ as

$l(\varphi,\psi):=\frac{1}{2}\sum\limits_{n\in\mathbb{Z}}\sum\limits_{m=\pm
1}\big{(}\varphi(n)-\varphi(n+m)\big{)}\big{(}\psi(n)-\psi(n+m)\big{)}-c(n)\varphi(n)\psi(n).$

We denote by $l(\varphi):=l(\varphi,\varphi)$ the induced quadratic form on
$l_{c}(\mathbb{Z})$. Furthermore, we write $l\geq 0$ if $l(\varphi)\geq 0$ for
all $\varphi\in l_{c}(\mathbb{Z})$.

######  Definition 2.1 ([37]).

Let $l$ be a quadratic form $l$ associated with the Schrödinger operator
$-\mathcal{L}$ such that $l\geq 0$. We say the form is critical if there does
not exist a positive $\varpi\in l^{\infty}_{loc}(\mathbb{Z})$(i.e. $\varpi\geq
0$ and $\varpi\not\equiv 0$ on $\mathbb{Z}$) such that
$l(\varphi)-\sum\limits_{n\in\mathbb{Z}}\varpi(n)\varphi(n)^{2}\geq 0$ for any
$\varphi\in l_{c}(\mathbb{Z})$.

The operator $\mathcal{L}$ is said to be critical, if the quadratic form $l$
associated with $-\mathcal{L}$ is critical.

The following well known formula reveals the connection between the operator
$\mathcal{L}$ and the associated quadratic form $l$.

######  Lemma 2.1 (Green formula[37]).

For all $\varphi,\psi\in l_{c}(\mathbb{Z})$, one has

$\displaystyle\frac{1}{2}\sum\limits_{n\in\mathbb{Z}}\sum\limits_{m=\pm
1}\big{(}\varphi(n)$
$\displaystyle-\varphi(n+m)\big{)}\big{(}\psi(n)-\psi(n+m)\big{)}-\sum\limits_{n\in\mathbb{Z}}c(n)\varphi(n)\psi(n)$
$\displaystyle=-\sum\limits_{n}(\mathcal{L}\varphi)(n)\psi(n)=-\sum\limits_{n}(\mathcal{L}\psi)(n)\varphi(n).$

That is, $l(\varphi,\psi)=\langle-\mathcal{L}\varphi,\psi\rangle$.

The critical operator has the following important property which is useful for
us to reveal the connection between the Lyapunov exponent and positive almost
periodic solution (c.f. Proposition 3.5):

######  Proposition 2.4.

[37] Let $\mathcal{L}$ be a critical operator. Then there exists a unique
positive function in $l^{\infty}_{loc}(\mathbb{Z})$ such that
$\mathcal{L}u\leq 0$ (up to scalar multiplication).

###  2.3. Properties of the almost periodic traveling front

Almost periodic traveling front also implies the almost periodicity of the
time recurrence in the sense of the following lemma.

######  Lemma 2.2.

Assume $u(t,n;g)$ is an almost periodic traveling front of (1.3). Then for any
$\epsilon>0$, there exist relative dense sets $\\{t_{k}\\}\subset\mathbb{R}$
and $\\{n_{k}\\}\subset\mathbb{Z}$ such that
$\sup\limits_{n\in\mathbb{Z}}|u(t_{k},n+n_{k};g)-u(0,n;g)|\leq\epsilon$.

###### Proof.

From Definition 1.1 (1), (2) and (4), for any $k>0$, there exists
$\\{n_{k}\\}\subset\mathbb{Z}$ such that

$\sup\limits_{n\in\mathbb{Z}}|u(0,n;g)-u(0,n;g\cdot
n_{k})|=\sup\limits_{n\in\mathbb{Z}}|u(0,n;g)-u(t(n_{k};g),n+n_{k};g)|\leq\epsilon,$

if $\sup\limits_{n}|g-g\cdot n_{k}|\leq\delta.$ Because $g$ is an almost
periodic sequence, we can always find such a relative dense set $\\{n_{k}\\}$
so that $\sup\limits_{n\in\mathbb{Z}}|g-g\cdot n_{k}|\leq\epsilon$. Take
$\\{t_{k}\\}$ as $\\{t(n_{k};g)\\}$, then it is sufficient to prove
$\\{t(n_{k};g)\\}$ is relative dense.

It follows from the Definition 1.1 (3) that

$t(n_{k+1};g)-t(n_{k};g)=t(n_{k+1}-n_{k};g\cdot n_{k}).$

Since $\\{n_{k}\\}$ is a relative dense set, i.e., there exists $L\geq 0$ such
that for any point $n$ in $\mathbb{Z}$, one has
$[n-L,n+L]\cap\\{n_{k}\\}\neq\emptyset.$ Denote $M:=\max\limits_{-L\leq n\leq
L,g\in\mathcal{H}(c)}t(n;g)$. Since $\\{t(n,g)\\}$ is a one-cover of
$\mathcal{H}(c)$ in $l_{loc}^{\infty}(\mathbb{Z})$ and $\mathcal{H}(c)$ is
compact, $M$ can be obtained. Hence for any $t\in\mathbb{R},$ one has
$[-M+t,M+t]\cap\\{t(n_{k};g)\\}\neq\emptyset$, as desired. ∎

The average wave speed of almost periodic traveling front relates with the
recurrence of $c$ in the following way.

######  Proposition 2.5.

The average wave speed $w(g)$ of the almost periodic traveling front is a
constant function on $\mathcal{H}(c)$.

###### Proof.

Recall that
$w(g)=\lim\limits_{|n-k|\rightarrow\infty}\frac{n-k}{t(n;g)-t(k;g)}$ exists,
and it is not zero. Then
$\frac{1}{w(g)}=\lim\limits_{|n-k|\rightarrow\infty}\frac{t(n;g)-t(k;g)}{n-k}$
exists. By Definition 1.1, $\\{t(1;g\cdot k)|k\in\mathbb{Z}\\}$ has a compact
closure in $l_{loc}^{\infty}(\mathbb{Z})$. Hence it is an almost periodic
sequence. Then for any $k\in\mathbb{Z}$,

$\displaystyle 1/w(g)$
$\displaystyle=\lim\limits_{|n|\rightarrow\infty}\frac{t(n+k;g)-t(k;g)}{n}=\lim\limits_{|n|\rightarrow\infty}\sum\limits_{i=k}^{n+k-1}\frac{t(i+1;g)-t(i;g)}{n}$
$\displaystyle=\lim\limits_{|n|\rightarrow\infty}\sum\limits_{i=k}^{n+k-1}\frac{t(1;g\cdot
i)}{n},$

The limit on the right-hand side exists, and it is independent of
$g\in\mathcal{H}(c)$. Then the proof is complete. ∎

An observation is that an almost traveling front is also a generalized
transition front.

######  Proposition 2.6.

An almost periodic traveling front of (1.3) with average wave speed $w(g)$ is
a generalized transition front of (1.3) with average speed $w(g)$.

###### Proof.

From Definition 1.1(3),
$w(g)=\lim\limits_{|n-k|\rightarrow\infty}\frac{n-k}{t(n;g)-t(k;g)}$ exists.
Without loss of generality, we will always assume that $w(g)>0$. For any
$k\in\mathbb{Z}$, there exists an absolute constant $L$ such that for any
$|n|\geq L$,

(2.2) $\frac{w(g)}{2}\leq\frac{n}{t(k+n;g)-t(k;g)}\leq\frac{3w(g)}{2}.$

Thus

(2.3) $\left\\{\begin{aligned} t(k+n;g)-t(k;g)\geq\frac{2n}{3w(g)}&\text{ for
}n>L,\\\ t(k+n;g)-t(k,g)\leq\frac{2n}{3w(g)}&\text{ for
}n<-L.\end{aligned}\right.$

From (2.3), if $n\rightarrow\pm\infty$, then $t(n;g)\rightarrow\pm\infty$.
Hence we can deduce that for any $t\in\mathbb{R}$, $\sharp\\{n:|t-t(n;g)|\leq
1\\}<\infty$. Then

$N(t):=\min\\{k:|t-t(k;g)|=\min\limits_{n}|t-t(n;g)|\\}$

is well defined.

Claim: For any $t\in\mathbb{R}$, there exists $M>0$ such that

(2.4) $|t-t(N(t);g)|\leq\sup\limits_{k}\sup\limits_{|j|\leq
M}|t(j+k;g)-t(k;g)|.$

Proof of Claim: As we have shown in the proof of Lemma 2.2, $\\{t(n;g)\\}$ is
a relative dense set, i.e. there exists $R>0$ such that for any
$t\in\mathbb{R}$, $(t-R/2,t+R/2)\cap\\{t(n;g)\\}\neq\emptyset$. Now by (2.3)
with $k=N(t)$, for any $|n|\geq\max\bigr{\\{}\frac{3w(g)}{2}R,L\bigr{\\}}$,

$|t(n+N(t);g)-t(N(t);g)|\geq R.$

Now taking $M=\max\\{\frac{3w(g)}{2}R,L\\}$, we finish the proof of the claim.

Combining (2.4) with Definition 1.1 (3), there exists an absolute constant $C$
such that

(2.5) $|t-t(N(t);g)|\leq\sup\limits_{k}\sup\limits_{|j|\leq M}|t(j;g\cdot
k)|\leq C,$

since $\mathcal{H}(c)$ is compact. At last, Definition 1.1 (2) shows that
$v(t,n;g)\rightarrow 1$ as $n\rightarrow-\infty$, $v(t,n;g)\rightarrow\infty$
as $n\rightarrow\infty$ uniformly in $g$. Thus,

$v(t,n+N(t);g)=v\big{(}t-t(N(t);g),n;g\cdot N(t)\big{)}\left\\{\begin{aligned}
\to 1&\text{ as }n\to-\infty,\\\ \to 0&\text{ as
}n\to+\infty,\end{aligned}\right.$

uniformly in $t$. Moreover, by (2.4) and (2.2), one has

$\displaystyle|t-s|$
$\displaystyle\leq|t-t(N(t);g)|+|s-t(N(s);g)|+|t(N(t);g)-t(N(s);g)|$
$\displaystyle\leq 2C+|t(N(t);g)-t(N(s);g)|\leq 2C+\frac{2}{w(g)}(N(t)-N(s)).$

It follows that $|t-s|\rightarrow\infty$ implies $N(t)-N(s)\rightarrow\infty.$
Now applying (2.4) again, we have

$\displaystyle\quad\lim\limits_{t-s\rightarrow\infty}\frac{N(t)-N(s)}{t-s}$
$\displaystyle=\lim\limits_{t-s\rightarrow\infty}\frac{N(t)-N(s)}{t(N(t);g)-t(N(s);g)}\cdot\frac{t(N(t);g)-t(N(s);g)}{t-s}$
$\displaystyle=\lim\limits_{t-s\rightarrow\infty}\frac{N(t)-N(s)}{t(N(t);g)-t(N(s);g)}\cdot\frac{(t(N(t);g)-t)-(t(N(s);g)-s)+(t-s)}{t-s}$
$\displaystyle=\lim\limits_{N(t)-N(s)\rightarrow\infty}\frac{N(t)-N(s)}{t(N(t);g)-t(N(s);g)}=w(g).$

∎

###  2.4.  $\mathrm{SL}(2,{\mathbb{R}})$ cocycles, Uniformly hyperbolic

Let $X$ be a compact metric space, $(X,\nu,T)$ be ergodic, and
$A:X\rightarrow\mathrm{SL}(2,\mathbb{R})$ be a continuous map. An
$\mathrm{SL}(2,\mathbb{R})$ cocycle over $(T,X)$ is an action defined on
$X\times\mathbb{R}^{2}$ such that

$(T,A):(x,v)\in X\times\mathbb{R}^{2}\mapsto(Tx,A(x)\cdot v)\in
X\times\mathbb{R}^{2}.$

For $n\in\mathbb{Z}$, $A_{n}$ is defined by $(T,A)^{n}=(T^{n},A_{n})$, where
$A_{0}(x)=\operatorname{id}$,

$\quad A_{n}(x)=\prod_{j=n-1}^{0}A(T^{j}x)=A(T^{n-1}x)\cdots A(Tx)A(x),\ n\geq
1,$

and $A_{-n}(x)=A_{n}(T^{-n}x)^{-1}$. The Lyapunov exponent is defined as

$L(T,A)=\lim_{n\rightarrow\infty}\frac{1}{n}\int_{X}\ln||A_{n}(x)||d\nu.$

We say an $\mathrm{SL}(2,\mathbb{R})$ cocycle $(T,A)$ is uniformly hyperbolic
if, for every $x\in X$, there exists a continuous splitting
$\mathbb{R}^{2}=E_{s}(x)\oplus E_{u}(x)$ such that for every $n\geq 0$,

(2.6) $\begin{split}|A_{n}(x)v(x)|&\leq Ce^{-cn}|v(x)|,\ v(x)\in E_{s}(x),\\\
|A_{-n}(x)v(x)|&\leq Ce^{-cn}|v(x)|,\ v(x)\in E_{u}(x),\end{split}$

for some constants $C,c>0$, and it holds that $A(x)E_{s}(x)=E_{s}(Tx)$ and
$A(x)E_{u}(x)=E_{u}(Tx)$ for every $x\in X$. Clearly, if $(T,A)$ is uniformly
hyperbolic, then $L(T,A)>0$. If $L(T,A)>0$ and the splitting is not
continuous, then we call $(T,A)$ is non-uniformly hyperbolic.

###  2.5.  Fibered rotation number

Assume $X$ is compact, and $(X,T)$ is uniquely ergodic with respect to its
unique invariant probability measure. For this kind of dynamically defined
cocycles $(T,A)$, one can define the rotation number of the cocycle. Let
$\mathbb{S}^{1}$ be the set of unit vectors of $\mathbb{R}^{2}$, consider a
projective cocycle $F_{T,A}$ on $X\times\mathbb{S}^{1}$:

$(x,v)\mapsto(Tx,\frac{A(x)v}{\|A(x)v\|}).$

If $A:X\rightarrow\mathrm{SL}(2,\mathbb{R})$ is continuous and homotopic to
the identity, then there exists a lift $\tilde{F}_{(T,A)}$ of $F_{(T,A)}$ to
$X\times\mathbb{R}$ such that
$\tilde{F}_{(T,A)}(x,y)=(Tx,y+\tilde{f}_{(T,A)}(x,y))$, where
$\tilde{f}_{(T,A)}:X\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous
lift that satisfies:

  * •

$\tilde{f}_{(T,A)}(x,y+1)=\tilde{f}_{(T,A)}(x,y)+1;$

  * •

for every $x\in X,\
\tilde{f}_{(T,A)}(x,\cdot):\mathbb{R}\rightarrow\mathbb{R}$ is a strictly
increasing homeomorphisim;

  * •

if $\pi_{2}$ is the projection map $X\times\mathbb{R}\rightarrow
X\times\mathbb{S}^{1}:(x,y)\mapsto(x,\mathrm{e}^{2\pi\mathrm{i}y})$, then
$F_{(T,A)}\circ\pi_{2}=\pi_{2}\circ\tilde{F}_{(T,A)}.$

Meanwhile, define the $n$-th iterate as

$\tilde{F}^{n}_{(T,A)}(x,y)=(T^{n}x,y+\tilde{f}^{n}_{(T,A)}(x,y)).$

Then there exists $\rho\in\mathbb{R}$ such that

$\frac{\tilde{f}^{n}_{(T,A)}(x,y)}{n}$

converges uniformly to $\rho$ in $(x,y)\in X\times\mathbb{R}$, and it is
independent on the lift of $F_{T,A}$, up to an addition of an integer [28].
Then $\rho$ is called fibered rotation number of $(T,A)$, and we denote it as
$\mathrm{rot}(T,A)$.

######  Remark 2.2.

In the following, we will always take $X={\mathbb{T}}^{d}$ where
$d\in{\mathbb{N}}_{+}$ or $\infty$ (we endow it with the product topology if
$d=\infty$), and consider the quasi-periodic (or almost-periodic) cocycle
$(\alpha,A)$.

The following two lemmas are are useful for us:

######  Lemma 2.3 ([1]).

For any $A\in\mathrm{SL}(2,{\mathbb{R}})$, we have

$|\mathrm{rot}(\alpha,Ae^{F})-\mathrm{rot}(\alpha,A)|<2\|A\|\|F\|_{0}^{\frac{1}{2}}.$

######  Lemma 2.4 ([36]).

The rotation number is invariant under the conjugation map which is homotopic
to the identity. More precisely, if
$A,B:{\mathbb{T}}^{d}\rightarrow\mathrm{SL}(2,{\mathbb{R}})$ is continuous and
homotopic to the identity, then

$\mathrm{rot}(\alpha,B(\cdot+\alpha)^{-1}A(\cdot)B(\cdot))=\mathrm{rot}(\alpha,A).$

###  2.6. Almost periodic Schrödinger operator

For the almost periodic sequence, we also have the following:

######  Proposition 2.7 ([19]).

A potential $c\in l^{\infty}(\mathbb{Z})$ is almost periodic if and only if it
can be represented as

$c(n)=V_{\omega}(n)=f(T^{n}\omega),$

where $\Omega$ is a compact abelian group, $\omega\in\Omega$,
$f:\Omega\rightarrow\mathbb{R}$ is continuous, and $T:\Omega\rightarrow\Omega$
is a minimal translation, say $T=\cdot+\alpha$.

As result, we define the almost periodic Schrödinger operator as a self-
adjoint operator on $l^{2}({\mathbb{Z}}):$

$(\mathcal{L}_{f,T,\omega}u)(n)=u(n+1)+u(n-1)-2u(n)+f(T^{n}\omega)u(n),\
\forall n\in\mathbb{Z}.$

It’s well known that the spectrum $\Sigma(\mathcal{L}_{f,T,\omega})$ is a
compact set of $\mathbb{R}$. Moreover $\Sigma(\mathcal{L}_{f,T,\omega})$ is
independent of $\omega$, and we shorten the notation as
$\Sigma(\mathcal{L}_{f,T})$. In particular, if $f=V,\ \Omega=\mathbb{T}^{d},\
T=\cdot+\alpha$, where $\alpha,\theta\in\mathbb{T}^{d}$, we denote

$(\mathcal{L}_{V,\alpha,\theta}u)(n):=u(n+1)+u(n-1)-2u(n)+V(n\alpha+\theta)u(n).$

We say $\theta$ is the phase, $V$ is the potential, and $\alpha$ is the
frequency.

For almost-periodic Schrödinger operator, one can define the integrated
density of states (shorten as IDS), denoted by $k(E)$, as follows:

$k(E)=\lim\limits_{L\rightarrow\infty}\frac{\sharp\\{\text{eigenvalues
(counting multiplicity) of $\mathcal{L}_{L}\leq$ }E\\}}{L-1},$

where $\mathcal{L}_{L}$ is the restriction of $\mathcal{L}_{V,\alpha,\omega}$
to the set $I=\\{1,\cdots,L-1\\}$ with boundary conditions
$\frac{u(0)}{u(1)}=cot\theta,u(L)=0.$ The integrated density of states will be
crucial for our study of the positive almost periodic solution. Furthermore,
we have the following elementary fact:

######  Remark 2.3.

Suppose $E$ is at the rightmost of the spectrum. Then by the well known
characterization of the biggest eigenvalue $\lambda^{L}$ and $E$

$\lambda^{{L}}=\sup\limits_{\begin{subarray}{c}u\in
l^{2}(\mathbb{Z}),u\not\equiv 0,\\\ \mathrm{supp}u\subset
I\end{subarray}}\frac{\langle\mathcal{L}u,u\rangle}{\langle u,u\rangle},\quad
E=\sup\limits_{u\in l^{2}(\mathbb{Z}),u\not\equiv
0}\frac{\langle\mathcal{L}u,u\rangle}{\langle u,u\rangle},$

the rightmost of the spectrum $E$ is always large than $\lambda^{L}$. Thus the
number of eigenvalues of $\mathcal{L}_{L}$ less than or equal to $E$ is always
$L-1$. By the definition of the $\mathrm{IDS}$, $k(E)=1.$

Note that a sequence $(u_{n})_{n\in{\mathbb{Z}}}$ is a formal solution of the
eigenvalue equation $\mathcal{L}_{f,T,\omega}u=Eu,$ if and only if

$\begin{pmatrix}u(n+1)\\\
u(n)\end{pmatrix}=S^{f}_{E}(T^{n}\omega)\begin{pmatrix}u(n)\\\
u(n-1)\end{pmatrix},$

where

$S_{E}^{f}(\omega)=\begin{pmatrix}E+2-f(\omega)&-1\\\ 1&0\end{pmatrix}.$

We call $(T,S^{f}_{E})$ an almost-periodic Schrödinger cocycle. In this paper,
we will mainly consider the following two kinds of Schrödinger cocycles.

  * •

$X_{1}=\mathbb{T}^{d},T_{1}\theta=\theta+\alpha,S_{E}^{V}=\begin{pmatrix}E+2-V(\theta)&-1\\\
1&0\end{pmatrix}$, where $d\in\mathbb{N}_{+}$, $\alpha\in\mathbb{T}^{d}$ is
rationally independent. Then $(X_{1},T_{1})$ is uniquely ergodic, and we call
$(\alpha,S_{E}^{V})$ a quasi-periodic Schrödinger cocycle.

  * •

$X_{2}=\mathbb{T}^{\infty}$, that is endowed with the product topology,
$T_{2}\theta=\theta+\alpha$, $S_{E}^{V}=\begin{pmatrix}E+2-V(\theta)&-1\\\
1&0\end{pmatrix}$, where $\alpha\in DC_{\infty}(\gamma,\tau)$, Then
$(X_{2},T_{2})$ is uniquely ergodic, and $(\alpha,S_{E}^{V})$ defines an
almost-periodic Schrödinger cocycle.

The study of the spectral properties of the almost-periodic Schrödinger
operator is closely related to the dynamics of almost-periodic Schrödinger
cocycle. For example, we will need the following two important facts:

######  Theorem 2.2 ([30]).

$\mathrm{IDS}$ of the Schrödinger operator relates with the fibered rotation
number of Schrödinger operator as follows:

$k(E)=1-2\mathrm{rot}(\alpha,S_{E}^{f})\quad(\mathrm{mod}\ \mathbb{Z}).$

######  Theorem 2.3 ([29]).

Let $\mathcal{L}_{f,T,\omega}$ be an almost periodic Schrödinger operator.
Then we have the following:

$\mathbb{R}\backslash\Sigma=\\{E\in\mathbb{R}|(T_{2},S_{E}^{f})\text{ is
uniformly hyperbolic}\\}.$

##  3\. Properties of the Linearized Problem

###  3.1. Generalized principal eigenvalue for more general operator

In this section, we will define and study the properties of generalized
principal eigenvalue of a more general operator since it will be needed in our
proof. Remarkably, generalized principal eigenvalue theory for elliptic
operator is of its own interest, and it turns out to be very useful in
studying maximum principle [9].

Now we consider the operator

$\mathcal{M}_{a,b,c}\phi(n):=a(n)\phi(n+1)+b(n)\phi(n-1)+c(n)\phi(n),$

with $a,b,c$ being almost periodic sequences and $\inf a>0,\ \inf b>0$. In
particular, $\mathcal{M}_{1,1,g-2}=\mathcal{L}_{g}$ defined in (1.5).

For any (maybe unbounded) interval $I\subset\mathbb{Z}$, we define the
generalized principal eigenvalue for $\mathcal{M}_{a,b,c}$ as

(3.1) $\lambda_{1}(\mathcal{M}_{a,b,c},I)=\inf\\{\lambda\in\mathbb{R}:\
\exists\phi>0\text{ in
}\mathbb{Z},\mathcal{M}_{a,b,c;I}\phi\leq\lambda\phi\text{ in }I\\},$

where $\mathcal{M}_{a,b,c;I}$ is the restriction of $\mathcal{M}_{a,b,c}$ to
the set $I\subset\mathbb{Z}$. If $I$ is bounded, it is exactly the classical
principal eigenvalue (the largest eigenvalue). In the case $I=\mathbb{Z},\
a=b\equiv 1,$ $c$ is replaced by $c-2$, we will show below that it coincides
with $\lambda_{1}=\max\Sigma(\mathcal{L})$. From (3.1),
$\lambda_{1}(\mathcal{M}_{a,b,c},I)$ is nondecreasing with respect to the
inclusion of intervals $I.$

######  Proposition 3.1.

There holds

(3.2)
$\lambda_{1}\bigr{(}\mathcal{M}_{a,b,c},(-N,N)\bigr{)}\nearrow\lambda_{1}(\mathcal{M}_{a,b,c},\mathbb{Z})\text{
as }N\rightarrow\infty.$

######  Remark 3.1.

The proof of this Proposition is similar to the proof in [9, Proposition 2.3]
and [12, Proposition 4.2] which works for continuous elliptic operator.

######  Proposition 3.2.

Let $b(n)=a(n-1),I$ be an interval in $\mathbb{Z}$. Then

(3.3) $\lambda_{1}(\mathcal{M}_{a,b,c},I)=\sup_{\begin{subarray}{c}v\in
l^{2}(\mathbb{Z}),v\not\equiv 0,\\\ \mathrm{supp}v\subset
I\end{subarray}}\frac{\langle M_{a,b,c}v,v\rangle}{\langle v,v\rangle}.$

where $\langle\cdot\rangle$ denotes the inner product on $l^{2}(\mathbb{Z})$.
Moreover,
$\lambda_{1}(\mathcal{L},\mathbb{Z})=\lambda_{1}=\max\Sigma(\mathcal{L})$.

###### Proof.

For simplicity, we shorten the notation
$\mathcal{M}_{a,b,c;N}:=\mathcal{M}_{a,b,c;[-N,N]},$
$\lambda_{1}^{N}:=\lambda_{1}(\mathcal{M}_{a,b,c};[-N,N])$.

We only prove the case $I=\mathbb{Z}.$ Since $b(n)=a(n-1)$, it is clear that
$\mathcal{M}_{a,b,c;N}$ is a bounded self-adjoint operator. Then for any
$N\in\mathbb{N}_{+},$

(3.4) $\lambda_{1}^{N}=\sup\limits_{\begin{subarray}{c}v\in
l^{2}(\mathbb{Z}),v\not\equiv 0,\\\
\mathrm{supp}v\subset[-N,N]\end{subarray}}\frac{\langle\mathcal{M}_{a,b,c;N}v,v\rangle}{\langle
v,v\rangle}=\|\mathcal{M}_{a,b,c;N}\|,$

here $\|\cdot\|$ denotes the norm in the Banach space of linear bounded
operators from $l^{2}(\mathbb{Z})$ to $l^{2}(\mathbb{Z}).$

Since $a,b,c$ is almost periodic, hence bounded. By direct calculation,

$\lim\limits_{N\rightarrow\infty}|\|\mathcal{M}_{a,b,c}\|-\|\mathcal{M}_{a,b,c;N}\||\leq\lim\limits_{N\rightarrow\infty}\|\mathcal{M}_{a,b,c}-\mathcal{M}_{a,b,c;N}\|=0.$

From (3.4) and Proposition 3.1, we have

$\displaystyle\lambda_{1}(\mathcal{M}_{a,b,c},\mathbb{Z})$
$\displaystyle=\lim\limits_{N\rightarrow\infty}\lambda_{1}(\mathcal{M}_{a,b,c;N})=\lim\limits_{N\rightarrow\infty}\|\mathcal{M}_{a,b,c;N}\|$
$\displaystyle=\|\mathcal{M}_{a,b,c}\|=\sup\limits_{v\in
l^{2}(\mathbb{Z}),v\not\equiv 0}\frac{\langle M_{a,b,c}v,v\rangle}{\langle
v,v\rangle}.$

In particular, $\lambda_{1}(\mathcal{M}_{a,b,c},\mathbb{Z})$ lies in the
rightmost of the spectrum. ∎

From Proposition 3.2, we can deduce that
$\lambda_{1}(\mathcal{L},\mathbb{Z})\geq\inf c.$ Indeed, taking the test
function $v_{2N}(n)=\left\\{\begin{aligned} 1&&|n|\leq 2N\\\ 0&&\text{ else
}\end{aligned}\right.$ in (3.3), we deduce the generalized principal
eigenvalue $\lambda_{1}(\mathcal{L},[-N,N])\geq\inf c.$ By Proposition 3.1, we
have $\lambda_{1}(\mathcal{L},\mathbb{Z})\geq\inf c,$ as desired.

Other generalizations of principal eigenvalue will be listed below, and they
are all indispensable to our proof. Define the following quantities:

(3.5)
$\displaystyle\underline{\lambda}_{1}(\mathcal{M}_{a,b,c}):=\sup\left\\{\lambda:\exists\phi\in\mathscr{S},\mathcal{M}_{a,b,c}\phi\geq\lambda\phi\text{
in }\mathbb{Z}\right\\};$
$\displaystyle\overline{\lambda}_{1}(\mathcal{M}_{a,b,c}):=\inf\left\\{\lambda:\exists\phi\in\mathscr{S},\mathcal{M}_{a,b,c}\phi\leq\lambda\phi\text{
in }\mathbb{Z}\right\\};$

(3.6)
$\displaystyle\underline{\lambda}_{1}^{\prime}(\mathcal{M}_{a,b,c}):=\sup\\{\lambda:\exists\phi>0,\
\phi\in l^{\infty}(\mathbb{Z}),\mathcal{M}_{a,b,c}\phi\geq\lambda\phi\text{ in
}\mathbb{Z}\\};$
$\displaystyle\lambda_{1}(\mathcal{M}_{a,b,c}):=\lambda_{1}(\mathcal{M}_{a,b,c};\mathbb{Z})=\inf\\{\lambda:\exists\phi>0,\mathcal{M}_{a,b,c}\phi\leq\lambda\phi\text{
in }\mathbb{Z}\\};$

(3.7)
$\displaystyle\underline{\mu}_{1}(\mathcal{M}_{a,b,c}):=\sup\\{\mu:\exists\phi\in
l^{\infty}(\mathbb{Z}),\
\inf\limits_{n\in\mathbb{Z}}\phi>0,\mathcal{M}_{a,b,c}\phi\geq\mu\phi\text{ in
}\mathbb{Z}\\};$
$\displaystyle\overline{\mu}_{1}(\mathcal{M}_{a,b,c}):=\inf\\{\mu:\exists\phi\in
l^{\infty}(\mathbb{Z}),\
\inf\limits_{n\in\mathbb{Z}}\phi>0,\mathcal{M}_{a,b,c}\phi\leq\mu\phi\text{ in
}\mathbb{Z}\\};$

where
$\mathscr{S}=\bigr{\\{}\phi>0:\lim\limits_{|n|\rightarrow+\infty}\frac{\log\phi(n)}{n}=0,\bigr{\\{}\frac{\phi(n+1)}{\phi(n)}\bigr{\\}}_{n\in\mathbb{Z}}\in
l^{\infty}(\mathbb{Z})\bigr{\\}}.$ We denote the hull of triple $(a,b,c)$ by
$\mathcal{H}(a,b,c)$:

$\biggr{\\{}(a^{*},b^{*},c^{*}):a\cdot n_{i}\rightarrow a^{*},b\cdot
n_{i}\rightarrow b^{*},c\cdot n_{i}\rightarrow c^{*}\text{ for some
}\\{n_{i}\\}_{i\in\mathbb{Z}}\biggr{\\}}.$

######  Proposition 3.3.

The following hold:

  * (1)

$\underline{\lambda}_{1}(\mathcal{M}_{a,b,c})=\overline{\lambda}_{1}(\mathcal{M}_{a,b,c})=\underline{\mu}_{1}(\mathcal{M}_{a,b,c})=\overline{\mu}_{1}(\mathcal{M}_{a,b,c})$.
Moreover, if $a(n)=b(n+1)$, then they all coincide with
$\lambda_{1}(\mathcal{M}_{a,b,c})$ and
$\underline{\lambda}_{1}^{\prime}(\mathcal{M}_{a,b,c})$.

  * (2)

$\lambda_{1}(\mathcal{M}_{a^{\prime},b^{\prime},c^{\prime}}),\
\underline{\mu}_{1}(\mathcal{M}_{a^{\prime},b^{\prime},c^{\prime}}),\
\overline{\mu}_{1}(\mathcal{M}_{a^{\prime},b^{\prime},c^{\prime}}),\
\overline{\lambda}_{1}(\mathcal{M}_{a^{\prime},b^{\prime},c^{\prime}})$ are
constant functions with respect to $(a,b,c)$ on $\mathcal{H}(a,b,c)$. If
$a(n)=b(n+1)$, then $\lambda_{1}(\mathcal{M}_{a,b,c})$,
$\underline{\lambda}^{\prime}_{1}(\mathcal{M}_{a,b,c})$ are also constant
functions.
In particular, $\lambda_{1}(\mathcal{L}_{g},\mathbb{Z})$ is a constant
function with respect to $g$ on $\mathcal{H}(c).$

###### Proof.

(1) It is straightforward to check by (3.5),(3.6) and (3.7) that

(3.8)
$\underline{\mu}_{1}(\mathcal{M}_{a,b,c})\leq\underline{\lambda}_{1}^{\prime}(\mathcal{M}_{a,b,c}),\
\lambda_{1}(\mathcal{M}_{a,b,c})\leq\overline{\mu}_{1}(\mathcal{M}_{a,b,c})$

and

(3.9)
$\underline{\mu}_{1}(\mathcal{M}_{a,b,c})\leq\underline{\lambda}_{1}(\mathcal{M}_{a,b,c}),\
\overline{\lambda}_{1}(\mathcal{M}_{a,b,c})\leq\overline{\mu}_{1}(\mathcal{M}_{a,b,c}).$

Moreover, by the standard argument in [9, Theorem 1.7], we can prove that
$\underline{\lambda}_{1}^{\prime}(\mathcal{M}_{a,b,c})\leq\lambda_{1}(\mathcal{M}_{a,b,c})$
if $a(n)=b(n+1)$. From [40, Proposition 2.1], we also have
$\underline{\lambda}_{1}(\mathcal{M}_{a,b,c})\leq\overline{\lambda}_{1}(\mathcal{M}_{a,b,c})$.

For any $\epsilon>0,$ [40, Lemma 5.2] guarantees that the existence of an
almost periodic function $u_{\epsilon}$ which satisfies

$\mathcal{M}_{a,b,c}\mathrm{e}^{u_{\epsilon}}=\epsilon
u_{\epsilon}\mathrm{e}^{u_{\epsilon}},\
-\frac{\|c\|_{l^{\infty}(\mathbb{Z})}}{\epsilon}<u^{\epsilon}<\frac{2+\|c\|_{l^{\infty}(\mathbb{Z})}}{\epsilon}.$

Now we choose $\mathrm{e}^{u_{\epsilon}}\in l^{\infty}(\mathbb{Z})$ with
$\inf\limits_{n\in\mathbb{Z}}\mathrm{e}^{u_{\epsilon}}>0$ as a test function
in (3.7). Meanwhile, one has $\epsilon u_{\epsilon}\rightarrow\lambda$
uniformly by [40, Lemma 5.2]. Thus for any $\delta>0$, there exists
$\epsilon_{\delta}$ such that
$|\epsilon_{\delta}u_{\epsilon_{\delta}}-\lambda|<\delta,$ and

$(\lambda-\delta)\mathrm{e}^{u_{\epsilon_{\delta}}}\leq\mathcal{M}_{a,b,c}\mathrm{e}^{u_{\epsilon_{\delta}}}=\epsilon_{\delta}u_{\epsilon_{\delta}}\mathrm{e}^{u_{\epsilon_{\delta}}}\leq(\lambda+\delta)\mathrm{e}^{u_{\epsilon_{\delta}}}.$

Then letting $\delta\rightarrow 0$,
$\underline{\mu}_{1}(\mathcal{M}_{a,b,c})\geq\lambda\geq\overline{\mu}_{1}(\mathcal{M}_{a,b,c})$
follows by the definition. Thus by (3.9), we obtain
$\lambda=\underline{\mu}_{1}(\mathcal{M}_{a,b,c})=\underline{\lambda}_{1}(\mathcal{M}_{a,b,c})=\overline{\lambda}_{1}(\mathcal{M}_{a,b,c})=\overline{\mu}_{1}(\mathcal{M}_{a,b,c})$.
Consequently, the desired result is obtained by (3.8).

(2). Analogous to the above argument, for any sequences
$\\{n_{i}\\}_{i\in\mathbb{Z}}$ such that

$a\cdot n_{i}\rightarrow a^{*},b\cdot n_{i}\rightarrow b^{*},c\cdot
n_{i}\rightarrow c^{*},$

we have

$\mathcal{M}_{a\cdot n_{i},b\cdot n_{i},c\cdot
n_{i}}\mathrm{e}^{u_{\epsilon}\cdot n_{i}}=\epsilon u_{\epsilon}\cdot
n_{i}\mathrm{e}^{u_{\epsilon}\cdot n_{i}}.$

Notice when $\epsilon\rightarrow 0$, $\epsilon u_{\epsilon}\rightarrow\lambda$
uniformly. Hence for any $\delta>0$, there exists $\epsilon_{\delta}$ such
that $|\epsilon_{\delta}u_{\epsilon_{\delta}}\cdot n-\lambda|\leq\delta$ holds
for any $n.$ Thereby

$(\lambda+\delta)\mathrm{e}^{u_{\epsilon_{\delta}}\cdot
n_{i}}\geq\mathcal{M}_{a\cdot n_{i},b\cdot n_{i},c\cdot
n_{i}}\mathrm{e}^{u_{\epsilon_{\delta}}\cdot
n_{i}}=\epsilon_{\delta}u_{\epsilon_{\delta}}\cdot
n_{i}\mathrm{e}^{u_{\epsilon_{\delta}}\cdot
n_{i}}\geq(\lambda-\delta)\mathrm{e}^{u_{\epsilon_{\delta}}\cdot n_{i}}.$

Note that $u_{\varepsilon}$ is almost periodic for any $\epsilon>0$. Passing
along a subsequence $i_{k}\rightarrow\infty$, for any $\delta>0$, it follows
from (1) that

$\lambda-\delta\leq\underline{\mu}_{1}(\mathcal{M}_{a^{*},b^{*},c^{*}})=\underline{\lambda}_{1}(\mathcal{M}_{a^{*},b^{*},c^{*}})=\overline{\lambda}_{1}(\mathcal{M}_{a^{*},b^{*},c^{*}})=\overline{\mu}_{1}(\mathcal{M}_{a^{*},b^{*},c^{*}})\leq\lambda+\delta.$

Finally, letting $\delta\rightarrow 0$,

$\lambda=\underline{\mu}_{1}(\mathcal{M}_{a^{*},b^{*},c^{*}})=\underline{\lambda}_{1}(\mathcal{M}_{a^{*},b^{*},c^{*}})=\overline{\lambda}_{1}(\mathcal{M}_{a^{*},b^{*},c^{*}})=\overline{\mu}_{1}(\mathcal{M}_{a^{*},b^{*},c^{*}}).$

If $a(n)=b(n+1)$, apply the above argument to
$\underline{\lambda}_{1}^{\prime}(\mathcal{M}_{a,b,c}),\
\lambda_{1}(\mathcal{M}_{a,b,c})$, then we are done. ∎

###  3.2. The Lyapunov exponent of the linearized operator

The Lyapunov exponent is crucial to determine the average wave speed of the
almost periodic traveling front. We will discuss its properties here and
illustrate the connection with the existence of positive almost periodic
solution. Recall

$(\mathcal{L}_{g}u)(n)=u(n+1)+u(n-1)-2u(n)+g(n)u(n),$

and $\mathcal{L}=\mathcal{L}_{c}$. Note that Proposition 3.3 tells us
$\lambda_{1}(\mathcal{L}_{g},\mathbb{Z})=\lambda_{1}(\mathcal{L},\mathbb{Z})$,
for any $g\in\mathcal{H}(c)$. As a consequence of Lemma 3.2,
$\lambda_{1}=\max\Sigma(\mathcal{L})=\lambda_{1}(\mathcal{L},\mathbb{Z})$.
Hence we do not distinguish them with a slight abuse of notation in the
forthcoming paragraphs. First we state the following technical lemma which
will be frequently used, and it is an immediate consequence of [40, Lemma
6.2].

######  Lemma 3.1.

Let $E>\lambda_{1}$, $n_{0}\in\mathbb{Z}$, and $\phi$ defined in
$[n_{0},+\infty)\cap\mathbb{Z}$ satisfy

$\mathcal{L}\phi\geq E\phi\text{ in
}(n_{0},+\infty)\cap\mathbb{Z},\lim_{n\rightarrow+\infty}\phi(n)=0.$

Then there are positive constants $C,\delta$ only depending on $E,\lambda_{1}$
and $\|c(n)\|_{l^{\infty}}$ such that

$\phi(n)\leq C\max\\{\phi(n_{0}),0\\}e^{-\delta(n-n_{0})}.$

Moreover, $\lim\limits_{E\to\lambda_{1}}\delta=0$ and
$\lim\limits_{E\to+\infty}\delta=+\infty$.

Now we study the Lyapunov exponent $L(E)$ of $\mathcal{L}$ defined in (1.6)
and our observation is stated as follows:

######  Proposition 3.4.

For all $E>\lambda_{1}$, there exists a unique positive solution
$\phi_{E}(\cdot;g)$ of

(3.10) $\mathcal{L}_{g}\phi=E\phi\text{ in }\mathbb{Z},\ \phi(0)=1,\
\lim\limits_{n\rightarrow\infty}\phi(n)=0.$

and the limit

$L(E)=-\lim\limits_{n\rightarrow\pm\infty}\frac{1}{n}\ln\phi_{E}(n;g)>0,\text{
for any }g\in\mathcal{H}(c),$

the convergence is uniform in $g\in\mathcal{H}(c).$

First we need the following lemma:

######  Lemma 3.2 ([55]).

Let $T:X\rightarrow X$ be a uniquely ergodic homeomorphism of the compact
metric space $X$ and $(T,A)$ be an $\mathrm{SL}(2,\mathbb{R})$ cocycle over
the probability space $(X,\mu)$. If $(T,A)$ is uniformly hyperbolic, then
$\frac{1}{n}\log\|A_{n}(x)\|$ converges uniformly to a constant.

###### Proof of Proposition 3.4.

The existence and uniqueness of the positive solution follows from [40, Lemma
6.3].

Now let $X=\mathcal{H}(c),T:\mathcal{H}(c)\rightarrow\mathcal{H}(c)$ with
$Tg=g\cdot 1$ for any $g\in\mathcal{H}(c)$,
$S^{E}(g)=\begin{pmatrix}E+2-g(0)&-1\\\ 1&0\end{pmatrix}\in
C^{0}(\mathcal{H}(c),\mathrm{SL}(2,\mathbb{R})).$ Then $(T,S^{E})$ is a
cocycle defined on $(X,\mu)$, where $\mu$ is the Haar measure on
$\mathcal{H}(c).$ Note that $T$ is uniquely ergodic on $\mathcal{H}(c)$
because $\mathcal{H}(c)$ is a compact Abelian group and $T$ is minimal.

Since $E>\lambda_{1}$, by Theorem 2.3, $(T,S^{E})$ is uniformly hyperbolic.
Then as a consequence of Lemma 3.2, the limit
$L^{E}:=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\ln\|S^{E}_{n}(g)\|$
exists, and it is independent of $g\in\mathcal{H}(c).$ Moreover, the
convergence is uniform in $g\in\mathcal{H}(c).$ Thus we can deduce that for
any $g\in\mathcal{H}(c)$,

$L^{E}=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\ln\|S^{E}_{n}(g)\|=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\int_{\mathcal{H}(c)}\ln\|S^{E}_{n}(g)\|d\mu=L(T_{2},A)=L(E)$

where the last equality follows from the direct examination of the definition.

Note that in the proof of Lemma 3.2, we can deduce that the limit
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}\ln|S_{n}^{E}(g)\cdot v|$ exists
for any $v\in\mathbb{R}^{2}\backslash\\{0\\}$. Now it follows from (2.6) that
uniform hyperbolicity implies that for any $g\in\mathcal{H}(c)$, and every
$n\geq 0$, only $v(g)\in E_{s}(g)\subset\mathbb{R}^{2}$ satisfies
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}\ln|S_{n}^{E}(g)v(g)|<0$.
Otherwise, all $v\in E_{s}(g)$ violates (2.6).

Denote $v^{g}_{E}=(\phi_{E}(1;g),\phi_{E}(0;g))$ for $g\in\mathcal{H}(c)$. By
Lemma 3.1 and (3.10), one has

(3.11) $\lim\limits_{n\rightarrow\infty}\frac{1}{n}\ln|S^{E}_{n}(g)\cdot
v_{E}^{g}|=\lim\limits_{n\rightarrow\infty}\frac{\ln(\phi_{E}^{2}(n;g)+\phi_{E}^{2}(n-1;g))}{2n}<0.$

Hence $v_{E}^{g}\in E_{s}(g)$, and
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}\ln|S_{n}^{E}(g)\cdot
v_{E}^{g}|=-L(E)$ since $S^{E}_{n}(g)$ takes value in
$\mathrm{SL}(2,\mathbb{R})$.

Meanwhile, since $S_{-n}^{E}(g)v_{E}^{g}\in E_{s}(T^{-n}g)$ in (2.6), one has

$|v_{E}^{g}|=|S_{n}^{E}(T^{-n}g)S_{-n}^{E}(g)v^{g}_{E}|\leq
Ce^{-nL(E)}|S_{-n}^{E}v_{E}^{g}|.$

Then we can deduce that

(3.12)
$\lim\limits_{n\rightarrow\infty}\frac{1}{n}|S_{-n}^{E}(g)v_{E}^{g}|=\lim\limits_{n\rightarrow\infty}\frac{\ln(\phi^{2}_{E}(-n)+\phi_{E}^{2}(-n+1))}{n}=L(E).$

From equation $(\mathcal{L}_{g}\phi_{E}(\cdot;g))(n)=E\phi_{E}(n;g)$, we have

$\frac{\phi_{E}(n+1;g)}{\phi_{E}(n;g)}+\frac{\phi_{E}(n-1;g)}{\phi_{E}(n;g)}=2+g(n)\leq
M,$

where $M$ depends on $\|g\|_{l^{\infty}}$. It follows that

$\frac{1}{M}\phi_{E}(n;g)\leq\phi_{E}(n+1;g)\leq M\phi_{E}(n;g).$

Inserting this inequality into (3.11) and (3.12),
$L(E)=\lim\limits_{n\rightarrow\pm\infty}-\frac{\ln\phi_{E}(n;g)}{n}$ follows
directly. Then the proof is complete. ∎

The concavity and monotonicity of the Lyapunov exponent $L(E)$ will be needed
in our construction of the almost periodic traveling front, and it is given
by:

######  Lemma 3.3.

The function $E\mapsto L(E)$ defined on $(\lambda_{1},+\infty)$ is concave,
nondecreasing and there exists $C>0$ such that, for $E>\lambda_{1}$,

(3.13) $\delta\leq L(E)\leq C\sqrt{E},$

where $\delta$ was given in Lemma 3.1 and
$L(E)>\underline{L}:=\lim\limits_{E\searrow\lambda_{1}}L(E).$

###### Proof.

$L(E)$ is nondecreasing and concave follows from [45, Lemma 2.5].

For $E_{1}<E_{2}$, let $\phi_{E_{1}}$ and $\phi_{E_{2}}$ be obtained in
Proposition 3.4. It is straightforward to check that $\phi_{E_{2}}$ is a
subsolution of the equation satisfied by $\phi_{E_{1}}$ in $[0,\infty)$.
Applying Lemma 3.1 to $\phi_{E_{2}}-\phi_{E_{1}}$, monotonicity follows
directly. Also Lemma 3.1 shows that,

$\displaystyle L(E)$
$\displaystyle=-\lim\limits_{n\rightarrow+\infty}\frac{1}{n}\ln\phi_{E}(n;g)\geq-\lim\limits_{n\rightarrow+\infty}\frac{C}{n}+\delta=\delta.$

Let $h_{E}(n;g)=e^{-n\sqrt{E-\inf c}}-\phi_{E}(n;g)$ with $\phi_{E}(\cdot;g)$
satisfying

$\mathcal{L}_{g}\phi=E\phi\text{ in }{\mathbb{Z}},\
\phi_{E}(0)=1,\lim\limits_{n\rightarrow\infty}\phi(n)=0.$

We can check $h_{E}$ satisfies $\mathcal{L}_{g}h_{E}\geq Eh_{E}$ and
$\lim\limits_{n\rightarrow\infty}h_{E}(n;g)=0.$ By Lemma 3.1, we have
$\phi_{E}(n;g)\geq e^{-n\sqrt{E-\inf c}}$, whence $L(E)\leq\sqrt{E-\inf c}\leq
C\sqrt{E}$, as desired. ∎

The existence of positive almost periodic solution always implies the Lyapunov
exponent $L(E)$ will decay to 0 as $E\searrow\lambda_{1}$, and it is crucial
for us to determine in which case we can establish the almost periodic
traveling front with average wave speed $w\in(w^{*},\infty)$ (c.f. Corollaries
1.2 and 1.3). First we need a preliminary lemma about critical operator.

######  Lemma 3.4.

Suppose that $\mathcal{L}$ admits a positive bounded solution $\varphi$ of
$\mathcal{L}\phi=E\phi$. Then the associated eigenvalue is the generalized
principal eigenvalue $\lambda_{1}$ and there hold:

  * (1)

$\mathcal{L}-\lambda_{1}$ is critical;

  * (2)

if $\inf\varphi>0$, then $\mathcal{L}_{g}-\lambda_{1}$ is critical for any
$g\in\mathcal{H}(c)$;

  * (3)

$\inf\varphi>0$ if and only if $\varphi$ is almost periodic.

###### Proof.

Taking $\varphi$ as the test function in (3.6), it follows from Proposition
3.3 the associated eigenvalue is exactly the generalized principal eigenvalue
$\lambda_{1}$.

(1). Assume by contradiction that $\mathcal{L}-\lambda_{1}$ is not critical.
Denote the quadratic form associated with $\lambda_{1}-\mathcal{L}$ by $h$.
Then by Lemma 2.1, there exists a positive function $\varpi\in
l^{\infty}_{loc}(\mathbb{Z})$ such that
$h(u)=\langle(\lambda_{1}-\mathcal{L})u,u\rangle\geq\langle\varpi u,u\rangle$
for any $u\in l_{c}.$ That is, $\mathcal{L}-\lambda_{1}\pm\varpi$ are bounded
above by 0(the supreme of the spectrum of $\mathcal{L}-\lambda_{1}$, see
Proposition 3.2, i.e. the ground state energe of $\mathcal{L}-\lambda_{1}$).
It follows from [20, Theorem 4] that $\varpi\equiv 0$, this is impossible
since $\varpi$ positive. Hence $\mathcal{L}-\lambda_{1}$ is critical.

The proofs of (2) and (3) are similar to that of [45, Propsition 1.7]. ∎

Although the positive solution of $\mathcal{L}\phi=E\phi$ may not be almost
periodic, almost periodicity can still be revealed in the following way:

######  Lemma 3.5.

For all $E>\lambda_{1}$, the function $\frac{\phi_{E}(n+1;g)}{\phi_{E}(n;g)}$
is almost periodic.

###### Proof.

The method is similar to the proof of [45, Lemma 2.4], and Lemma 3.1 is needed
in the proof. ∎

Once we have this, the following result follows:

######  Proposition 3.5.

Assume that $\mathcal{L}$ admits a positive almost periodic solution $\phi$ of
$\mathcal{L}\phi=\lambda_{1}\phi$. Then we have
$\underline{L}:=\lim\limits_{E\rightarrow\lambda_{1}}L(E)=0.$

###### Proof.

Denote $\phi_{E}(\cdot):=\phi_{E}(\cdot;c)$ which can be obtained in
Proposition 3.4. Consider the analogue $\tilde{\phi}_{E}$ of $\phi_{E}$ but
with the initial condition as follows:

$\mathcal{L}\tilde{\phi}_{E}=E\tilde{\phi}_{E}\text{ in }\mathbb{Z},\
\tilde{\phi}_{E}(0)=1,\
\lim\limits_{n\rightarrow-\infty}\tilde{\phi}_{E}(n)=0.$

The function $\tilde{\phi}_{E}(-n)$ shares the same properties with
$\phi_{E}$. Particularly, the limit
$\tilde{L}(E):=\lim\limits_{n\rightarrow\pm\infty}\frac{1}{n}\ln\tilde{\phi}_{E}(n)$
exists and it is positive. Let $\varphi_{E}:=\sqrt{\tilde{\phi}_{E}\phi_{E}}$.
Then by the direct computation that

Thus,

(3.14) $(\mathcal{L}-E)\varphi_{E}=-\frac{1}{2}\varphi_{E}q_{E},$

with
$q_{E}=-\left(\frac{\phi_{E}^{\frac{1}{2}}(n+1)}{\phi_{E}^{\frac{1}{2}}(n)}-\frac{\tilde{\phi}_{E}^{\frac{1}{2}}(n+1)}{\tilde{\phi}_{E}^{\frac{1}{2}}(n)}\right)^{2}-\left(\frac{\phi_{E}^{\frac{1}{2}}(n-1)}{\phi_{E}^{\frac{1}{2}}(n)}-\frac{\tilde{\phi}_{E}^{\frac{1}{2}}(n-1)}{\tilde{\phi}_{E}^{\frac{1}{2}}(n)}\right)^{2}.$

We claim that $q_{E}$ converges uniformly to 0 in $\mathbb{R}$ as
$E\rightarrow\lambda_{1}$. Otherwise, there exist $\epsilon>0$ and two
sequences $\\{E_{i}\\}_{i}$ and $\\{n_{i}\\}_{i}$ such that
$E_{i}\rightarrow\lambda_{1}$ and $|q_{E_{i}}(n_{i})|\geq\epsilon$ for all
$i\in\mathbb{N}$. According to Lemma 3.5,
$\frac{\phi_{E}(\cdot+1)}{\phi_{E}(\cdot)}$ is almost periodic, hence
$\\{q_{E_{i}}\\}$ is uniformly bounded. Passing along a subsequence
$n_{i_{k}}$ in (3.14),
$\frac{\varphi_{E}(\cdot+n_{i_{k}})}{\varphi(n_{i_{k}})}$ converges pointwise
to a positive solution $\varphi^{*}$ of

$(\mathcal{L}_{g_{*}}-\lambda_{1})\varphi^{*}=-\frac{1}{2}\varphi^{*}q$

where $g_{*}=\lim\limits_{k\rightarrow\infty}c\cdot n_{i_{k}}$ and $q$ is the
limit of $\\{q_{E_{i_{k}}}(n+n_{i_{k}})\\}_{k}$. Since $|q(0)|\geq\epsilon$,
$\mathcal{L}_{g_{*}}$ admits a supersolution which is not a solution. And from
the assumption, Lemma 3.4 (2),(3) and Proposition 2.4, $\mathcal{L}_{g_{*}}$
admits a unique positive supersolution. However, as will see, $\varphi$ and
$\phi$ are both positive supersolutions. This is impossible! Therefore we have
$q_{E}\rightarrow 0$ unifromly as $E\rightarrow\lambda_{1}$. Finally we have

$\tilde{L}(E)+L(E)=\lim\limits_{n\rightarrow+\infty}\frac{1}{n}\sum\limits_{0}^{n}\left(-\ln\frac{\phi_{E}(n+1)}{\phi_{E}(n)}+\ln\frac{\tilde{\phi}_{E}(n+1)}{\tilde{\phi}_{E}(n)}\right)$

tends to 0 as $E\rightarrow\lambda_{1}$. Then they both tend to 0 as
$E\rightarrow\lambda_{1}$ due to the positivity of $L(E)$ and $\tilde{L}(E)$.
∎

##  4\. Construction of the fronts

###  4.1. Construction of almost periodic fronts

In this section, we start to prove Theorem 1.1. First we will consider (1) in
Theorem 1.1 in this subsection. The basic idea is to apply the super-sub
solution method.

From now on, we set

$\sigma_{E}(n;g):=-\ln\frac{\phi_{E}(n+1;g)}{\phi_{E}(n;g)}\text{ for any
}n\in\mathbb{Z},\ g\in\mathcal{H}(c),$

where $\phi_{E}(n;g)$ is given by Proposition 3.4. Notice that
$\sigma_{E}(n;g)$ is almost periodic by Lemma 3.5.

Before constructing the the almost periodic traveling front, we should first
notice the following facts:

######  Lemma 4.1.

Let $w^{*},\ \underline{w}$ be defined in Theorem 1.1. Then

  1. (1)

$w^{*}<\underline{w}$ provided $g$ in the equation (1.3) replaced by $g+c_{0}$
with $c_{0}$ large enough.

  2. (2)

$w^{*}$ is a minimum provided $w^{*}<\underline{w}$. Moreover, for all
$w\in(w^{*},\underline{w}),$ there exists $E>\lambda_{1}$ such that
$w=\frac{E}{L(E)}$ and $w>\frac{E^{\prime}}{L(E^{\prime})}$ for
$E^{\prime}-E>0$ small enough.

###### Proof.

The proofs can be referred to [45, Lemmas 1.6 and 3.2]. ∎

Now afterwards, we take $w\in(w^{*},\underline{w})$, and let $E>\lambda_{1}$
be as in Lemma 4.1. For a given almost periodic sequence $\sigma$, we define
the operator

(4.1) $\displaystyle\mathcal{L}^{\sigma}_{g}\phi$
$\displaystyle:=\mathrm{e}^{\sum\limits_{0}^{n-1}\sigma}\mathcal{L}_{g}(\mathrm{e}^{-\sum\limits_{0}^{n-1}\sigma}\phi)$
$\displaystyle=\mathrm{e}^{-\sigma(n)}\phi(n+1)+\mathrm{e}^{\sigma(n-1)}\phi(n-1)+g(n)\phi(n).$

######  Proposition 4.1.

For any $g\in\mathcal{H}(c)$, there exist absolute constants $\delta>0,\
\epsilon\in(0,1)$ and a function $\theta\in l^{\infty}(\mathbb{Z})$ satisfying

$\inf\limits_{\mathbb{Z}}\theta>0,\
-\mathcal{L}^{(1+\epsilon)\sigma_{E}(\cdot;g)}_{g}\theta\geq(\delta-(1+\epsilon)E)\theta\text{
in }\mathbb{Z}.$

###### Proof.

As we choose $E$ in the Lemma 4.1, there exists $\epsilon\in(0,1)$ such that

$\frac{E}{L(E)}>\frac{(1+\epsilon)E}{L((1+\epsilon)E)}.$

Now we define

$F(\kappa):=\frac{1}{L(E)}-\frac{1+\epsilon}{L(E+\kappa)}.$

Then $F(E\epsilon)=\frac{1}{L(E)}-\frac{1+\epsilon}{L((1+\epsilon)E)}>0$ and
$F(0)=-\frac{\epsilon}{L(E)}<0$. As $L(E)$ is concave in
$(\lambda_{1},\infty)$, it is continuous. Hence $F$ is continuous. Then there
exists $\kappa\in(0,\epsilon E)$ so that $F(\kappa)=0.$

Consider now the function

$\zeta(n;g):=\frac{\phi_{E+\kappa}(n;g)}{\phi_{E}^{1+\epsilon}(n;g)}\text{ for
all }n\in\mathbb{Z}.$

First, since
$e^{-(1+\epsilon)\sum\limits_{0}^{n-1}\sigma_{E}(\cdot;g)}=\phi_{E}^{1+\epsilon}(n;g),$
the positive function $\zeta$ satisfies

$\displaystyle\mathcal{L}_{g}^{(1+\epsilon)\sigma_{E}(\cdot;g)}\zeta$
$\displaystyle=\mathrm{e}^{(1+\epsilon)\sum\limits_{0}^{n-1}\sigma_{E}(\cdot;g)}\mathcal{L}_{g}\biggr{(}\mathrm{e}^{-(1+\epsilon)\sum\limits_{0}^{n-1}\sigma_{E}(\cdot;g)}\zeta\biggr{)}$
$\displaystyle=\frac{\mathcal{L}_{g}(\zeta\phi_{E}^{1+\epsilon}(\cdot;g))}{\phi_{E}^{1+\epsilon}(n;g)}=\frac{\mathcal{L}_{g}(\phi_{E+\kappa}(\cdot;g))}{\phi_{E}^{1+\epsilon}(n;g)}=(E+\kappa)\zeta.$

Then it follows that

$-(\mathcal{L}_{g}^{(1+\epsilon)\sigma_{E}(\cdot;g)}-(1+\epsilon)E)\zeta=(\epsilon
E-\kappa)\zeta\text{ in }\mathbb{Z}.$

Moreover,

$\displaystyle\lim\limits_{n\rightarrow\pm\infty}\frac{1}{n}\ln\zeta(n;g)$
$\displaystyle=\lim\limits_{n\rightarrow\pm\infty}\frac{1}{n}\ln\phi_{E+\kappa}(n;g)-\lim\limits_{n\rightarrow\pm\infty}\frac{1+\epsilon}{n}\ln\phi_{E}(n;g)$
$\displaystyle=-L(E+\kappa)+(1+\epsilon)L(E)=0,$

where the last equality follows from the definition of $\kappa$.

Next, it follows from Lemma 3.5 that
$\frac{\zeta(\cdot+1;g)}{\zeta(\cdot;g)}\in l^{\infty}(\mathbb{Z})$, and the
coefficients of
$\mathcal{M}_{a,b,c}=\mathcal{L}_{g}^{(1+\epsilon)\sigma_{E}(\cdot;g)}-(1+\epsilon)E$
are almost periodic. Applying Proposition 3.3 with $\mathcal{M}_{a,b,c}$, then
$\lambda:=\overline{\lambda}_{1}(\mathcal{M}_{a,b,c})=\underline{\lambda}_{1}(\mathcal{M}_{a,b,c})=\kappa-\epsilon
E$. Let $\delta^{\prime}=\epsilon E-\kappa>0$. Thus for any
$0<\delta<\delta^{\prime}$, there exists a function
$\theta:=\mathrm{e}^{u_{\epsilon^{\prime}}(\cdot;g)}$ with
$|\epsilon^{\prime}u_{\epsilon^{\prime}}-\lambda|<\delta^{\prime}-\delta$
which is defined in the proof of Proposition 3.3 satisfies

$\mathcal{L}_{g}^{(1+\epsilon)\sigma_{E}(\cdot;g)}\theta-(1+\epsilon)E\theta=\epsilon^{\prime}u_{\epsilon^{\prime}}\theta\leq(\lambda+\delta^{\prime}-\delta)\theta\leq-\delta\theta\text{
in }\mathbb{Z}.$

Moreover
$0<\inf\limits_{\mathbb{R}}\theta\leq\sup\limits_{\mathbb{R}}\theta<+\infty$
since $u_{\epsilon^{\prime}}(\cdot;g)$ is bounded. ∎

It is clear that the choice of $\theta$ is not unique. Let us now define
$\\{\theta(\cdot;g)\\}_{g\in\mathcal{H}(c)}$ in the following certain way.
Note that the function $\theta=e^{u_{\epsilon^{\prime}}}$ with
$|\epsilon^{\prime}u_{\epsilon^{\prime}}-\lambda|<\delta^{\prime}-\delta$ in
Proposition 4.1 is the unique almost periodic solution (see [40, Lemma 5.1])
of

(4.2)
$\big{(}\mathcal{L}^{(1+\epsilon)\sigma_{E}(n;g)}_{g}-(1+\epsilon)E\big{)}\theta=\epsilon^{\prime}u_{\epsilon^{\prime}}\theta\text{
in }\mathbb{Z}.$

Indeed, almost periodicity follows from the inequality

$\|u_{\epsilon^{\prime}}\cdot n_{i}-u_{\epsilon^{\prime}}\cdot
n_{j}\|_{l^{\infty}}\leq\frac{C}{\epsilon^{\prime}}\max\\{\|a\cdot
n_{i}-a\cdot n_{j}\|_{l^{\infty}},\|b\cdot n_{i}-b\cdot
n_{j}\|_{l^{\infty}},\|c\cdot n_{i}-c\cdot n_{j}\|_{l^{\infty}}\\},$

for any sequence $\\{n_{i}\\}_{i},\ \\{n_{j}\\}_{j}$, where
$a=\mathrm{e}^{-(1+\epsilon)\sigma_{E}(n;g)},\
b=\mathrm{e}^{(1+\epsilon)\sigma_{E}(n-1;g)}$. The details can be found in the
proof of [40, Lemma 5.2]. Since
$\epsilon^{\prime}u_{\epsilon^{\prime}}(n)\to\lambda$ uniformly in $n$ which
follows from [40, Lemma 5.2], one can choose a fixed $\epsilon_{0}>0$
sufficiently small such that
$|\epsilon_{0}u_{\epsilon_{0}}-\lambda|<\delta^{\prime}-\delta$, and it is
independent of $g\in\mathcal{H}(c)$. In this way, we always denote
$\theta(n;g)=\mathrm{e}^{u_{\epsilon_{0}}(n;g)}$.

Define for all $(t,n)\in\mathbb{R}\times\mathbb{Z},g\in\mathcal{H}(c)$:

(4.3) $\displaystyle\overline{u}(t,n;g)$
$\displaystyle:=\min\\{1,\phi_{E}(n;g)\mathrm{e}^{Et}\\},$
$\displaystyle\underline{u}(t,n;g)$
$\displaystyle:=\max\\{0,\phi_{E}(n;g)\mathrm{e}^{Et}-A\theta(n;g)\phi_{E}^{1+\epsilon}(n;g)\mathrm{e}^{(1+\epsilon)Et}\\},$

where $\epsilon$ and $\theta(\cdot;g)$ are given by Propositions 4.1 and $A$
is a positive constant that is to be specified. Notice that $\epsilon$ is
independent of $g$.

Denote

$\mathcal{S}_{g}=\\{\tilde{u}\text{ is an entire solution of
}\eqref{h(c)}|\underline{u}(t,n;g)\leq\tilde{u}(t,n)\leq\overline{u}(t,n;g)\text{
in }\mathbb{R}\times\mathbb{Z}.\\}$

######  Proposition 4.2.

There exists a solution $u$ of (1.3) satisfying $u\in\mathcal{S}_{g}$.
Moreover, $u=u(t,n;g)$ is increasing in $t.$

###### Proof.

By the calculation, $\phi_{E}(n;g)\mathrm{e}^{Et}$ is a supersolution on the
$\mathbb{R}\times\mathbb{Z}$ of $(\ref{1})$. Then $\overline{u}$ is also a
supersolution of the equation (1.2). Take $(t,n)\in\mathbb{R}\times\mathbb{Z}$
so that $\underline{u}(t,n;g)>0$ and set
$\zeta:=\phi_{E}(n;g)\mathrm{e}^{Et}$. Then we have:

$\displaystyle\underline{u}_{t}(t,n;g)-\underline{u}(t,n+1;g)-\underline{u}(t,n-1;g)+2\underline{u}(t,n;g)-g(n)\underline{u}(t,n;g)$
$\displaystyle=$
$\displaystyle-(1+\epsilon)AE\theta(\cdot;g)\phi_{E}^{1+\epsilon}(\cdot;g)\mathrm{e}^{(1+\epsilon)Et}+Ae^{(1+\epsilon)Et}\phi_{E}^{1+\epsilon}(\cdot;g)\mathcal{L}_{g}^{(1+\epsilon)\sigma_{E}(\cdot;g)}\theta(\cdot;g)$
$\displaystyle=$ $\displaystyle
A\zeta^{1+\epsilon}[\mathcal{L}_{g}^{(1+\epsilon)\sigma_{E}(\cdot;g)}\theta(\cdot;g)-(1+\epsilon)E\theta(\cdot;g)]$
$\displaystyle\leq$ $\displaystyle-A\delta\theta\zeta^{1+\epsilon}.$

Therefore, as 0 obviously solves (1.2), for $\underline{u}$ to be a
subsolution it is sufficient to choose $A$ so large that, for all $(t,n)$ such
that $\underline{u}(t,n;g)>0$, one has

$A\delta\theta(\cdot;g)\zeta^{1+\epsilon}\geq g\zeta^{2}.$

Observe that $\underline{u}(t,n;g)>0$ if and only if
$A\theta(\cdot;g)\zeta^{\epsilon}(t,n;g)<1$, that is,
$\zeta^{\epsilon-1}(t,n)>(A\theta(n;g))^{\frac{1}{\epsilon}-1}$. Therefore, we
can choose $A$ so that
$A\geq\frac{\sup_{\mathbb{R}}g^{\epsilon}}{\delta^{\epsilon}\inf_{\mathbb{R}}\theta(\cdot;g)}.$

The above argument also shows that
$\underline{u}<(A\theta(\cdot;g))^{-\frac{1}{\epsilon}}$. $A$ can be chosen
such that $\underline{u}<1,$ whence $\underline{u}\leq\overline{u}$.

Define the sequence $\\{u_{i}\\}$ as follows: $u_{i}$ is the solution of (1.3)
for $t>-i$ with initial condition $u_{i}(-i,n;g)=\overline{u}(-i,n;g).$ By the
comparison principle Proposition 2.3, $u_{i}$ satisfies

$\forall t>-i,\ n\in\mathbb{Z},\ \underline{u}(t,n;g)\leq
u_{i}(t,n;g)\leq\overline{u}(t,n;g).$

Thus, for $i,j\in\mathbb{N}$ with $j<i$ and for any $0<h<1$, using the
monotonicity of $\overline{u}$, we will get

$u_{j}(-j,n;g)=\overline{u}(-j,n;g)\geq\overline{u}(-j-h,n;g)\geq
u_{i}(-j-h,n;g).$

Note that $u_{i}(\cdot-h,\cdot;g)$ is also a solution of (1.3). The comparison
principle Proposition 2.3 gives us

$\forall j<i,\ 0<h<1,t>-j,\ n\in\mathbb{Z},\ g\in\mathcal{H}(c)\
u_{j}(t,n;g)\geq u_{i}(t-h,n;g).$

Now by the arguments before, we can prove $\\{u_{i}\\}_{i}$ converges locally
uniformly to a global function $\underline{u}\leq u\leq\overline{u}$ of (1.3).
Then passing to the limit as $i,j\rightarrow\infty$, $u(t,n;g)\geq u(t-h,n;g)$
for all $(t,n)\in\mathbb{R}\times\mathbb{Z}$ and $0<h<1$,
$g\in\mathcal{H}(c)$. This means that $u$ is nondecreasing in $t$. If the
monotonicity were not strict, then the parabolic maximum principle Corollary
2.1 would imply that $u$ is constant in time, which contradicts
$\underline{u}\leq u\leq\overline{u}$. Then we finish the proof. ∎

One should notice the following fact:

######  Proposition 4.3.

For any $\tilde{u}\in\mathcal{S}_{g}$, either $\tilde{u}(t,n)<u(t,n;g)$, or
$\tilde{u}(t,n)\equiv u(t,n;g)$ in $\mathbb{R}\times\mathbb{Z}$.

###### Proof.

Since $\tilde{u}\in\mathcal{S}_{g}$, then, as constructed in Proposition 4.2,

$\forall j\in\mathbb{Z}_{+},\
\tilde{u}(-j,n)\leq\overline{u}(-j,n;g)=u_{j}(-j,n;g).$

Therefore, the comparison principle Proposition 2.3 gives $\tilde{u}(t,n)\leq
u_{j}(t,n;g)$ for $t>-j$. Taking $j\to\infty$, we have

$\tilde{u}(t,n)\leq u(t,n;g),\ \forall(t,n)\in\mathbb{R}\times\mathbb{Z}.$

Let $\underline{u}=\tilde{u}(t,n),\overline{u}=u(t,n;g)$. By Proposition 2.3,
one has either $\tilde{u}(t,n)<u(t,n;g)$, or $\tilde{u}(t,n)\equiv u(t,n;g)$
in $\mathbb{R}\times\mathbb{Z}$. Thus the proof is complete. ∎

In fact, $u(t,n;g)$ is an almost periodic traveling front of (1.3) as we will
show afterwards. To prove it, we shall notice some facts which will be
represented below.

######  Proposition 4.4.

Let $E>\lambda_{1}$. For any $g\in\mathcal{H}(c)$, the following properties
hold:

  1. (1)

$\phi_{E}(n;g\cdot k)=\frac{\phi_{E}(n+k;g)}{\phi_{E}(k;g)}$.

  2. (2)

$\\{\phi_{E}(n;g)|g\in\mathcal{H}(c)\\}$ is a one-cover of $\mathcal{H}(c)$ in
$l^{\infty}_{loc}(\mathbb{Z})$.

  3. (3)

$\big{\\{}\frac{\phi_{E}(n+1;g)}{\phi_{E}(n;g)}|g\in\mathcal{H}(c)\big{\\}}$
is a one-cover of $\mathcal{H}(c)$ in $l^{\infty}(\mathbb{Z})$.

  4. (4)

$\theta(n+k;g)=\theta(n;g\cdot k)$.

  5. (5)

$\\{\theta(n;g)|g\in\mathcal{H}(c)\\}$ is a one-cover of $\mathcal{H}(c)$ in
$l^{\infty}(\mathbb{Z})$.

Before proving this proposition, we give an observation about one-cover.

######  Remark 4.1.

We claim that $\\{\Phi_{g}|g\in\mathcal{H}(c)\\}$ is said to be a one-cover of
$\mathcal{H}(c)$ in some metric space $X$ if and only if

(4.4) $\Phi_{g\cdot n_{k}}\to\Phi_{g^{*}}\ \text{ in $X$ provided that }g\cdot
n_{k}\to g^{*}.$

In fact, (4.4) holds straightforward if $\\{\Phi_{g}|g\in\mathcal{H}(c)\\}$ is
a one-cover. Assume that $g_{k}\to g^{*}$ and $\varrho$ is the metric of $X$.
Then for any $k\in{\mathbb{Z}}$, there exists $\\{n_{k}\\}$ such that
$\varrho(\Phi_{g\cdot n_{k}},\Phi_{g_{k}})+\|g\cdot
n_{k}-g_{k}\|_{\ell^{\infty}}<1/k$. Therefore, $g\cdot n_{k}\to g^{*}$ since
$g_{k}\to g^{*}$, and thus $\Phi_{g\cdot n_{k}}\to\Phi_{g^{*}}$ by (4.4).
Hence $\Phi$ is continuous since

$\varrho(\Phi_{g^{*}},\Phi_{g_{k}})\leq\varrho(\Phi_{g\cdot
n_{k}},\Phi_{g_{k}})+\varrho(\Phi_{g\cdot n_{k}},\Phi_{g^{*}})\to 0\text{ as
}k\to\infty.$

###### Proof of Proposition 4.4.

(1) follows from the uniqueness of $\phi_{E}(n;g)$ which follows from
Proposition 3.4.

(2). By Remark 4.1, it is sufficient to prove $\Phi_{g\cdot
n_{k}}\to\Phi_{g^{*}}\ \text{ provided that }g\cdot n_{k}\to g^{*}.$ For
$g\cdot n_{k}\to g^{*}$, $\phi_{E}(n;g\cdot n_{k})$ converges locally
uniformly, up to some subsequence, to some function $\tilde{\phi}_{E}(n)$.
Moreover, we have $\tilde{\phi}_{E}(n)\leq Ce^{-\delta n}$ for $n>0$ since
$\phi_{E}(n;g\cdot n_{k})\leq Ce^{-\delta n}$ from Lemma 3.1. Therefore
$\tilde{\phi}$ satisfies

$\mathcal{L}_{g^{*}}\phi=E\phi\text{ in }{\mathbb{Z}},\ \phi(0)=1,\
\lim\limits_{n\rightarrow\infty}\phi(n)=0,$

which yields that $\tilde{\phi}=\phi_{E}(n;g^{*})$ by uniqueness. That is to
say, all the convergence subsequences converge to the same limit. Hence
$\phi_{E}(n;g\cdot n_{k})\to\phi_{E}(n;g^{*})$ locally uniformly in
$n\in{\mathbb{Z}}$.

(3). We want to prove that $\frac{\phi_{E}(\cdot+1;g\cdot
n_{k})}{\phi_{E}(\cdot;g\cdot
n_{k})}\to\frac{\phi_{E}(\cdot+1;g^{*})}{\phi_{E}(\cdot;g^{*})}$ in
$l^{\infty}(\mathbb{Z})$ if $g\cdot n_{k}\to g^{*}$. If not, then there exists
a sequence $\\{m_{k}\\}_{k}$ such that, up to extraction,

$\lim\limits_{k\rightarrow\infty}\bigg{|}\frac{\phi_{E}(m_{k}+1;g\cdot
n_{k})}{\phi_{E}(m_{k};g\cdot
n_{k})}-\frac{\phi_{E}(m_{k}+1;g^{*})}{\phi_{E}(m_{k};g^{*})}\bigg{|}>0,$

i.e., by (1),

(4.5)
$\lim\limits_{k\rightarrow\infty}\big{|}\phi_{E}(1;g\cdot(m_{k}+n_{k}))-\phi_{E}(1;g^{*}\cdot
m_{k})\big{|}>0.$

Note that
$\lim\limits_{k\to\infty}g\cdot(m_{k}+n_{k})=\lim\limits_{k\to\infty}\lim\limits_{l\to\infty}g\cdot
n_{l}\cdot m_{k}=\lim\limits_{k\to\infty}g^{*}\cdot m_{k}$ since $g$ is almost
periodic. Then using (2), we have

$\lim\limits_{k\rightarrow\infty}\big{|}\phi_{E}(1;g\cdot(m_{k}+n_{k}))-\phi_{E}(1;g^{*}\cdot
m_{k})\big{|}\to 0$

as $k\to\infty$, which contradicts (4.5).

(4). $\theta(n+k;g)=\theta(n;g\cdot k)$ follows from the uniqueness.

(5). By the similar argument in (2), $\\{\theta(n;g)|g\in\mathcal{H}(c)\\}$ is
a one-cover of $\mathcal{H}(c)$ in $l^{\infty}(\mathbb{Z})$. ∎

With this at hand, we can prove that $u$ satisfies (4) of Definition 1.1.

######  Lemma 4.2.

For any $g\in\mathcal{H}(c)$, we have

$u(t+t(k;g),n+k;g)=u(t,n;g\cdot k)\ \forall t\in\mathbb{R},\
n,k\in\mathbb{Z},$

where $t(k;g)=-\frac{1}{E}\ln\phi_{E}(k;g)$.

###### Proof.

From the definitions of $\overline{u}(t,n;g)$ and $\underline{u}(t,n;g)$,
together with Proposition 4.4 (1), we can verify that

$\overline{u}(t+t(k;g),n+k;g)=\overline{u}(t,n;g\cdot k),$
$\underline{u}(t+t(k;g),n+k;g)=\underline{u}(t,n;g\cdot k).$

Combining these two equations, we can check that

$u(t+t(k;g),n+k;g)\in\mathcal{S}_{g\cdot k},u(t-t(k;g),n-k;g\cdot
k)\in\mathcal{S}_{g}.$

Therefore, $u(t+t(k;g),n+k;g)\leq u(t,n;g\cdot k)$ and $u(t-t(k;g),n-k;g\cdot
k)\leq u(t,n;g)$ by Proposition 4.3, which gives
$u(t+t(k;g),n+k;g)=u(t,n;g\cdot k)$ for any
$(t,n)\in\mathbb{R}\times\mathbb{Z}$. ∎

To prove $u$ satisfying (2) of Definition 1.1, we will consider $u(t,0;g\cdot
n)$.

######  Lemma 4.3.

The function $u(t,0;g\cdot n)$ satisfies

(4.6) $\lim\limits_{t\rightarrow-\infty}u(t,0;g\cdot n)=0,\
\lim\limits_{t\rightarrow+\infty}u(t,0;g\cdot n)=1,\ \text{ uniformly in
}n\in\mathbb{Z}.$

###### Proof.

For any $(t,n)\in\mathbb{R}\times\mathbb{Z}$, we have

$u(t,0;g\cdot n)\leq\overline{u}(t,0;g\cdot n)\leq\phi_{E}(0;g\cdot
n)e^{Et}=e^{Et},$

Meanwhile,

$\displaystyle u(t,0;g\cdot n)$ $\displaystyle\geq\underline{u}(t,0;g\cdot
n)\geq e^{Et}\phi_{E}(0;g\cdot n)-A\theta(0;g\cdot
n)\phi_{E}^{1+\epsilon}(0;g\cdot n)e^{(1+\epsilon)Et}$ $\displaystyle\geq
e^{Et}-A(\sup\limits_{n}\theta(n;g))e^{(1+\epsilon)Et},$

where the last inequality follows from $\theta(n+k;g)=\theta(n;g\cdot k).$
Therefore,

(4.7) $\forall(t,n)\in\mathbb{R}\times\mathbb{Z},\ e^{Et}(1-Me^{\epsilon
Et})\leq u(t,0;g\cdot n)\leq e^{Et},$

for some positive constant $M$. From the second inequality, we deduce that

$\lim\limits_{t\rightarrow-\infty}u(t,0;g\cdot n)=0\ \text{ uniformly in
}n\in\mathbb{Z}.$

From the first inequality of (4.7), we can see that
$\inf_{n\in\mathbb{Z}}u(t,0;g\cdot n)>0$ for $t<0$ small enough. Therefore, it
follows from the monotonicity of $u$ in $t$ that

(4.8) $\forall t\in\mathbb{R},\ \inf_{[t,+\infty)\times\mathbb{Z}}u(t,0;g\cdot
n)>0,$

and the quantity
$\vartheta:=\lim\limits_{t\rightarrow+\infty}\inf\limits_{n\in\mathbb{Z}}u(t,0;g\cdot
n)>0$ is well defined. To conclude the proof we only need to prove
$\vartheta=1$. Let $\\{n_{k}\\}_{k\in\mathbb{Z}_{+}}\subset\mathbb{Z}$ be such
that $u(k,0;g\cdot n_{k})\rightarrow\vartheta$ as $k\rightarrow\infty.$
Consider the family of functions $\\{p^{k}\\}_{k\in\mathbb{Z}_{+}}$

$p^{k}(t,n):=u(t+k,n;g\cdot n_{k})=u\big{(}t+k-t(n;g\cdot
n_{k}),0;g\cdot(n+n_{k})\big{)},$

where the last equality follows from Lemma 4.2. Then we have
$p^{k}(0,0)=u(k,0;g\cdot n_{k})\rightarrow\vartheta$ as $k\rightarrow\infty$,
and, for any $(t,n)\in\mathbb{R}\times\mathbb{Z},$

(4.9)
$\liminf\limits_{k\rightarrow\infty}p^{k}(t,n)=\liminf\limits_{k\rightarrow\infty}u\big{(}t+k-t(n;g\cdot
n_{k}),0;g\cdot(n+n_{k})\big{)}\geq\vartheta$

by (4.8). Moreover, the sequence $\\{p^{k}\\}_{k}$ converges, up to sequences,
to a function $p$ satisfying

$p_{t}(t,n)-p(t,n+1)-p(t,n-1)+2p(t,n)=g^{*}(n)p(t,n)(1-p(t,n))\text{ in
}\mathbb{R}\times\mathbb{Z},$

where $g^{*}\in\mathcal{H}(c)$. Note that $p$ reaches its minimum $\vartheta$
at $(0,0)$. Then we have $g^{*}(0)\vartheta(1-\vartheta)\leq 0.$ Therefore,
$\vartheta=1$ since $g^{*}(0)\geq\inf_{\mathbb{Z}}c>0$ and $0<\vartheta\leq
1.$ ∎

Now we need the uniform convergence in $\mathbb{R}\times\mathbb{Z}$ to explain
why $u(t,n;g)$ is an almost periodic traveling front. Before that, the
following lemma is needed.

######  Lemma 4.4.

Let two bounded uniformly continuous functions $u^{1}$ and $u^{2}$ be a
subsolution and a supersolution of (1.3), respectively, i.e.,

$u_{t}^{1}(t,n)-u^{1}(t,n+1)-u^{1}(t,n-1)+2u^{1}(t,n)\leq
g(n)u^{1}(t,n)(1-u^{1}(t,n)),\text{ }t\in\mathbb{R},$
$u_{t}^{2}(t,n)-u^{2}(t,n+1)-u^{2}(t,n-1)+2u^{2}(t,n)\geq
g(n)u^{2}(t,n)(1-u^{2}(t,n)),\text{ }t\in\mathbb{R},$

and satisfy $0\leq u^{1}\leq u^{2}$ in $\mathbb{Z}\times\mathbb{R}$.

If $\inf\limits_{n\in\mathbb{Z}}(u^{2}-u^{1})(t_{0}+t(n;g),n)=0$ for some
$t_{0}\in\mathbb{R}$, then

$\forall t<t_{0},\ \inf_{n\in\mathbb{Z}}(u^{2}-u^{1})(t+t(n;g),n)=0.$

###### Proof.

Let $\\{n_{k}\\}_{k}\subset{\mathbb{Z}}$ be such that

$u^{2}(t_{0}+t(n_{k};g),n_{k})-u^{1}(t_{0}+t(n_{k};g),n_{k})\rightarrow
0\text{ as }k\rightarrow\infty.$

Consider

$u_{k}^{i}(t,n):=u^{i}(t+t(n_{k};g),n+n_{k}),\ i=1,2.$

Then the nonnegative function $w_{k}:=u_{k}^{2}-u_{k}^{1}$ satisfies
$\lim\limits_{k\to\infty}w_{k}(t_{0},0)=0$ and

$\frac{d}{dt}w_{k}(t,n)-\mathcal{D}w_{k}(t,n)\geq(g\cdot
n_{k})(n)(1-u^{2}_{k}(t,n)-u^{1}_{k}(t,n))w_{k}(t,n),$

where $\mathcal{D}w_{k}(t,n)=w_{k}(t,n+1)-w_{k}(t,n-1)+2w_{k}(t,n).$ Take $M$
large enough such that $M+\sup\limits_{n\in\mathbb{Z}}(g\cdot
n_{k})(1-u^{2}_{k}-u^{1}_{k})>1$. Then the nonnegative function
$v_{k}(t,n):=e^{Mt}w_{k}(t,n)$ satisfies

$\frac{d}{dt}v_{k}(t,n)-v_{k}(t,n+1)-v_{k}(t,n-1)+2v_{k}(t,n)\geq v_{k}(t,n).$

Now from Proposition 2.2 we have $v_{k}(t,n)\leq C(t_{0}-t)v_{k}(t_{0},n)$,
where $C(t_{0}-t)$ is a constant which is independent of $k$. Therefore,
$w_{k}(t,n)\leq e^{M(t_{0}-t)}C(t_{0}-t)w_{k}(t_{0},n)$, and this yields that

$0\leq\lim\limits_{k\to\infty}w_{k}(t,0)\leq\lim\limits_{k\to\infty}e^{M(t_{0}-t)}C(t_{0}-t)w_{k}(t_{0},0)=0.$

Therefore,

$\displaystyle 0$
$\displaystyle\leq\inf_{n\in\mathbb{Z}}(u^{2}-u^{1})(t+t(n;g),n)\leq\inf_{k\in\mathbb{Z}}(u^{2}-u^{1})(t+t(n_{k};g),n_{k})$
$\displaystyle=\inf_{k\in\mathbb{Z}}w_{k}(t,0)\leq\lim\limits_{k\to\infty}w_{k}(t,0)=0.$

Thus the proof is complete. ∎

Using Lemma 4.4, we have the following result about uniform convergence.

######  Theorem 4.1.

Assume that $g^{*}=\lim\limits_{k\to\infty}g\cdot n_{k}$ for some sequence
$\\{n_{k}\\}_{k\in\mathbb{Z}_{+}}$. Then $u(t+t(n;g\cdot n_{k}),n;g\cdot
n_{k})\to u(t+t(n;g^{*}),n;g^{*})$ uniformly in $\mathbb{R}\times\mathbb{Z}$.

###### Proof.

First, considering a sequence $\\{n_{k}\\}\subset{\mathbb{Z}}$ where
$c(\cdot+n_{k})$ converges uniformly in $\mathbb{R}$, we prove that
$u(t+t(n;g\cdot n_{k}),n;g\cdot n_{k})$ converges uniformly in
$\mathbb{R}\times\mathbb{Z}$. Assume by contradiction that it is false. Then
there exist $\\{t_{k}\\}_{k}\subset\mathbb{R}$,
$\\{m_{k}\\}_{k}\subset\mathbb{Z}$, and two subsequences
$\\{n_{k}^{1}\\}_{k},\\{n_{k}^{2}\\}_{k}$ of $\\{n_{k}\\}_{k}$ such that

$\liminf\limits_{k\rightarrow\infty}\big{(}u(t_{k}+t(m_{k};g\cdot
n^{1}_{k}),m_{k};g\cdot n^{1}_{k})-u(t_{k}+t(m_{k};g\cdot
n^{2}_{k}),m_{k};g\cdot n^{2}_{k})\big{)}>0,$

i.e.,
$\liminf\limits_{k\rightarrow\infty}\big{(}u(t_{k},0;g\cdot(m_{k}+n^{1}_{k}))-u(t_{k},0;g\cdot(m_{k}+n^{2}_{k}))\big{)}>0$.
By Lemma 4.3, $\\{t_{k}\\}$ is bounded. Let $\zeta$ be a limit point of
$\\{t_{k}\\}_{k}$. Then by $0\leq u_{t}\leq 4+\|c\|_{l^{\infty}(\mathbb{Z})}$
which follows from (1.3), $u$ is uniform continuous in $t$. Thus,

$\liminf\limits_{k\rightarrow\infty}\big{(}u(\zeta,0;g\cdot(m_{k}+n^{1}_{k}))-u(\zeta,0;g\cdot(m_{k}+n^{2}_{k}))\big{)}>0,$

Consider for $i=1,2,$

$p_{k}^{i}(t,n):=u(t,n;g\cdot(m_{k}+n^{i}_{k}))=u\big{(}t-t(n;g\cdot(m_{k}+n^{i}_{k})),0;g\cdot(n+m_{k}+n^{i}_{k})\big{)},$

where the second equality follows from Lemma 4.2. Then, up to subsequences,
$p_{k}^{i}(t,n)$ and $g\cdot(m_{k}+n^{i}_{k})(i=1,2)$ converge locally
uniformly to $p^{i}$ and $g^{**}$ ,respectively, as $k\to\infty$. Moreover
$p^{1}(\zeta,0)-p^{2}(\zeta,0)>0,\text{ and}$

$p^{i}_{t}(t,n)=p^{i}(t,n+1)+p^{i}(t,n-1)-2p^{i}(t,n)+g^{**}(n)p^{i}(t,n)(1-p^{i}(t,n)),\
i=1,2.$

Recall $t(n;g)=-\frac{1}{E}\ln\phi_{E}(n;g)$. Thanks to (2) of Proposition 4.4
and the fact that $u$ is continuous in $t$, we have for $i=1,2$,

$\displaystyle\lim\limits_{k\to\infty}u(t,0;g\cdot(n+m_{k}+n^{i}_{k}))$
$\displaystyle=\lim\limits_{k\to\infty}p_{k}^{i}(t+t(n;g\cdot(m_{k}+n^{i}_{k})),n)$
$\displaystyle=p^{i}(t+t(n;g^{**}),n)$

Combining this with (4.6) and (4.7), we have
$p^{1}(t+t(n;g^{**}),n)/p^{2}(t+t(n;g^{**}),n)\rightarrow 1$ as
$t\rightarrow\pm\infty$ uniformly in $n\in\mathbb{Z}$. Note that by (4.8),

$\kappa^{*}:=\sup_{\mathbb{R}\times\mathbb{Z}}\frac{p^{1}(t+t(n;g^{**}),n)}{p^{2}(t+t(n;g^{**}),n)}$

is finite and $\kappa^{*}>1$ since $p^{1}(\zeta,0)-p^{2}(\zeta,0)>0$.
Moreover, by the uniform continuity of $p^{1}$ and $p^{2}$, we can find some
finite $\overline{t}$ such that

$\sup_{n\in\mathbb{Z}}\frac{p^{1}(\overline{t}+t(n;g^{**}),n)}{p^{2}(\overline{t}+t(n;g^{**}),n)}=\sup\limits_{\mathbb{R}\times\mathbb{Z}}\frac{p^{1}(t+t(n;g^{**}),n)}{p^{2}(t+t(n;g^{**}),n)}=\kappa^{*}.$

By the direct computation, we can show that $\kappa^{*}p^{2}$ is a
supersolution of (1.3) with $g$ replaced by $g^{**}$ since $\kappa^{*}>1$. Now
we can apply Lemma 4.4 to deduce that

$\forall
t<\overline{t},\inf\limits_{n\in\mathbb{Z}}(\kappa^{*}p^{2}(t+t(n;g^{**}),n)-p^{1}(t+t(n;g^{**}),n))=0$

which contradicts (4.7) when $t$ is sufficiently large. Hence $u(t+t(n;g\cdot
n_{k}),n;g\cdot n_{k})$ converges uniformly in $\mathbb{R}\times\mathbb{Z}$.

Let us now show that $\lim\limits_{k\to\infty}u(t+t(n;g\cdot n_{k}),n;g\cdot
n_{k})=u(t+t(n;g^{*}),n;g^{*})$. Denote
$v(t+t(n;g^{*}),n):=\lim\limits_{k\to\infty}u(t+t(n;g\cdot n_{k}),n;g\cdot
n_{k})$. Using (2) of Proposition 4.4 again, we have

$\lim\limits_{k\to\infty}u(t,n;g\cdot
n_{k})=\lim\limits_{k\to\infty}u\big{(}t-t(n;g\cdot n_{k})+t(n;g\cdot
n_{k}),n;g\cdot n_{k}\big{)}=v(t,n).$

Hence $v\in\mathcal{S}_{g^{*}}$, and thus $v(t,n)\leq u(t,n;g^{*})$ by
Proposition 4.3. We claim that $v(t,n)\equiv u(t,n;g^{*})$. Otherwise,
$v(t,n)<u(t,n;g^{*})$. Note that by Lemma 4.2,

$v(t+t(n;g^{*}),n)=\lim\limits_{k\to\infty}u(t+t(n;g\cdot n_{k}),n;g\cdot
n_{k})=\lim\limits_{k\to\infty}u(t,0;g\cdot(n+n_{k})).$

Then, by the similar reasons as before, we deduce that

$\kappa^{\prime}:=\sup_{\mathbb{R}\times\mathbb{Z}}\frac{u(t+t(n;g^{*}),n)}{v(t+t(n;g^{*}),n)}$

is finite, $\kappa^{\prime}>1$ since $v(t,n)<u(t,n;g^{*})$, and
$\kappa^{\prime}v$ is a supersolution of (1.3) with $g$ replaced by $g^{*}$.
Applying Lemma 4.4 as before, we obtain a contradiction. Hence $v(t,n)\equiv
u(t,n;g^{*})$, that is to say,

$u(t+t(n;g\cdot n_{k}),n;g\cdot n_{k})\to u(t+t(n;g^{*}),n;g^{*})\text{ as
}k\to\infty$

uniformly in $(t,n)\in\mathbb{R}\times\mathbb{Z}$. ∎

We are now in the position to prove the existence of almost periodic traveling
front.

######  Theorem 4.2.

$u(t,n;g)$ is an almost periodic traveling front with the average wave speed
$w\in(w^{*},\underline{w})$.

###### Proof.

We prove that $u(t,n;g)$ satisfies (1) of Definition 1.1 first. Consider
$g\cdot n_{k}\to g^{*}$, and note that
$\\{t(n;g)|g\in\mathcal{H}(c)\\}=\\{-\frac{1}{E}\ln\phi_{E}(n;g)|g\in\mathcal{H}(c)\\}$
is a one-cover of $\mathcal{H}(c)$ in $l^{\infty}_{loc}(\mathbb{Z})$ by
Proposition 4.4 (2). Then combining Theorem 4.1, we have

$\displaystyle u(t,n;g\cdot n_{k})$ $\displaystyle=u\big{(}t-t(n;g\cdot
n_{k})+t(n;g\cdot n_{k}),n;g\cdot n_{k}\big{)}$ $\displaystyle\to
u\big{(}t-t(n;g^{*})+t(n;g^{*}),n;g^{*}\big{)}$ $\displaystyle=u(t,n;g^{*})$

locally uniformly in $(t,n)\in{\mathbb{R}}\times{\mathbb{Z}}$. Hence
$\\{u(t,n;g)|g\in\mathcal{H}(c)\\}$ is a one-cover of
$C(\mathbb{R}\times\mathbb{Z})$.

Note that $u(t,n;g)=u(t-t(n;g),0;g\cdot n)$ by Lemma 4.2. From the choice of
$t(n,g)$, we have $t(n;g)\rightarrow\pm\infty$ as $n\rightarrow\pm\infty$.
Then by Lemma 4.3, we deduce that $u(t,n;g)$ satisfies (2) of Definition 1.1.

By Proposition 3.4, one has

$\displaystyle w(g)$
$\displaystyle=\lim\limits_{|n|\rightarrow\infty}\frac{n}{t(n+k;g)-t(k;g)}$
$\displaystyle=\lim\limits_{|n|\rightarrow\infty}\frac{n}{t(n;g\cdot
k)}=-\lim\limits_{|n|\rightarrow\infty}\frac{nE}{\ln\phi_{E}(n;g\cdot
k)}=\frac{E}{L(E)}\in(w^{*},\underline{w}).$

Now combining this with Proposition 4.4, (3) of Definition 1.1 is proved.
Finally, Lemma 4.2 gives rise to (4). Hence we finish the proof. ∎

Now we ends the proof of (1) in Theorem 1.1.

###  4.2. Non-existence of Fronts with Speed Less than $w^{*}$

Next we turn to prove (3) in Theorem 1.1, i.e., there is even no generalized
transition front with average speed $w<w^{*}$. Compared to the previous
section, we only consider the generalized transition front of (1.2). However,
the similar arguments can be applied to (1.3) with minor modification. From
now on, for the sake of simplicity, we denote
$\mathcal{D}\phi(t,n)=\phi(t,n+1)+\phi(t,n-1)-2\phi(t,n),\
t,n\in\mathbb{R}\times\mathbb{Z}$ and $u(t,n;s,u_{0})$, $t\geq
s,n\in\mathbb{Z}$ a solution of (1.2) with initial value $u_{0}$ starting at
time $s$.

Now we begin with a lemma which will provide a lower bound of the average
speed of generalized transition front.

######  Lemma 4.5.

Let $u(t,n;0,u^{(0)})$ be a solution of (1.2) with its initial value
$u(0,n;0,u^{(0)})=u^{(0)}$, where

$u^{(0)}(n)=0\ \text{if}\ n>0,\ \text{and}\ \inf\limits_{n\leq
0}u^{(0)}(n)=\alpha\in(0,1].$

Then $\lim\limits_{t\to\infty}\inf\limits_{n\leq wt}u(t,n;0,u^{(0)})=1,\
\forall 0\leq w<w^{*}.$

To prove this lemma, we need

######  Proposition 4.5.

Let $g\in\mathcal{H}(c)$ and $u^{g}(t,n;0,u_{0})$ be the solution of

(4.10) $\left\\{\begin{aligned}
u_{t}(t,n)-\mathcal{D}u(t,n)=g(n)u(t,n)(1-u(t,n))&\text{ in
}\mathbb{R}\times\mathbb{Z},\\\ u(0,n)=u_{0}(n)\text{ with
}u_{0}(0)=\alpha\in(0,1]\text{ and }u_{0}(n)=0,&\text{ if }n\neq
0.\end{aligned}\right.$

Then for any $0\leq w<w^{*}$, we have
$\lim\limits_{t\to\infty}\inf\limits_{0\leq n\leq wt}u^{g}(t,n;0,u_{0})=1$
exists uniformly in $g$.

###### Proof.

Step 1: Show that $\lim\limits_{t\to\infty}\inf\limits_{0\leq n\leq
wt}u^{g}(t,n;0,u_{0})=1$ for any $0\leq w<w^{*}$.

Denote $\mathcal{L}_{g,p}\phi:=e^{p\cdot}\mathcal{L}_{g}(e^{-p\cdot}\phi)$,
where $\mathcal{L}_{g}$ is the linearized operator given by
$(\mathcal{L}_{g}\phi)(n)=\phi(n+1)+\phi(n-1)-2\phi(n)+g(n)\phi(n)$. Then as
proved in [40, Theorem 2.1], we have

$\lim\limits_{t\to+\infty}\inf\limits_{0\leq n\leq
wt}u^{g}(t,n;0,u_{0})=1\text{ for any }0\leq
w<\inf\limits_{p>0}\frac{\underline{\lambda}_{1}(\mathcal{L}_{g,p})}{p},$

where $\underline{\lambda}_{1}(\mathcal{L}_{g,p})$ is defined as (3.5) with
$\mathcal{M}_{a,b,c}$ replaced by $\mathcal{L}_{g,p}$. Proposition 3.3 yields
that
$\underline{\lambda}_{1}(\mathcal{L}_{g,p})={\lambda}_{1}(\mathcal{L}_{c,p})$
for any $g\in\mathcal{H}(c)$, where $\lambda_{1}(\mathcal{L}_{g,p})$ is
defined as (3.6), and then we denote it by $k(p)$. It still needs to show that
$\inf\limits_{p>0}\frac{k(p)}{p}=w^{*}=\inf\limits_{E>\lambda_{1}}\frac{E}{L(E)}$
with $L(E)$ given by (1.6).

In fact, by similar arguments to the proof of [40, Theorem 6.1], we can deduce
that the map $L:(\lambda_{1},+\infty)\to(\underline{L},+\infty)$, where
$\underline{L}=\lim\limits_{E\searrow\lambda_{1}}L(E)\geq 0$, admits an
inverse, which is exactly $k:(\underline{L},+\infty)\to(\lambda_{1},+\infty)$.
If $\underline{L}>0$, then $k(p)\equiv k(0)=\lambda_{1}$ for any
$p\in[0,\underline{L}]$. Hence

$w^{*}=\inf\limits_{E>\lambda_{1}}\frac{E}{L(E)}=\inf\limits_{p>\underline{L}}\frac{k(p)}{p}=\inf\limits_{p>0}\frac{k(p)}{p}.$

Step 2: Construct a appropriate subsolution to obtain the uniform convergence.

Suppose by contradiction that there exist $\delta\in(0,1)$ and a sequence
$\\{g_{l},t_{l},n_{l}\\}\subset\mathcal{H}(c)\times\mathbb{R}_{+}\times\mathbb{Z}$
with $0\leq n_{l}\leq wt_{l}$ and $t_{l}\to+\infty$ such that

$u^{g_{l}}(t_{l},n_{l};0,u_{0})\leq 1-\delta.$

Without loss of generality, we assume that $g_{l}\to g^{*}$ in
$l^{\infty}(\mathbb{Z})$. Let
$g^{s}(n)=(1-s)g^{*}(n)+s\inf\limits_{n\in\mathbb{Z}}c(n)/2$ and
$u^{g_{s}}(t,n;0,u_{0})$ be the solution of

(4.11) $\left\\{\begin{aligned}
&u_{t}(t,n)-\mathcal{D}u(t,n)=g^{s}(n)u(t,n)(1-u(t,n))\text{ in
}\mathbb{R}\times\mathbb{Z},\\\ &u(0,\cdot)=u_{0}.\end{aligned}\right.$

Then we choose $N$ large enough such that for $l\geq N$ and $s\in(0,1)$,
$\|g_{l}-g^{*}\|_{l^{\infty}}\leq s\inf\limits_{n\in\mathbb{Z}}c(n)/2$. Hence
for $l\geq N$,

(4.12) $g_{l}(n)\geq g^{*}(n)-\|g_{l}-g^{*}\|_{l^{\infty}}\geq
g^{*}(n)-s\inf\limits_{n\in\mathbb{Z}}c(n)/2\geq g^{s}(n)$

since $\inf\limits_{n\in\mathbb{Z}}g^{*}(n)=\inf\limits_{n\in\mathbb{Z}}c(n)$.
From this, for $l\geq N$, we have

$\displaystyle u_{t}^{g^{s}}(t,n;0,u_{0})-\mathcal{D}u^{g^{s}}(t,n;0,u_{0})$
$\displaystyle=g^{s}(n)u^{g^{s}}(t,n;0,u_{0})(1-u^{g^{s}}(t,n;0,u_{0}))$
$\displaystyle\leq g_{l}(n)u^{g^{s}}(t,n;0,u_{0})(1-u^{g^{s}}(t,n;0,u_{0})).$

That is to say, $u^{g^{s}}(t,n;0,u_{0})$ is a subsolution of (4.10) with $g$
replaced by $g_{l}$ for any $l\geq N$.
Step 3: End the proof.

As proved in [40, Theorem 2.1], we have

$\lim\limits_{t\to+\infty}\inf\limits_{0\leq n\leq
wt}u^{g^{s}}(t,n;0,u_{0})=1\text{ for any }0\leq
w<\inf\limits_{p>0}\frac{\underline{\lambda}_{1}(\mathcal{L}_{g^{s},p})}{p}.$

[40, Proposition 3.1] also tells us that

$\lim\limits_{s\to
0}\inf\limits_{p>0}\frac{\underline{\lambda}_{1}(\mathcal{L}_{g^{s},p})}{p}=\inf\limits_{p>0}\frac{\underline{\lambda}_{1}(\mathcal{L}_{g,p})}{p}=\inf\limits_{E>\lambda_{1}}\frac{E}{L(E)}=w^{*}.$

Hence for any $0\leq w<w^{*}$, we can take $s>0$ small such that $0\leq
w<\inf\limits_{p>0}\frac{\underline{\lambda}_{1}(\mathcal{L}_{g^{s},p})}{p}$.
Moreover, we have $\lim\limits_{t\to\infty}\inf\limits_{0\leq n\leq
wt}u^{g^{s}}(t,n;0,u_{0})=1$. On the other hand, since
$u^{g^{s}}(t,n;0,u_{0})$ is a subsolution, it follows from Proposition 2.3
that we have $u^{g^{s}}(t,n;0,u_{0})\leq u^{g_{l}}(t,n;0,u_{0})$ for $l$ large
enough. Therefore,

$\displaystyle\lim\limits_{t\to\infty}\inf\limits_{0\leq n\leq
wt}u^{g^{s}}(t,n;0,u_{0})$
$\displaystyle\leq\limsup\limits_{l\to\infty}u^{g^{s}}(t_{l},n_{l};0,u_{0})$
$\displaystyle\leq\limsup\limits_{l\to\infty}u^{g_{l}}(t_{l},n_{l};0,u_{0})\leq
1-\delta.$

That’s impossible. Hence $\lim\limits_{t\to\infty}\inf\limits_{0\leq n\leq
wt}u^{g}(t,n;0,u_{0})=1$ exists uniformly in $g\in\mathcal{H}(c)$. ∎

###### Proof of Lemma 4.5.

Consider the solution $u_{k}(t,n)$ of

$\left\\{\begin{aligned}
&u_{t}(t,n)-\mathcal{D}u(t,n)=c(n+k)u(t,n)(1-u(t,n)),&(t,n)\in\mathbb{R}\times\mathbb{Z},\\\
&u(0,0)=\alpha,\ \text{and }u(0,n)=0&\text{ if }n\neq 0.\end{aligned}\right.$

Then for any $0\leq w<w^{*}$, we have
$\lim\limits_{t\to\infty}\inf\limits_{0\leq n\leq wt}u_{k}(t,n)=1$ exists
uniformly in $k$ by Proposition 4.5. Therefore, the solution
$u(t,n;0,u^{(0)}_{k})$ of

$\left\\{\begin{aligned}
u_{t}(t,n)-\mathcal{D}u(t,n)&=c(n)u(t,n)(1-u(t,n)),&(t,n)\in\mathbb{R}\times\mathbb{Z},\\\
u(0,n)&=u^{(0)}_{k}(n),&n\in\mathbb{Z},\end{aligned}\right.$

where $u^{(0)}_{k}(-k)=\alpha\ \text{and }u(0,n)=0\text{ if }n\neq-k$,
satisfies

$\lim\limits_{t\to\infty}\inf\limits_{-k\leq n\leq
wt-k}u(t,n;0,u^{(0)}_{k})=1\text{ for any }w<w^{*}\ \text{ uniformly in
}k\in\mathbb{N},$

since $u(t,n;0,u^{(0)}_{k})=u_{k}(t,n-k)$ by Theorem 2.1. Then it follows from
Proposition 2.3 that

$\lim\limits_{t\to\infty}\inf\limits_{n\leq wt}u(t,n;0,u^{(0)})=1,\ \forall
0\leq w<w^{*}.$

∎

Using Lemma 4.5, we finish the proof of (3) of Theorem 1.1.

######  Proposition 4.6.

Let $u$ be a generalized transition front of equation (1.2) and let $N$ be
such that (1.4) holds. Then

$\liminf\limits_{t-s\rightarrow+\infty}\frac{N(t)-N(s)}{t-s}\geq w^{*}.$

Particularly, there exists no generalized transition front with average speed
$w<w^{*}$.

###### Proof.

First by (1.4) and Proposition 2.3, we can check that

(4.13) $\alpha:=\inf\limits_{t\in\mathbb{R},n\leq 0}u(t,n+N(t))>0\ \text{and
}\beta:=\sup\limits_{t\in\mathbb{R},n>0}u(t,n+N(t))<1.$

It is clear that $\alpha<1,\beta>0$. Assume that, by contradiction, there
exist $t_{k}$ and $s_{k}$ such that

$\lim\limits_{k\rightarrow\infty}t_{k}-s_{k}=+\infty\text{ and
}\lim\limits_{k\rightarrow\infty}\frac{N(t_{k})-N(s_{k})}{t_{k}-s_{k}}=w<w^{*}.$

Set $v_{k}(t,n):=u(t+s_{k},n+N(s_{k}))$. It is clear that $v_{k}(0,n)\geq
u^{(0)}(n)$, where $u^{(0)}(n)=0\ \text{if}\ n>0,\ \text{ and }\
u^{(0)}(n)=\alpha\ \text{if}\ n\leq 0$. Thus by Proposition 2.3, we have
$v_{k}(t,n)\geq u(t,n;0,u^{(0)})$ for $t\geq 0$ and $n\in\mathbb{Z}$, which
yields

(4.14) $\displaystyle u(t_{k},N(t_{k})+1)$
$\displaystyle=v_{k}(t_{k}-s_{k},N(t_{k})-N(s_{k})+1)$ $\displaystyle\geq
u(t_{k}-s_{k},N(t_{k})-N(s_{k})+1;0,u^{(0)}).$

For the left-hand side of (4.14), we have $u(t_{k},N(t_{k})+1)\leq\beta<1$ by
(4.13). But from Lemma 4.5, the right-hand side of (4.14) converges to $1$ as
$k\to\infty$ since
$\lim\limits_{k\rightarrow\infty}\frac{N(t_{k})-N(s_{k})+1}{t_{k}-s_{k}}=w<w^{*},$
which is a contradiction. ∎

In all, we have proved (3) in Theorem 1.1.

###  4.3. Construction of the critical fronts

At last, to verify (2) of Theorem 1.1, we only need to consider (1.3) with
$g=c$, i.e., (1.2). First we want to construct the critical front with average
speed $w^{*}$. By critical front we mean that

######  Definition 4.1.

We say that an entire solution $u$ of (1.2) with $0<u<1$, is a critical
traveling front (to the right) if for all
$(t_{0},n_{0})\in\mathbb{R}\times\mathbb{Z}$, $v$ is an entire solution of
(1.2) such that $v(t_{0},n_{0})=u(t_{0},n_{0})$ and $0<v<1,$ then

$u(t_{0},n)\geq v(t_{0},n)\text{ if }n\leq n_{0}\text{ and }u(t_{0},n)\leq
v(t_{0},n)\text{ if }n>n_{0}.$

Before going any further, we introduce some useful lemmas.

######  Lemma 4.6.

Let $u(t,n)$ be an entire solution of (1.2) with $0<u<1$. There exists
$\delta\in(0,1),$ such that $|u(t,n)-u(t,n+1)|\leq 1-\delta\ \text{for all}\
t,n\in\mathbb{R}\times\mathbb{Z}.$

###### Proof.

Suppose that the conclusion fails. Then for any $k\in\mathbb{Z}_{+}$, there
exist $t_{k}$ and $n_{k}$ such that $|u(t_{k},n_{k})-u(t_{k},n_{k}+1)|>1-1/k$.
After passing a subsequence, we have
$u(s_{k},m_{k})-u(s_{k},m_{k}+1)>1-\varepsilon_{k}$ or
$u(s_{k},m_{k}+1)-u(s_{k},m_{k})>1-\varepsilon_{k}$ for some $s_{k}$, $m_{k}$,
and $\varepsilon_{k}$ with $\varepsilon_{k}\to 0$. We only prove the former
case. Note that $0<u(t,n)<1$. Then

(4.15) $u(s_{k},m_{k})>1-\varepsilon_{k}\text{ and
}u(s_{k},m_{k}+1)<\varepsilon_{k}.$

Note also that $|\frac{d}{dt}u|\leq 4+\|g\|_{l^{\infty}(\mathbb{Z})}$. Then
there exists $S$ which is independent of $k$ such that $u(s_{k}-S,m_{k})\geq
3/4-\varepsilon_{k}>1/2$ for $k$ large. On the other hand, from Proposition
2.2, there exists a constant $C(S)$ which only depends on $S$ such that

$1/2<u(s_{k}-S,m_{k})\leq C(S)u(s_{k},m_{k}+1)\leq C(S)\varepsilon_{k}\to 0\
\text{as }k\to 0.$

This yields a contradiction. ∎

With Lemma 4.6 at hand, we have the following equivalent definition of
generalized transition front.

######  Lemma 4.7.

Let $u(t,n)$ be a solution of (1.2) with $0<u<1$ and

$u(t,n)\to 0\ \text{as}\ n\to\infty,u(t,n)\to 1\ \text{as}\ n\to-\infty\
\text{for any}\ t\in\mathbb{R}.$

Then $u$ is a generalized transition front if and only if

(4.16) $\sup\limits_{t\in\mathbb{R}}diam\\{n\in\mathbb{Z}|\varepsilon\leq
u(t,n)\leq 1-\varepsilon\\}<\infty\ \text{for any}\ \varepsilon\in(0,1/2).$

###### Proof.

As we see, (4.16) is satisfied if $u$ is a generalized transition front. Next
we prove that $u$ is a generalized transition front provided (4.16) holds. Set
$N(t):=\sup\\{n|u(t,n)\geq 1/4\\}$.

Now we claim that
$1/4\leq\inf\limits_{t\in\mathbb{R}}u(t,N(t))\leq\sup\limits_{t\in\mathbb{R}}u(t,N(t))<1.$
In fact, if $\sup\limits_{t\in\mathbb{R}}u(t,N(t))=1$, then there exists a
sequence $\\{t_{k}\\}\subset{\mathbb{Z}}$ such that $u(t_{k},N(t_{k}))\to 1$.
After passing to a subsequence, $u(t+t_{k},n+N(t_{k}))$ converges locally
uniformly to a function $v(t,n)$. Then $v(0,0)=1$, and thus $v(t,n)\equiv 1$
by Corollary 2.1. On the other hand, the definition of $N(t)$ gives
$v(0,1)\leq 1/4$, which is a contradiction.

For any
$\varepsilon<\varepsilon_{0}:=\min\\{1-\sup\limits_{t\in\mathbb{R}}u(t,N(t)),1/4,\delta/2\\}$,
where $\delta$ was given in Lemma 4.6, we have

$N(t)\in\\{n\in\mathbb{Z}|\varepsilon_{0}\leq u(t,n)\leq
1-\varepsilon_{0}\\}\subset\\{n\in\mathbb{Z}|\varepsilon\leq u(t,n)\leq
1-\varepsilon\\}.$

Denote
$L_{\epsilon}:=\sup\limits_{t}\mathrm{diam}\\{n\in\mathbb{Z}|\varepsilon\leq
u(t,n)\leq 1-\varepsilon\\}$. Therefore,

$n+N(t)\notin\\{n\in\mathbb{Z}|\varepsilon\leq u(t,n)\leq
1-\varepsilon\\}\text{ for all }t\in\mathbb{R}$

if $n>L_{\varepsilon}+1$ or $n<-L_{\varepsilon}-1$. Note that $u(t,n)\to 0\
\text{as}\ n\to\infty\text{ and }u(t,n)\to 1\ \text{as}\ n\to-\infty\
\text{for any}\ t\in\mathbb{R}$. Combining this with Lemma 4.6, we have
$u(t,n+N(t))\leq\varepsilon\text{ for }n>L_{\varepsilon}+1,$ and
$u(t,n+N(t))\geq 1-\varepsilon\text{ for }n<-L_{\varepsilon}-1.$ Therefore,
(1.4) holds. ∎

Let us now construct critical front. Fix any $\theta\in(0,1)$. For any
$k\in\mathbb{Z}_{+}$, we define

$H_{k}(n)=\left\\{\begin{aligned} &1&\text{ if }&n\leq-k\\\ &0&\text{ if
}&n>-k.\end{aligned}\right.$

Then by Lemma 4.5 and the continuity of $u(t,n;0,H_{k})$ with respect to $t$,
we can define $s_{k}:=\min\\{s|u(s,0;0,H_{k})=\theta\\}>0$. In particular,
$u(s_{k},0;0,H_{k})=\theta$. Note that by Theorem 2.1,
$u(t,n;s,H_{k})=u(t-s,n;0,H_{k})$ for any $t\geq s$. Then
$u(0,0;-s_{k},H_{k})=\theta$ for any $k\in\mathbb{Z}_{+}$. The idea is to take
the limit of some subsequence from $\\{u(t,n;-s_{k},H_{k})\\}$, and prove the
resulting function is exactly the critical traveling front. Moreover, it is
exactly a generalized transition front with average speed $w^{*}$. Before
that, an observation about some important properties of $s_{k}$ is given by

######  Lemma 4.8.

$s_{k}$ is strictly increasing and converges to $+\infty$.

###### Proof.

We first show that $s_{k}$ is strictly increasing. Assume by contradiction,
that $s_{k}\geq s_{k+1}$ for some $k\in\mathbb{Z}_{+}$. Note that

$u(t+s_{k}-s_{k+1},n;-s_{k+1},H_{k+1})\Large|_{t=-s_{k}}=H_{k+1}\leq
H_{k}=u(t,n;-s_{k},H_{k})\Large|_{t=-s_{k}}.$

Then by Proposition 2.3, we have

$u(t+s_{k}-s_{k+1},n;-s_{k+1},H_{k+1})<u(t,n;-s_{k},H_{k})\text{ for
}t>-s_{k}.$

In particular,
$\theta=u(0,0;-s_{k+1},H_{k+1})<u(s_{k+1}-s_{k},0;-s_{k},H_{k})$. Notice that
$u(-s_{k},0;-s_{k},H_{k})=0$. Then by intermediate value theorem there exists
$-s_{k}<\tau<s_{k+1}-s_{k}\leq 0$ such that

$\theta=u(\tau,0;-s_{k},H_{k})=u(\tau+s_{k},0;0,H_{k}).$

Therefore the definition of $s_{k}$ gives $s_{k}\leq\tau+s_{k}$. That is
impossible since $\tau<0$. Hence $s_{k}<s_{k+1}$.

Next, we prove that $\lim\limits_{k\to\infty}s_{k}=+\infty$. Suppose by
contradiction that after passing to a subsequence,
$\lim\limits_{k\to\infty}s_{k}=s_{\infty}<+\infty$. Let $\phi_{E,k}$ be a
solution of

$\phi(n+1)+\phi(n-1)-2\phi(n)+c(n-k)\phi(n)=E\phi(n),\ n\in\mathbb{Z},$

with $\phi(0)=1,\ \lim\limits_{n\rightarrow+\infty}\phi(n)=0,$ where
$E>\lambda_{1}.$ Then
$\overline{u}_{k}(t,n+k):=\min\\{1,\phi_{E,k}(n+k)e^{Et}\\}$ is a
supersolution of

$u_{t}(n)-u(n+1)-u(n-1)+2u(n)=c(n)u(n)(1-u(n)),\
(t,n)\in\mathbb{R}\times\mathbb{Z}.$

Note that for any $t\leq 0$,

$\overline{u}_{k}(t,k)=\min\\{1,\phi_{E,k}(k)e^{Et}\\}\leq\phi_{E,k}(k)\leq
Ce^{-\delta k},$

for some constant $C$ only depending on $E$, $\lambda_{1}$ and
$\|g\|_{l^{\infty}}$, and the last inequality follows from Lemma 3.1. Then we
can take $K$ large such that $\overline{u}_{K}(t,K)\leq\theta/2$ for any
$t<0$. Moreover, combining the Proposition 3.4, there exists $K_{1}$ such that
for any $t\geq-s_{\infty}$ and $n\in\mathbb{Z}$,

$H_{K_{1}}(n)\leq\overline{u}_{K}(-s_{\infty},n+K)\leq\overline{u}_{K}(t,n+K).$

Then
$u(-s_{K_{1}},n;-s_{K_{1}},H_{K_{1}})\leq\overline{u}_{K}(-s_{\infty},n+K)\leq\overline{u}_{K}(-s_{K_{1}},n+K)$.
Thus for any $t\geq-s_{K_{1}}$ and $n\in\mathbb{Z},$ it follows from
Proposition 2.3

$u(t,n;-s_{K_{1}},H_{K_{1}})\leq\overline{u}_{K}(t,n+K).$

In particular, we have
$\theta=u(0,0;-s_{K_{1}},H_{K_{1}})\leq\overline{u}_{K}(0,K)\leq\theta/2$,
which is a contradiction. Then we complete the proof. ∎

As $s_{k}\to\infty$, there exists a subsequence of $\\{u(t,n;-s_{k},H_{k})\\}$
such that it converges to some entire solution $u(t,n)$. Moreover, $u(t,n)$ is
”steeper” than any other entire solution in the following sense (see [21] for
continuous case).

######  Lemma 4.9.

Let $u(t,n)$ be a limit of some subsequences of $u(t,n;-s_{k},H_{k})$. Assume
that $v$ is an entire solution of (1.2) with $v(t,n)\in(0,1)$ on
$\mathbb{R}\times\mathbb{Z}$. Then for any $t\in\mathbb{R}$, there exists
$n_{t}\in\mathbb{Z}\cup\\{\pm\infty\\}$ such that

$u(t,n)\geq v(t,n)\text{ if }n\leq n_{t},\ \text{and }u(t,n)\leq v(t,n)\text{
if }n>n_{t}.$

First we need the following proposition whose proof can be found in [25, Lemma
4]:

######  Proposition 4.7 ([25]).

Consider the solution $u_{1}(t,n)$ and $u_{2}(t,n)$ of (1.2). Denote
$w(t,n)=u_{1}(t,n)-u_{2}(t,n).$ If $w(t_{0},n_{0})>0$ for some
$(t_{0},n_{0})\in\mathbb{R}\times\mathbb{Z}$, $w(t_{0},n)>0$ for $n<n_{0}$,
$w(t_{0},n)<0$ for $n>n_{0}$, then the following hold:

  1. (1)

For any $t\geq t_{0},$ if $w(t,n)>0$ for some $n\in\mathbb{Z}$, then
$w(t,m)>0$ for any $m<n$.

  2. (2)

For any $t\geq t_{0},$ if $w(t,n)<0$ for some $n\in\mathbb{Z}$, then
$w(t,m)<0$ for any $m>n$.

###### Proof of Lemma 4.9.

First we prove that if $u(t_{0},n_{0})<v(t_{0},n_{0})$ for some
$(t_{0},n_{0})\in\mathbb{R}\times\mathbb{Z}$, then $u(t_{0},m)\leq v(t_{0},m)$
for any $m>n_{0}$.

Suppose by contradiction that $u(t_{0},m)>v(t_{0},m)$ for some $m>n_{0}$. Then
we have

(4.17) $u(t_{0},n_{0};-s_{k},H_{k})<v(t_{0},n_{0}),\text{ and
}u(t_{0},m;-s_{k},H_{k})>v(t_{0},m)$

for some $k$ large enough. It is clear to check that

$\left\\{\begin{aligned} 1=u(-s_{k},n;-s_{k},H_{k})>v(-s_{k},n),\ &n\leq-k,\\\
0=u(-s_{k},n;-s_{k},H_{k})<v(-s_{k},n),\ &n>-k.\end{aligned}\right.$

Applying Proposition 4.7 with $w(t,n):=u(t,n;-s_{k},H_{k})-v(t,n)$, we can
conclude that for $t\geq-s_{k}$ and $n\in\mathbb{Z}$ where $w(t,n)<0$, one has
$w(t,m)<0$ for any $m>n$. This contradicts (4.17).

Similarly, if $u(t_{0},n_{0})>v(t_{0},n_{0})$ for some
$(t_{0},n_{0})\in\mathbb{R}\times\mathbb{Z}$, then $u(t_{0},m)\geq v(t_{0},m)$
for any $m<n_{0}$. Therefore the existence of $n_{t}$ follows directly. ∎

Now we begin to construct an entire solution by taking the limit of
$\\{u(t,n;-s_{k},H_{k})\\}$:

######  Lemma 4.10.

The limit $u(t,n):=\lim\limits_{k\to\infty}u(t,n;-s_{k},H_{k})$ exists locally
uniformly in $(t,n)\in\mathbb{R}\times\mathbb{Z}$. Moreover, $u$ is an entire
solution of (1.2) with $u(0,0)=\theta$.

###### Proof.

Define $w(t,n):=u(t,n;-s_{k},H_{k})-u(t,n;-s_{k+1},H_{k+1})$ for $t\geq-s_{k}$
and $n\in\mathbb{Z}$. Then for $t>-s_{k},\ n\in\mathbb{Z}$, $w$ satisfies

$w_{t}(t,n)-w(t,n+1)-w(t,n-1)+2w(t,n)=c(n)f_{k}(t,n)w(t,n),$

where $f_{k}(t,n)=1-u(t,n;-s_{k},H_{k})-u(t,n;-s_{k+1},H_{k+1})$. Now we prove
this lemma in the following two steps.
Step 1: Show that $w(0,n)\geq 0$ for $n<0$, and $w(0,n)\leq 0$ for $n>0$.

Assume, by contradiction, that there is $n_{1}<0$ such that $w(0,n_{1})<0$
(The proof is similar in the case where there is $n_{1}>0$ such that
$w(0,n_{1})>0$). Note that

$\left\\{\begin{aligned} &\forall
n\leq-k,w(-s_{k},n)=1-u(-s_{k},n;-s_{k+1},H_{k+1})>0,\\\ &\forall
n>-k,w(-s_{k},n)=-u(-s_{k},n;-s_{k+1},H_{k+1})<0.\end{aligned}\right.$

We can deduce from Proposition 4.7 that if $w(t,n)<0$ for some $t>-s_{k}$ and
$n\in\mathbb{Z}$, then $w(t,m)<0$ for any $m>n$ (similarly, if $w(t,n)>0$ for
some $t>-s_{k}$ and $n\in\mathbb{Z}$, then $w(t,m)>0$ for $m<n$). Therefore,
$w(0,m)<0$ for $m>n_{1}$ contradicts with $w(0,0)=0$.
Step 2: Show that $u_{1}(t,n)=u_{2}(t,n)$, where $u_{i}(t,n)(i=1,2)$ are any
two limits of different subsequences of $u(t,n;-s_{k},H_{k})$.

From Step 1, we have $u(0,n;-s_{k},H_{k})\geq u(0,n;-s_{k+1},H_{k+1})$ for
$n<0$ and $u(0,n;-s_{k},H_{k})\leq u(0,n;-s_{k+1},H_{k+1})$ for $n>0$. That is
to say, the sequence $\\{u(0,n;-s_{k},H_{k})\\}_{k}$ is nonincreasing if $n<0$
and is nondecreasing if $n>0$. Thus the limit
$u_{0}(n):=\lim\limits_{k\to\infty}u(0,n;-s_{k},H_{k})$ is well defined for
$n\in\mathbb{Z}$. Hence $u_{1}(0,n)=u_{2}(0,n)$, which yields that
$u_{1}(t,n)=u_{2}(t,n)$ for $t>0$ and $n\in\mathbb{Z}$ by Proposition 2.3.

Now we prove $u_{1}(t,n)=u_{2}(t,n)$ for all
$(t,n)\in\mathbb{R}\times\mathbb{Z}$. From Lemma 4.9, there exist
$n^{i}_{t}\in\mathbb{Z}\cup\\{\pm\infty\\},\ (i=1,2)$ such that

$u_{1}(t,n)\geq u_{2}(t,n)\text{ if }n\leq n^{1}_{t},\ \text{and
}u_{1}(t,n)\leq u_{2}(t,n)\text{ if }n>n^{1}_{t},$ $u_{2}(t,n)\geq
u_{1}(t,n)\text{ if }n\leq n^{2}_{t},\ \text{and }u_{2}(t,n)\leq
u_{1}(t,n)\text{ if }n>n^{2}_{t}.$

It’s clear that $u_{1}(t,n)=u_{2}(t,n)$ if $n^{1}_{t}=n^{2}_{t}$ for any
$t<0$. Now we assume, without loss of generality, that
$n^{1}_{\tau}<n^{2}_{\tau}$ for some $\tau<0$. It follows directly that

(4.18) $\left\\{\begin{aligned} &u_{1}(\tau,n)=u_{2}(\tau,n)\text{ if }n\leq
n^{1}_{\tau}\text{ or }n>n^{2}_{\tau},\\\ &u_{1}(\tau,n)\leq
u_{2}(\tau,n)\text{ if }n^{1}_{\tau}<n\leq n^{2}_{\tau}.\end{aligned}\right.$

It’s still needed to prove $u_{1}(\tau,n)=u_{2}(\tau,n)$ for
$n^{1}_{\tau}<n\leq n^{2}_{\tau}$. If not, then $u_{1}(t,n)<u_{2}(t,n)$ for
$t>\tau$ and $n\in\mathbb{Z}$ by Proposition 2.3. That is impossible since
$u_{1}(t,n)=u_{2}(t,n)$ for any $t>0$ and $n\in\mathbb{Z}$. Hence
$u_{1}(t,n)=u_{2}(t,n)$ in $\mathbb{R}\times\mathbb{Z}$ with $u(0,0)=\theta$
which follows from $u(0,0;-s_{k},H_{k})=\theta$ for any $k\in\mathbb{Z}_{+}$.
∎

Now we prove (2) of Theorem 1.1 by verifying that the above solution is
exactly a time increasing generalized transition front with average speed
$w^{*}$.

######  Theorem 4.3.

$u(t,n)$ in Lemma 4.10 is a critical traveling front and a time increasing
generalized transition front with average speed $w^{*}$.

###### Proof.

We prove that $u(t,n)$ is a critical traveling front first. Assume that $v$ is
an entire solution of (1.2) such that $v(t_{0},n_{0})=u(t_{0},n_{0})$ and
$0<v<1$. Then by Lemma 4.9, we have

$u(t_{0},n)\geq v(t_{0},n)\text{ if }n\leq n_{0}\text{ and }u(t_{0},n)\leq
v(t_{0},n)\text{ if }n>n_{0}.$

That is to say, $u(t,n)$ is a critical traveling front.

Let $u_{w}(t,n)$ be the almost periodic traveling front $u(t,n;c)$ obtained in
Theorem 4.2 with average wave speed $w>w^{*}$. Thus by Proposition 2.6,
$u_{w}(t,n)$ is also a generalized transition front with average speed $w$.
Then a similar argument to that of [44, Theorem 3.1] yields that
$\sup\limits_{t\in\mathbb{R}}diam\\{n\in\mathbb{Z}|\varepsilon\leq u(t,n)\leq
1-\varepsilon\\}<\infty$ for any $\varepsilon\in(0,1/2)$. Therefore, it
follows from Lemma 4.7 that $u(t,n)$ is a generalized transition front.

It remains to show that $u(t,n)$ possesses an average speed $w^{*}$. Set
$N(t):=\sup\\{n|u(t,n)\geq 1/4\\}$ and $M_{w}(t):=\sup\\{n|u_{w}(t,n)\geq
1/4\\}$. Then by similar arguments to that of [44, Theorem 3.6], we can find
$L$ such that, for any $s\in\mathbb{R}$, there exists $\tilde{s}\in\mathbb{R}$
satisfying

$N(s+\tau)-N(s)\leq M_{w}(\tilde{s}+\tau)-M_{w}(\tilde{s})+L,\ \forall\
\tau>0.$

Then

$\lim\limits_{\tau\to+\infty}\sup\limits_{s}\frac{N(s+\tau)-N(s)}{\tau}\leq\lim\limits_{\tau\to+\infty}\sup\limits_{\tilde{s}}\frac{M_{w}(\tilde{s}+\tau)-M_{w}(\tilde{s})+L}{\tau}=w,$

for all $w^{*}<w<\underline{w}$, which gives
$\limsup\limits_{t-s\rightarrow+\infty}\frac{N(t)-N(s)}{t-s}\leq w^{*}$. On
the other hand, since $u(t,n)$ is a generalized transition front and $N(t)$
satisfies (1.4), we have
$\liminf\limits_{t-s\rightarrow+\infty}\frac{N(t)-N(s)}{t-s}\geq w^{*}$ by
Proposition 4.6. Hence $w^{*}$ is the average speed of $u$.

At last, we need to prove $u(t,n)$ is a time increasing traveling front.
Otherwise, there exist some $t^{\prime},\tau,n^{\prime}$ such that
$u(t^{\prime}+\tau,n^{\prime})<u(t^{\prime},n^{\prime})$. Since
$\alpha:=\inf\limits_{n<0}u(0,n)\in(0,1)$, then by the similar argument in the
proof of Proposition 4.6, we can deduce that $u(t,n)\geq u(t,n;0,u^{(0)})$
with $u^{(0)}(n)=0$ for $n>0$ and $u^{(0)}(n)=\alpha$ for $n\leq 0$. Thus, we
have $\limsup\limits_{t\rightarrow\infty}u(t,n)=1.$ Combining this with the
intermediate value theorem, there exists $T_{0}>\tau$ such that
$u(t^{\prime}+T_{0},n^{\prime})=u(t^{\prime},n^{\prime})$.

As $u(t,n)$ is a critical traveling front, so is $v(t,n):=u(t+T_{0},n)$.
Combining $u(t^{\prime},n^{\prime})=v(t^{\prime},n^{\prime})$ and the
definition of critical traveling front, one has

$u(t^{\prime},n)\geq v(t^{\prime},n)\text{ if }n<n^{\prime}\text{ and
}u(t^{\prime},n)\leq v(t^{\prime},n)\text{ if }n>n^{\prime},$
$v(t^{\prime},n)\geq u(t^{\prime},n)\text{ if }n<n^{\prime}\text{ and
}v(t^{\prime},n)\geq u(t^{\prime},n)\text{ if }n>n^{\prime}.$

Hence $u(t^{\prime},n)=v(t^{\prime},n)$ for $n\in\mathbb{Z}$. Now we get the
conclusion that $u(t,n)=v(t,n)=u(t+T_{0},n)$ for any
$(t,n)\in\mathbb{R}\times\mathbb{Z}$ by the similar arguments in Step 2 of
Lemma 4.10. This means $u$ is a time periodic transition front. Then the
maximum 1 of $u(t,0)$ can be attained since
$\lim\sup\limits_{t\in\mathbb{R}}u(t,0)=1$. This contradicts Corollary 2.1.
Then the proof is complete. ∎

Summing up Theorem 4.2, Proposition 4.6 and Theorem 4.3, we have proved
Theorem 1.1.

##  5\. Positive almost periodic solution of the Discrete Schrödinger equation

###  5.1. Criterion of existence of positive almost periodic solution

Theorem 1.1 is an incomplete answer since we cannot determine whether (1.2)
has an almost periodic traveling front with average wave speed
$w\geq\underline{w}$, even the generalized transition front. However, in the
case $\underline{w}=\infty$, we can get a complete answer by Theorem 1.1. In
this section, we will provide some conditions on $c$ in (1.2) to guarantee
$\underline{w}=\infty$. The idea is to apply Proposition 3.5 and to use KAM
method for constructing the positive almost periodic solution of
$(\mathcal{L}_{V,\alpha,\theta}u)(n)=u(n+1)+u(n-1)-2u(n)+V(n\alpha+\theta)u(n)=\lambda_{1}u(n)$
with $V,\alpha,\theta$ to be specified and
$\lambda_{1}=\max\Sigma(\mathcal{L}_{V,\alpha,\theta})$.

We will first provide a simple criterion, which says that if the Schrödinger
cocycle can be reduced to constant parabolic cocycle, and the conjugacy is
close to the identity, then the corresponding Schrödinger equation has a
positive almost-periodic solution. Recall that an almost-periodic cocycle
$(\alpha,A)$ is said to be $C^{s}$ reducible where $\ 0\leq
s\leq\infty,\omega$, if there exist $B(\cdot)\in
C^{s}({\mathbb{T}}^{d},\mathrm{SL}(2,\mathbb{R}))$ and a constant matrix
$\tilde{A}$ such that

$B(\theta+\alpha)^{-1}A(\theta)B(\theta)=\tilde{A}.$

Denote the norm in $C^{s}(\mathbb{T}^{d},\mathrm{SL}(2,\mathbb{R}))$ as
$\|F\|_{s}:=\sup\limits_{|l|\leq
s,\theta\in\mathbb{T}^{d}}\|\partial^{l}F(\theta)\|$. Then we have the
following:

######  Lemma 5.1.

Suppose that there exists $B\in
C^{0}({\mathbb{T}}^{d},\mathrm{SL}(2,\mathbb{R}))$ with
$\|B-\operatorname{id}\|_{0}\leq\frac{1}{100}$ such that

(5.1) $B^{-1}(\theta+\alpha)S^{V}_{E}(\theta)B(\theta)=\tilde{A},$

where $\tilde{A}\in\mathrm{SL}(2,\mathbb{R})$ is a parabolic matrix (i.e. the
trace $|tr(\tilde{A})|=2$). Then

(5.2)
$(\mathcal{L}_{V,\alpha,\theta}u)(n)=u(n+1)+u(n-1)-2u(n)+V(\theta+n\alpha)u(n)=Eu(n)$

has an almost-periodic positive solution.

###### Proof.

Without loss of generality, we assume $tr(\tilde{A})=2$. Then one can find
some $R_{\eta}:=\begin{pmatrix}\cos\eta&-\sin\eta\\\
\sin\eta&\cos\eta\end{pmatrix}\in SO(2,\mathbb{R})$ with $\eta\in[0,2\pi)$
such that

(5.3) $R_{\eta}\begin{pmatrix}1&p\\\ 0&1\end{pmatrix}R_{-\eta}=\tilde{A}$

for some $p\in\mathbb{R}.$ Denote $B(\theta):=\begin{pmatrix}B_{11}&B_{12}\\\
B_{21}&B_{22}\end{pmatrix}$ and
$C(\theta):=R_{\eta}B(\theta)=\begin{pmatrix}C_{11}(\theta)&C_{12}(\theta)\\\
C_{21}(\theta)&C_{22}(\theta)\end{pmatrix}$. It directly follows that

(5.4) $C^{-1}(\theta+\alpha)S^{V}_{E}(\theta)C(\theta)=\begin{pmatrix}1&p\\\
0&1\end{pmatrix}.$

Now we can write (5.4) as

$\begin{pmatrix}E+2-V(\theta)&-1\\\
1&0\end{pmatrix}C(\theta)=C(\theta+\alpha)\begin{pmatrix}1&p\\\
0&1\end{pmatrix},$

which gives us

$\displaystyle(E+2-V(\theta))C_{11}(\theta)-C_{21}(\theta)$ $\displaystyle=$
$\displaystyle C_{11}(\theta+\alpha),$ $\displaystyle C_{11}(\theta)$
$\displaystyle=$ $\displaystyle C_{21}(\theta+\alpha),$

that is

$C_{11}(\theta+\alpha)+C_{11}(\theta-\alpha)-2C_{11}(\theta)+V(\theta)C_{11}(\theta)=EC_{11}(\theta).$

Hence $u(n)=C_{11}(n\alpha+\theta)=C_{21}((n+1)\alpha+\theta)$ is an almost
periodic solution. Now we distinguish this into the following two cases:

Case 1: If $\cos^{2}\eta>\frac{1}{2}$, $\sin^{2}\eta<\frac{1}{2}$. Moreover if
$\cos\eta>0$, then by the assumption
$\|B-\operatorname{id}\|_{C^{s}}\leq\frac{1}{100}$,

$C_{11}(\theta)=\cos\eta B_{11}(\theta)-\sin\eta
B_{21}(\theta)>\frac{\sqrt{2}}{4}\text{ for any }\theta\in\mathbb{T}^{d}.$

This implies $C_{11}(n\alpha+\theta)$ is an positive almost periodic solution
of (5.2); If $\cos\eta<0$, then $-C_{11}(n\alpha+\theta)$ is an positive
almost periodic solution of (5.2).

Case 2: If $\cos^{2}\eta\leq\frac{1}{2}$, $\sin^{2}\eta\geq\frac{1}{2}$.
Moreover if $\sin\eta>0$, one can similarly verify that
$C_{21}(\theta)=\cos\eta B_{21}(\theta)+\sin\eta
B_{11}(\theta)\geq\frac{\sqrt{2}}{4}$ for any $\theta\in\mathbb{T}^{d}$. Hence
$C_{21}(n\alpha+\theta)$ is an positive almost periodic solution; If
$\sin\eta<0$, $-C_{21}(n\alpha+\theta)$ is an positive almost periodic
solution of (5.2).

In all, the proof is complete. ∎

###  5.2. Analytic quasi-periodic potential

Motivated by Lemma 5.1, to prove the existence of positive almost periodic
solution of (5.2), we only need to prove that the corresponding Schrödinger
cocycle is reducible to a parabolic constant cocycle:

$\mathrm{e}^{-Y(\theta+\alpha)}S^{V}_{E}(\theta)\mathrm{e}^{Y(\theta)}=\tilde{A}.$

Moreover, the conjugation is close to constant.

First we state the reducibility result for analytic quasi-periodic potential.
To prove this, we first need a non-resonance cancelation lemma. The result
will be the basis of our proof, and we will also use this when we deal with
analytic almost periodic potentials.

Let $\mathcal{B}$ be a $\mathrm{sl}(2,\mathbb{R})$ valued Banach algebra.
Assume that for any given $\eta>0,\alpha\in\mathbb{T}^{d}$ where
$d\in\mathbb{N}_{+}\cup\\{\infty\\}$ and $A\in\mathrm{SL}(2,\mathbb{R}),$ we
have a decomposition of the Banach space $\mathcal{B}$ into non-resonant
spaces and resonant spaces, i.e.
$\mathcal{B}=\mathcal{B}^{nre}(\eta)\oplus\mathcal{B}^{re}(\eta)$. Here
$\mathcal{B}^{nre}(\eta)$ is defined in the following way: for any
$Y\in\mathcal{B}^{nre}(\eta),$ we have

$A^{-1}Y(\theta+\alpha)A\in\mathcal{B}^{nre}(\eta),\
|A^{-1}Y(\theta+\alpha)A-Y(\theta)|\geq\eta|Y(\theta)|,$

where $|\cdot|$ is the norm of the Banach space $\mathcal{B}$.

Once we have this, we have the following:

######  Lemma 5.2.

Assume that
$A\in\mathrm{SL}(2,\mathbb{R}),\epsilon\leq(4\bigr{\|}A\bigr{\|})^{-4}$ and
$\eta\geq 13\|A\|^{2}\epsilon^{\frac{1}{2}}$. For any $F\in\mathcal{B}$ with
$|F|\leq\epsilon,$ there exist $Y\in\mathcal{B}$ and
$F^{re}\in\mathcal{B}^{re}(\eta)$ such that

$e^{-Y(\theta+\alpha)}Ae^{F(\theta)}e^{Y(\theta)}=Ae^{F^{re}(\theta)}.$

Moreover, we have the estimates $|Y|\leq\epsilon^{\frac{1}{2}}$ and
$|F^{re}|\leq 2\epsilon.$

######  Remark 5.1.

The proof of the lemma for
$\mathcal{B}:=C^{\omega}_{r}(\mathbb{T}^{d},\mathrm{su}(1,1))$ with
$d\in\mathbb{N}_{+}$ could be found in [17, Lemma 3.1], and we can easily see
that the proof works for any other Banach algebra.

In our application, we will set
$\mathcal{B}:=C^{\omega}_{r}(\mathbb{T}^{d},\mathrm{sl}(2,\mathbb{R}))$ where
$d\in\mathbb{N}_{+}\cup\\{\infty\\},r>0$ (the definition of
$C^{\omega}_{r}({\mathbb{T}}^{\infty},\mathrm{sl}(2,{\mathbb{R}}))$ will be
introduced later). Define the norm in $\mathcal{B}$ as
$|F|_{r}:=\sup\limits_{|\Im\theta|\leq r}|F(\theta)|$. The non-resonant space
$\mathcal{B}^{nre}$ will take the truncating operator $\mathcal{T}_{K}$ on
$\mathcal{B}$: For any $K>0$, we define

$\mathcal{T}_{K}F(\theta)=\sum\limits_{k\in\mathbb{Z}^{d},|k|<K}\hat{F}(k)\mathrm{e}^{\mathrm{i}\langle
k,\theta\rangle}$

and define $\mathcal{R}_{K}$ as

$\mathcal{R}_{K}F(\theta)=\sum\limits_{k\in\mathbb{Z}^{d},|k|\geq
K}\hat{F}(k)\mathrm{e}^{\mathrm{i}\langle k,\theta\rangle}.$

Obviously, $\mathcal{T}_{K}F+\mathcal{R}_{K}F=F$. Now as a direct application
of Lemma 5.2, we have the following:

######  Proposition 5.1.

Let $\alpha\in\mathrm{DC}_{d}(\gamma,\tau),\ r,\delta,\gamma>0,\ \tau>d\geq
1.$ Suppose that $A\in\mathrm{SL(2,\mathbb{R})}$, $F\in
C^{\omega}_{r}(\mathbb{T}^{d},\mathrm{sl(2,\mathbb{R})})$. For any
$r^{\prime}\in(0,r),$ there exists $c=c(\gamma,\tau,d)$ such that if
$|\mathrm{rot}(\alpha,A)|\leq 2\|A\|\epsilon^{\frac{1}{2}}$, and

(5.5)
$|F|_{r}\leq\epsilon<\frac{c(r-r^{\prime})^{6(1+\delta)\tau}}{\|A\|^{6}},$

then there exist $Y,F^{\prime}\in
C^{\omega}_{r^{\prime}}(\mathbb{T}^{d},\mathrm{sl(2,\mathbb{R})})$ and
$A^{\prime}\in\mathrm{SL}(2,\mathbb{R})$ such that

$\mathrm{e}^{-Y(\theta+\alpha)}A\mathrm{e}^{F(\theta)}\mathrm{e}^{Y(\theta)}=A^{\prime}\mathrm{e}^{F^{\prime}(\theta)}.$

Moreover, we have the following estimates:

$|Y|_{r^{\prime}}\leq\epsilon^{\frac{1}{2}},\quad|F^{\prime}|_{r^{\prime}}\leq
4\epsilon^{2},\quad\|A-A^{\prime}\|\leq 2\epsilon\|A\|.$

###### Proof.

We only need to apply the non-resonance cancelation lemma (Lemma 5.2). In this
case, we will define

$\Lambda_{K}=\\{f\in
C^{\omega}_{r}({\mathbb{T}}^{d},\mathrm{sl}(2,{\mathbb{R}}))\mid
f(\theta)=\sum_{k\in{\mathbb{Z}}^{d},0<\lvert k\rvert<K}\hat{f}(k)e^{i\langle
k,\theta\rangle}\\},$

where $K=\frac{2}{r-r^{\prime}}|\ln\epsilon|$, and prove that for any
$Y\in\Lambda_{K}$, the operator

$Y\rightarrow A^{-1}Y(\theta+\alpha)A-Y(\theta)$

has a bounded inverse.

Thus we only need to consider the equation

$A^{-1}Y(\theta+\alpha)A-Y(\theta)=\mathcal{T}_{K}G(\theta)-\hat{G}(0).$

Without loss of generality, we assume that
$A=\begin{pmatrix}\mathrm{e}^{\mathrm{i}\rho}&p\\\
0&\mathrm{e}^{-\mathrm{i}\rho}\end{pmatrix},$ where
$\mathrm{e}^{\pm\mathrm{i}\rho}$ are the two eigenvalues of $A$,
$p\in\mathbb{R}$, and write

$Y=\begin{pmatrix}Y_{11}&Y_{12}\\\ Y_{21}&-Y_{11}\end{pmatrix},\qquad
G=\begin{pmatrix}G_{11}&G_{12}\\\ G_{21}&-G_{11}\end{pmatrix}.$

Taking the Fourier transformation for the above equation and comparing the
Fourier coefficients, we can get

$\left\\{\begin{aligned}
\hat{Y}_{11}(k)&=\frac{\hat{G}_{11}(k)+p\mathrm{e}^{\mathrm{i}(\rho+\langle
k,\alpha\rangle)}\hat{Y}_{21}(k)}{\mathrm{e}^{\mathrm{i}\langle
k,\alpha\rangle}-1},\\\
\hat{Y}_{12}(k)&=\frac{\hat{G}_{12}(k)+p^{2}\mathrm{e}^{\mathrm{i}\langle
k,\alpha\rangle}\hat{Y}_{21}(k)-2p\mathrm{e}^{\mathrm{i}(\langle
k,\alpha\rangle-\rho)}\hat{Y}_{11}(k)}{e^{\mathrm{i}\langle
k,\alpha\rangle-2\rho}-1},\\\
\hat{Y}_{21}(k)&=\frac{\hat{G}_{21}(k)}{\mathrm{e}^{\mathrm{i}(\langle
k,\alpha\rangle+2\rho)}-1}.\end{aligned}\right.$

Note for any $k\in\mathbb{Z}^{d}$ with $0<|k|\leq K$, if $\epsilon$ satisfies
(5.5), we have

$|\langle
k,\alpha\rangle|\geq\frac{\gamma}{|k|^{\tau}}\geq\frac{\gamma}{|K|^{\tau}}\geq\frac{\gamma\epsilon^{\frac{1}{6(1+\delta)}}}{|\ln\epsilon|^{\tau}}\geq\epsilon^{\frac{1}{6(1+\frac{\delta}{2})}},$
$\displaystyle|2\mathrm{rot}(\alpha,A)-\langle
k,\alpha\rangle|\geq\frac{\gamma}{|K|^{\tau}}-4\|A\|\epsilon^{\frac{1}{2}}\geq\epsilon^{\frac{1}{6(1+\frac{\delta}{2})}},$

which implies that

$|Y(\theta)|_{r}\leq\epsilon^{-\frac{1}{2+\delta}}|\mathcal{T}_{K}G(\theta)|_{r},$

and then
$\Lambda_{K}\subset\mathcal{B}_{r}^{nre}(\epsilon^{\frac{1}{2+\delta}})$.

Since $\epsilon^{\frac{1}{2+\delta}}\geq 13\|A\|^{2}\epsilon^{\frac{1}{2}}$,
it follows from Lemma 5.2 there exist $Y,F^{re}\in
C^{\omega}_{r}(\mathbb{T}^{d},\mathrm{sl(2,\mathbb{R})})$ such that

$e^{Y(\theta+\alpha)}A\mathrm{e}^{F(\theta)}\mathrm{e}^{-Y(\theta)}=A\mathrm{e}^{F^{re}(\theta)},$

where $\mathcal{T}_{K}F^{re}=\hat{F}^{re}(0),\
\mathcal{R}_{K}F^{re}=\sum\limits_{|k|>K}\hat{F}^{re}(k)\mathrm{e}^{\mathrm{i}\langle
k,\theta\rangle}.$ Moreover, we have the estimates
$|Y|_{r}\leq\epsilon^{\frac{1}{2}},|F^{re}|_{r}\leq 2\epsilon$. Consequently,
for any $r^{\prime}\in(0,r)$, we have