# Dynamics of nonlocal and local discrete Ginzburg-Landau equations: global
attractors and their congruence

Dirk Hennig  Nikos I. Karachalios  Department of Mathematics, University of
Thessaly, Lamia GR35100, Greece

###### Abstract

Discrete Ginzburg-Landau (DGL) equations with non-local nonlinearities have
been established as significant inherently discrete models in numerous
physical contexts, similar to their counterparts with local nonlinear terms.
We study two prototypical examples of non-local and local DGLs on the one-
dimensional infinite lattice. For the non-local DGL, we identify distinct
scenarios for the asymptotic behavior of the globally existing in time
solutions depending on certain parametric regimes. One of these scenarios is
associated with a restricted compact attractor according to J. K. Hale’s
definition. We also prove the closeness of the solutions of the two models in
the sense of a ”continuous dependence on their initial data” in the $l^{2}$
metric under general conditions on the intrinsic linear gain or loss
incorporated in the model. As a consequence of the closeness results, in the
dissipative regime we establish the congruence of the attractors possessed by
the semiflows of the non-local and of the local model respectively, for
initial conditions in a suitable domain of attraction defined by the non-local
system.

##  I Introduction

In this paper, we study the following two discrete complex Ginzburg-Landau
equations:

$\dot{u}_{n}=(1-\delta)u_{n}+(1+i\alpha)(u_{n+1}-2u_{n}+u_{n-1})-(1+i\beta)\,|u_{n}|^{2}u_{n}+g_{n},\,\,\,n\in{\mathbb{Z}},$
(1.1)

with $u_{n}\in{\mathbb{C}}$ and initial conditions:

$u_{n}(0)=u_{n,0},\,\,\,n\in{\mathbb{Z}},$ (1.2)

and

$\dot{v}_{n}=(1-\delta)v_{n}+(1+i\alpha)(v_{n+1}-2v_{n}+v_{n-1})-\frac{1}{2}(1+i\beta)|v_{n}|^{2}(v_{n+1}+v_{n-1})+g_{n},\,\,\,n\in{\mathbb{Z}},$
(1.3)

with $v_{n}\in{\mathbb{C}}$ and initial conditions:

$v_{n}(0)=v_{n,0},\,\,\,n\in{\mathbb{Z}}.$ (1.4)

In what follows we refer to the systems (1.1) and (1.3) as the L-DGL and NL-
DGL, respectively. For real parameter $\delta>0$, the first term on the right-
hand side of (1.1) and (1.3) is of intrinsic linear gain [loss (damping)]
nature if $\delta<1$ ($\delta>1$), while $g=(g_{n})_{n\in{\mathbb{Z}}}\neq 0$
acts in both systems as a general external force. Further, we assume
$\alpha,\beta\in{\mathbb{R}}_{+}$.

Nonlinear lattices of the form (1.1) or (1.3), are encountered in several
diverse branches of physics, ranging from the dynamics of coupled laser
systems and waveguides in 1D UGL1 ,UGL2 ,BUGL2 , and 2D-setups UGL3 ,UGL4
,BUGL4 , to the description of the dynamical behavior of low dimensional fluid
dynamical systems UGL5 . For instance, models of the form (1.3) with nonlocal
nonlinearities determine the evolution of the amplitude of the separatrices
between a linear array of vortices or coupled waves observed in hydrodynamic
systems of low dimensionality AGL1 ,AGL2 . Furthermore, the NL-DGL (1.3) and
other variants with generalized nonlocal nonlinearities have proven to be
relevant models for the construction of soliton solutions in the presence of
gain and loss AGL3 ,AGL4  and investigations of their potential stability. In
the above physical contexts, other relevant discrete DGL equations involve
saturable nonlinearities UGL6 .

Notice that in (1.1) and (1.3) the nonlinear terms are both of cubic order,
and an important common feature of both systems, is that in the continuum
limit approximate the Ginzburg-Landau partial differential equation (PDE),

$\partial_{t}u=(1-\delta)u+(1+i\alpha)\partial_{xx}u-(1+i\beta)\,|u|^{2}u+g(x),$
(1.5)

which is a spatially driven counterpart of the complex Ginzburg-Landau
equation; a spatial driver may appear after application of a usual gauge-time
transformation in the case of a spatiotemporal driving force with harmonic in
time driving dGLE1 -dGLE5 . However, in the discrete regime, the systems (1.1)
and (1.3) are markedly different in the sense that the nonlinear terms in
(1.3) are of nonlocal nature compared to the local terms in (1.1).

Herein, we compare the solution properties of the two infinite lattice
dynamical systems presented by (1.1) and (1.3) with particular attention to
their global asymptotic behaviour. We introduce the distance variable
$\Delta=(\Delta_{n})_{n\in\mathbb{Z}}$ where $\Delta_{n}=u_{n}-v_{n}$. The
main results are the following:

  1. 1.

Closeness of the solutions. Solutions of both systems remain close when
initiated from sufficiently close initial data according to

###### Theorem I.1

Assume that the initial condition $(u_{n}(0))_{n\in{\mathbb{Z}}}=u^{0}$ of the
L-DGL (1.1)-(1.2) and the initial condition
$(v_{n}(0))_{n\in{\mathbb{Z}}}=v^{0}$ of the NL-DGL (1.3)-(1.4) satisfy

$\displaystyle||u(0)-v(0)||_{l^{2}}$ $\displaystyle\leq$ $\displaystyle
C_{0}\epsilon^{3},$ (1.6) $\displaystyle||u(0)||_{l^{2}}$ $\displaystyle\leq$
$\displaystyle C_{u,0}\epsilon:=R_{u}(\epsilon),\,\,\,||v(0)||_{l^{2}}\leq
C_{v,0}\epsilon:=R_{v}(\epsilon),$ (1.7)

for some positive constants $C_{u,0}$, $C_{v,0}>0$. Then, we have the
following cases for the closeness of the corresponding solutions:

    1. (a)

Case $\delta>1$. For any $t\in(0,\infty]$, the corresponding solutions to Eqs.
(1.1) and (1.3) fulfill the estimate

$||u(t)-v(t)||_{l^{2}}\leq C\epsilon^{3},$ (1.8)

where the constant $C(C_{0},C_{u,0},C_{v,0},\beta,\delta)$ is independent of
$t$. Furthermore, there exists a constant
$C_{1}(C_{0},C_{u,0},C_{v,0},\beta,\delta)$, still independent of $t$, such
that

$\displaystyle\limsup_{t\rightarrow\infty}||\Delta(t)||_{l^{2}}\leq
C_{1}\epsilon^{3}.$ (1.9)

    2. (b)

Case $\delta\leq 1$. For any arbitrary $0<T_{f}<\infty$ there exists a
constant $C_{2}(C_{0},C_{u,0},C_{v,0},\beta,\delta,T_{f})$ such that the the
corresponding solutions to Eqs. (1.1) and (1.3) fulfill the estimate

$||u(t)-v(t)||_{l^{2}}\leq C_{2}\epsilon^{3}.$ (1.10)

A first application of Theorem I.1 is in the context of dissipative discrete
solitons, mentioned above: Localised solutions have been constructed for the
NL-DGL (1.3),AGL3 ,AGL4 ,Abdul2  as deformed continuations of the exact
soliton solutions of the integrable Ablowitz-Ladik lattice AL , Ablowitz .
Theorem I.1 rigorously establishes that the L-DGL model (1.1) admits
$\mathcal{O}(\epsilon)$ solutions that stay $\mathcal{O}(\epsilon^{3})$ close
to the solution of the NL-DGL (1.3). In the undamped case $\delta=1$ we prove
that the growth of the distance $||\Delta(t)||_{l^{2}}$ is at most linear for
$t\in[0,T_{f}]$, while in the case of the intrinsic forcing $\delta<1$ we
found that the growth is at most exponential; the latter case is similar to
the exponential growth of the distance function when the inviscid limit of the
complex Ginzburg-Landau PDE the Nonlinear Schrödinger PDE is considered JWU
,OG .

  2. 2.

Congruence of attractors in the dissipative regime $\delta>1$. A second
application of Theorem I.1 is in the context of infinite dimensional dynamical
systems Hale ,Temam ,Robinson ,Chueshov  concerning the existence of
attractors possessed by the L-DGL and NL-DGL systems.

It is useful for completeness and in order to highlight their differences, to
recall the definitions of the global and restricted attractor from Hale ,
stated for a generic dynamical system in $l^{2}$ denoted by
$\left\\{S(t)\right\\}_{t\geq 0}$ representing one of the systems (1.1) and
(1.3).

###### Definition I.1

A set ${\cal A}\subset l^{2}$ is called a global attractor for the semigroup
$\left\\{S(t)\right\\}_{t\geq 0}$ in $l^{2}$ if (i) ${\cal A}\neq 0$ is a
compact subset of $l^{2}$, (ii) an invariant set, that is, $S(t){\cal A}={\cal
A}$ for all $t\geq 0$, and, (iii) an attracting set for
$\left\\{S(t)\right\\}_{t\geq 0}$ in $l^{2}$, viz. for all bounded
$\mathcal{B}\subset l^{2}$, it holds that
$\lim_{t\rightarrow\infty}\,dist(S(t)\mathcal{B},{\cal A})=0$, where the
Hausdorff semi-distance between two nonempty subsets $\mathcal{U},\mathcal{V}$
of $l^{2}$ is determined by

$dist(\mathcal{U},\mathcal{V})=\sup_{u\in\mathcal{U}}\,\inf_{v\in\mathcal{V}}\,d(u,v)_{l^{2}}.$
(1.11)

###### Definition I.2

We say that ${\cal A}_{u}\subset l^{2}$ is a restricted global attractor for
$\left\\{S(t)\right\\}_{t\geq 0}$ in $l^{2}$ if for some closed, nonempty
subset $\mathcal{U}$ of $l^{2}$, $S(t):\mathcal{U}\rightarrow\mathcal{U}$
($t\geq 0$) is a semigroup on $\mathcal{U}$ s.t. ${\cal A}_{u}$ is a global
attractor for $\left\\{S(t)\right\\}_{t\geq 0}$ restricted to $\mathcal{U}$,
viz. (i) $S(t){\cal A}_{u}={\cal A}_{u}$ for $t\geq 0$, (ii) ${\cal A}_{u}$ is
compact, and (iii) ${\cal A}_{u}$ attracts solutions of bounded subsets of
$\mathcal{U}$.

When $\delta>1$ the constant $C$ in the closeness of solutions estimate (1.8)
is independent of $t$ and moreover, as $t\rightarrow\infty$, the distance
function becomes uniformly bounded by $\mathcal{O}(\epsilon^{3})$, as dictated
by the limit (1.9). This is the key element in proving that the global
attractor $\mathcal{A}_{u}$ belonging to the dynamical system L-DGL and the
restricted attractor $\mathcal{A}_{v}$ in the sense of Hale  belonging to the
dynamical system NL-DGL coincide for a suitable choice of their initial
conditions: Let $\mathcal{B}_{v,0}$ denote the absorbing set for restricted
initial data $v^{0}$ (i.e. satisfying a certain smallness condition) of the
dynamical system NL-DGL. We prove

###### Theorem I.2

Let $\delta>1$ and assume that the initial conditions $u^{0}$ of the L-DGL
(1.1)-(1.2) and the initial condition $v^{0}$ of the NL-DGL (1.3)-(1.4)
satisfy (1.7) and are such that $u^{0},\,v^{0}\in\mathcal{B}_{v,0}$ . Then the
attractors ${\cal A}_{u}$ and ${\cal A}_{v}$ in $\ell^{2}$ coincide according
to

${\rm dist}\left({\cal A}_{v},{\cal A}_{u}\right)=0.$ (1.12)

In (1.12), the distance between the two sets is the usual Haussdorff semi-
distance Temam .

On route to the proof of Theorem 1.12 we shall discover much richer scenarios
for the dynamics of the NL-DGL concerning the global existence of solutions
and invariant sets than for the L-DGL, highlighting the impact of the nonlocal
nonlinearity. For instance, a careful manipulation of a Ricatti differential
inequality (instead of Gronwall’s lemma and its variants) reveals the
emergence of distinct regimes for the existence of global in time solutions
for all initial data, the existence of an invariant annulus with a potential
convergence to a particular state, the existence of a non-escaping ball
depending on the size of the initial data, as well as the existence of the
restricted absorbing ball giving rise to the restricted attractor. To
establish the compactness of the attracting sets we employ an alternative
contradiction argument to prove the asymptotic tail property instead of the
tail-estimates method Bates ,Zhou0 ,Zhou ,Zhou1 ,Zhou3 ,Wang ,Zhou2 ,Nikos
,NAE2007 ,Du ; the tail-estimates method has been established as a
fundamental, generic approach to prove asymptotic compactness not only in
autonomous but also in non-autonomous, delayed or stochastic lattice dynamical
systems, see Stoch1 ,Stoch2 ,Stoch3 ,Stoch4  and references therein. In the
case of the NL-DGL (1.3), this contradiction argument is combined with
generalised comparison principles for ordinary differential equations
McNabb1986 .

The presentation of the paper is as follows: Section II contains global
existence results and asymptotic estimates for the solutions for the
considered GLEs. For local existence of solutions, we combine both systems in
a physically significant dissipative version of the Salerno model Salerno  or
Integrable-Nonintegrable-DNLS (IN-DNLS) Cai94 . For global existence each of
the L-DGL and NL-DGL systems is treated separately, revealing in the case of
the NL-DGL the relevant distinct parametric regimes and the associated
asymptotic estimates. Section III contains the results on the existence of
attracting sets, while section IV contains the proofs of the main results,
Theorems I.1 and 1.12. Section V summarizes the findings and provide a brief
plan for further relevant studies still to be performed.

##  II Global existence and asymptotic estimates for the solutions

For the current study of existence and uniqueness of a global solution of the
NL-DGL and L-DGL, we combine them in a single system

$\dot{u}_{n}-(1+i\alpha)(u_{n+1}+u_{n-1}-2u_{n})=(1-\delta)u_{n}-(1+i\beta)\left[\gamma|u_{n}|^{2}u_{n}+\frac{\mu}{2}(u_{n+1}+u_{n-1})|u_{n}|^{2}\right]+g_{n},\,\,\,n\in{\mathbb{Z}},$
(2.1)

with $u_{n}\in{\mathbb{C}}$ and initial conditions:

$u_{n}(0)=u_{n,0},\,\,\,n\in{\mathbb{Z}}.$ (2.2)

Note that for $\gamma=0$ and $\mu=1$ ($\mu=0$ and $\gamma=1$) the L-DGL (NL-
DGL) results from (2.1). We remark that the combined system (2.1) is of
physical significance itself as it can be considered as an extended
dissipative and driven version of the Salerno model Salerno  or Integrable-
Nonintegrable-DNLS (IN-DNLS) Cai94 . In fact, the latter is obtained from
(2.1) in the limit $\alpha,\beta\rightarrow\infty$, $\delta=0$ and $g=0$.

##### Preliminaries.

The functional setting of the initial value problem for the infinite system of
ordinary differential equations in (2.1)-(2.2) consists of the sequence spaces

$l^{p}=\left\\{u=(u_{n})_{n\in{\mathbb{Z}}}\,\in{\mathbb{C}}\,\,\,|\,||u||_{l^{p}}=\left(\sum_{n\in{\mathbb{Z}}}|u_{n}|^{p}\right)^{1/p}\right\\}.$
(2.3)

When $p=2$, the Hilbert space $l^{2}$ is endowed with the inner product

$(u,v)_{l^{2}}=\sum_{n\in{\mathbb{Z}}}u_{n}\overline{v}_{n},$ (2.4)

for any $u=(u_{n})_{n\in{\mathbb{Z}}},v=(v_{n})_{n\in{\mathbb{Z}}}\in l^{2}$
(where $\overline{v}_{n}$ denotes the conjugate of $v_{n}$) and induced norm

$||u||_{l^{2}}^{2}=(u,u).$ (2.5)

In some cases, when convenient, we shall use the inner product

$\displaystyle(u,v)_{\ell^{2}}=\mathrm{Re}\sum_{{n\in\mathbb{Z}}}u_{n}\overline{v_{n}},\;\;u,\,v\in\ell^{2},$

without a change in notation. The continuous embeddings

$l^{r}\subset l^{s},\,\,\,||w||_{l^{s}}\leq||w||_{l^{r}},\,\,\,1\leq r\leq
s\leq\infty,$ (2.6)

will be thoroughly used in the sequel.

Next, let us recall some basic properties of the linear difference operators
involved in the models. For any $u\in l^{2}$, the 1D-discrete Laplacian
$\Delta_{d}:l^{2}\rightarrow l^{2}$,

$(\Delta_{d}u)_{n}=u_{n+1}-2u_{n}+u_{n-1},$ (2.7)

is bounded, self-adjoint on $D(\Delta_{d})=l^{2}$ and $\Delta_{d}\leq 0$, due
to the relations

$\begin{split}(\Delta_{d}u,v)_{l^{2}}&=(u,\Delta_{d}v)_{l^{2}},\quad\phi,\;\psi\in
l^{2},\\\
(\Delta_{d}u,u)_{l^{2}}&=-\sum_{n\in\mathbb{Z}}|u_{n+1}-u_{n}|^{2}\leq 0,\\\
||\Delta_{d}u||_{l^{2}}^{2}&\leq 4||u||_{l^{2}}^{2}.\end{split}$ (2.8)

With the properties (2.8), it is established in (NAE2007, , Lemma 2.2, pg.
715), that the operator $\mathbf{L}:l^{2}\rightarrow l^{2}$,
$(\mathbf{L}u)_{n\in\mathbb{Z}}=(1+i\alpha)(\Delta_{d}u)_{n\in\mathbb{Z}}$,
generates a one parameter semigroup $U(t)$ on $\ell^{2}$, as a perturbation of
$-\Delta_{d}$ by the bounded (skew-adjoint) linear operator
$A_{1}:\ell^{2}\rightarrow\ell^{2}$,
$(A_{1}u)_{n\in\mathbb{Z}}=-i\alpha(\Delta_{d}u)_{n\in\mathbb{Z}}$ Pazy83 .
The semigroup $U(t)$ solves the underlying linear equation
$\dot{u}=\mathbf{L}u$, namely $U(t)=\exp(\mathbf{L}t)$.

##### Local existence of solutions.

With the preliminaries stated above, the initial value problem (2.1)-(2.2) can
be considered in the form of the integral equation

$\displaystyle
u(t)=U(t)u^{0}+\int_{0}^{t}U(t-s)F(u(s))ds,\;\;\;u^{0}=\\{u_{n,0}\\}_{n\in\mathbb{Z}}.$
(2.9)

where the operator $F$ is defined by

$\left(F(u)\right)_{n}=(1-\delta)u_{n}+(1+i\alpha)(u_{n+1}+u_{n-1}-2u_{n})-(1+i\beta)\left[\gamma|u_{n}|^{2}u_{n}+\frac{\mu}{2}(u_{n+1}+u_{n-1})|u_{n}|^{2}\right]+g_{n}.$
(2.10)

In order to apply subsequently the local solvability theory to the integral
equation (2.9), the following lemma is a prerequisite:

###### Lemma II.1

Let $g=(g_{n})_{n\in{\mathbb{Z}}}\in l^{2}$ and
$\alpha,\beta,\delta,\mu\in\mathbb{R}$. The operator $F$ defined (2.10) is
well defined as an operator $F:\,l^{2}\rightarrow l^{2}$ and is Lipschitz
continuous on bounded sets of $l^{2}$.

Proof: Let $u\in B_{R}$, with $B_{R}$ a closed ball of $l^{2}$ of center $0$
and radius $R$. We only need to consider the nonlinear operator
$N:l^{2}\rightarrow l^{2}$,
$(N(u))_{n}=-(1+i\beta)\left[\gamma|u_{n}|^{2}u_{n}+\frac{\mu}{2}(u_{n+1}+u_{n-1})|u_{n}|^{2}\right]$.
Recalling that
$\sum_{n}|\theta_{n}|\,|\theta_{n+1}|\leq\sum_{n}|\theta_{n}|^{2}$, for any
$\theta\in l^{2}$, we have:

$\displaystyle||N(u)||_{l^{2}}^{2}$ $\displaystyle=$
$\displaystyle\sum_{n}\left|(1+i\beta)\left[\gamma|u_{n}|^{2}u_{n}+\frac{\mu}{2}(u_{n+1}+u_{n-1})|u_{n}|^{2}\right]\right|^{2}$
$\displaystyle\leq$
$\displaystyle(1+\beta^{2})\sum_{n}\left(|\gamma||u_{n}|^{2}|u_{n}|+\left|\frac{\mu}{2}\right||u_{n+1}+u_{n-1}||u_{n}|^{2}\right)^{2}$
$\displaystyle=$
$\displaystyle\gamma^{2}(1+\beta^{2})\sum_{n}|u_{n}|^{4}|u_{n}|^{2}+\frac{\mu^{2}}{4}(1+\beta^{2})\sum_{n}|u_{n}|^{4}|u_{n+1}+u_{n-1}|^{2}+|\gamma||\mu|\sum_{n}|u_{n}|^{2}|u_{n}||u_{n+1}+u_{n-1}||u_{n}|^{2}$
$\displaystyle\leq$
$\displaystyle\gamma^{2}(1+\beta^{2})\sup_{n}|u_{n}|^{4}\sum_{n}|u_{n}|^{2}+\mu^{2}(1+\beta^{2})\sup_{n}|u_{n}|^{4}\sum_{n}|u_{n}|^{2}+2|\mu||\gamma|\sup_{n}|u_{n}|^{4}\sum_{n}|u_{n}|^{2}.$

Then, since $l^{2}\subset l^{\infty}$ with a continuous embedding, we derive
the existence of a constant $K(\beta,\gamma,\mu,R)$, such that

$\displaystyle||N(u)||_{l^{2}}^{2}\leq K(\beta,\gamma,\mu,R)||u||^{2}_{l^{2}}$

Hence, $N:\,l^{2}\rightarrow l^{2}$ is bounded on bounded sets of $l^{2}$. For
$u,v\in B_{R}$ we have

$\displaystyle||N(v)-N(u)||_{l^{2}}^{2}=\sum_{n\in{\mathbb{Z}}}\bigg{|}(1+i\beta)\left[\gamma|v_{n}|^{2}v_{n}+\frac{\mu}{2}(v_{n+1}+v_{n-1})|v_{n}|^{2}\right]-(1+i\beta)\left[\gamma|u_{n}|^{2}u_{n}+\frac{\mu}{2}(u_{n+1}+u_{n-1})|u_{n}|^{2}\right]\bigg{|}^{2}.$

Note that in the difference $|u_{n}|^{2}u_{n}-|v_{n}|^{2}v_{n}$, when adding
and subtracting simultaneously the term $|v_{n}|^{2}u_{n}$, we may write that

$\displaystyle\left||u_{n}|^{2}u_{n}-|v_{n}|^{2}v_{n}\right|\leq|v_{n}|^{2}|u_{n}-v_{n}|+|u_{n}|\left(|v_{n}|+|u_{n}|\right)|u_{n}-v_{n}|.$

Similarly, for the difference
$(v_{n+1}+v_{n-1})|v_{n}|^{2}-(u_{n+1}+u_{n-1})|u_{n}|^{2}$, when adding and
subtracting simultaneously the term $(v_{n+1}+v_{n-1})|u_{n}|^{2}$, we may
write

$\displaystyle\left|(v_{n+1}+v_{n-1})|v_{n}|^{2}-(u_{n+1}+u_{n-1})|u_{n}|^{2}\right|$
$\displaystyle\leq$
$\displaystyle|v_{n+1}+v_{n-1}|\left(|v_{n}|+|u_{n}|\right)|v_{n}-u_{n}|$
$\displaystyle+|u_{n}|^{2}|v_{n+1}-u_{n+1}|+|u_{n}|^{2}|v_{n-1}-u_{n-1}|.$

Then, we may treat $||N(v)-N(u)||_{l^{2}}^{2}$ with a similar algebra used
above for the local boundedness of $N$ in $l^{2}$, to derive the existence of
a constant $L(\beta,\gamma,\mu,R)$ such that

$\displaystyle||N(v)-N(u)||_{l^{2}}^{2}\leq
L(\beta,\gamma,\mu,R)||v-u||_{l^{2}}^{2},$

verifying that the map $N:\,l^{2}\rightarrow l^{2}$ is Lipschitz continuous on
bounded sets of $l^{2}$. $\square$

With Lemma II.1 in hand, the local existence result of solutions can be stated
in the following proposition as an application of the local existence results
for abstract evolution equations in Banach spaces Pazy83 ,Zeidler .

###### Proposition II.1

Assume that $g_{n}\in l^{2}$, $\alpha,\beta,\delta,\mu\in\mathbb{R}$, and let
$u^{0}\in l^{2}$, arbitrary. There exists some $T^{*}(u^{0})>0$ such that the
integral equation (2.9) has a unique solution $u\in C^{1}([0,T],l^{2})$ for
all $0<T<T^{*}(u^{0})$. In addition, the following alternatives hold: Either
$T^{*}(u^{0})=\infty$ (global existence) or $T^{*}(u^{0})<\infty$ and
$\lim_{t\uparrow T^{*}(U^{0})}||U(t)||_{\ell^{2}}=\infty$ (collapse).
Furthermore the solution $u$ depends continuously on the initial condition
$u^{0}\in l^{2}$, with respect to the norm of $C([0,T],l^{2})$.

##### Global existence and asymptotic estimates.

Proposition II.1 for the extended DGL (2.1)-(2.2) establishes local existence
of solutions for its limits, the problems (1.1)-(1.2) and (1.3)-(1.4). For
global existence of solutions we start with the case of the L-DGL (1.1)-(1.2).

###### Lemma II.2

For the L-DGL (1.1) assume $\delta>1$ and $\beta>0$. For every $u(0)=u^{0}\in
l^{2}$, the system (1.1) possesses a unique global solution $u(t)$ on
$[0,\infty)$ belonging to $C^{1}([0,\infty),l^{2})$.

Proof: For the L-DGL (1) we multiply the equation by $\overline{u}_{n}$,
summing over $\mathbb{Z}$ keeping the real parts. The result is the energy
balance equation:

$\displaystyle\frac{d}{dt}\sum_{n\in{\mathbb{Z}}}|u_{n}|^{2}+2(\delta-1)||u||_{l^{2}}^{2}+2\sum_{n\in\mathbb{Z}}|u_{n+1}-u_{n}|^{2}=-2\sum_{n\in{\mathbb{Z}}}|u_{n}|^{4}+2\mathrm{Re}\sum_{n\in\mathbb{Z}}g_{n}\overline{u}_{n}.$
(2.11)

Using the Cauchy-Schwarz inequality and Young’s inequality for the last term
on the right-hand side of (2.11)

$\bigg{|}\sum_{n\in\mathbb{Z}}g_{n}\overline{u}_{n}\bigg{|}\leq\frac{(\delta-1)}{2}||u||^{2}_{l^{2}}+\frac{1}{2(\delta-1)}||g||^{2}_{l^{2}},$
(2.12)

we obtain

$\frac{d}{dt}||u(t)||_{l^{2}}^{2}+(\delta-1)||u(t)||_{l^{2}}^{2}\leq\frac{1}{\delta-1}||g||_{l^{2}}^{2}.$
(2.13)

By the assumption that $\delta>1$ it follows that $u\in
L^{\infty}([0,\infty),l^{2})$. With the use of Gronwall’s inequality we
obtain:

$||u(t)||_{l^{2}}^{2}\leq||u(0)||_{l^{2}}^{2}\exp[-(\delta-1)t]+\frac{1}{(\delta-1)^{2}}||g||_{l^{2}}^{2}\left(1-\exp[-(\delta-1)t]\right).$
(2.14)

Asymptotically for $t\rightarrow\infty$, yields the following upper bound:

${\lim\sup}_{t\rightarrow\infty}||u(t)||_{l^{2}}^{2}\leq\frac{1}{(\delta-1)^{2}}||g||_{l^{2}}^{2},$
(2.15)

implying the global existence of solutions for For the L-DGL (1.1) $\square$

The parametric global existence regimes for the the NL-DGL (1.3) _i_ s much
richer than for the L-DGL (1.1), as it is shown in the lemma below.

###### Lemma II.3

For the NL-DGL (1.3) we assume that $\delta>1$, $\beta>0$, and consider the
following cases for $\beta$, $||g||_{l^{2}}$ and the initial data
$v(0)=v^{0}\in l^{2}$:

  1. 1.

Assume that $\delta$, $\beta$ and $||g||_{l^{2}}$ satisfy

$(\delta-1)^{3}<8\sqrt{1+\beta^{2}}||g||^{2}_{l^{2}}.$ (2.16)

Then, for any $v^{0}\in l^{2}$, the solutions exist globally in time and are
uniformly bounded, satisfying

$||v(t)||_{l^{2}}^{2}\leq||v^{0}||_{l^{2}}^{2}.$ (2.17)

  2. 2.

We consider the constants $B,R_{1},R_{2}>0$ defined explicitly by the
relations (2.24), (2.26) below. Assume that $\delta$, $\beta$ and
$||g||_{l^{2}}$ satisfy

$(\delta-1)^{3}>8\sqrt{1+\beta^{2}}||g||^{2}_{l^{2}},$ (2.18)

and that the initial data $v^{0}\in l^{2}$ satisfy

$\displaystyle||v^{0}||_{l^{2}}^{2}\leq R_{1}.$ (2.19)

Then, solutions exist globally for all $t>0$ and have the following asymptotic
behaviour:

$\displaystyle R_{2}<||v(t)||_{l^{2}}^{2}\leq||v^{0}||^{2}\leq
R_{1}\;\;\mbox{for
all}\;\;t>0\;\;\mbox{and}\lim_{t\rightarrow\infty}||v(t)||_{l^{2}}^{2}=R_{2}.$
(2.20)

Proof: We multiply equation (1.3) by $\overline{v}_{n}$, sum over
$\mathbb{Z}$, keeping the real parts, to get the following energy balance
equation:

$\frac{1}{2}\frac{d}{dt}||v||_{l^{2}}^{2}+(\delta-1)||v||_{l^{2}}^{2}+\sum_{n\in\mathbb{Z}}|v_{n+1}-v_{n}|^{2}=-\frac{1}{2}\mathrm{Re}(1+i\beta)\sum_{n\in\mathbb{Z}}|v_{n}|^{2}(v_{n+1}+v_{n})\overline{v}_{n}+\mathrm{Re}\sum_{n\in\mathbb{Z}}g_{n}\overline{v}_{n}.$
(2.21)

The right-hand side of (2.21) can be estimated from above, as follows:

$\bigg{|}\frac{1}{2}\mathrm{Re}(1+i\beta)\sum_{n\in\mathbb{Z}}|v_{n}|^{2}(v_{n+1}+v_{n})\overline{v}_{n}\bigg{|}+\bigg{|}\sum_{n\in\mathbb{Z}}g_{n}\overline{v}_{n}\bigg{|}\leq\sqrt{1+\beta^{2}}||v||^{4}_{l^{4}}+\frac{(\delta-1)}{2}||v||_{l^{2}}^{2}+\frac{1}{2(\delta-1)}||g||^{2}_{l^{2}}.$
(2.22)

Then, combining (2.21) and (2.22), we arrive at the differential inequality

$\frac{d}{dt}||v||_{l^{2}}^{2}+(\delta-1)||v||^{2}_{l^{2}}-2\sqrt{1+\beta^{2}}\big{(}||v||_{l^{2}}\big{)}^{2}-\frac{||g||^{2}_{l^{2}}}{\delta-1}\leq
0,$ (2.23)

which shall be explored further. For this purpose, we set
$\chi(t)=||v(t)||^{2}_{l^{2}}$, and

$A=(\delta-1),\;\;B=2\sqrt{1+\beta^{2}},\;\;C=\frac{||g||^{2}_{l^{2}}}{\delta-1},$
(2.24)

and (2.23) can be written as

$\dot{x}(t)+A\chi(t)-B\chi(t)^{2}-C\leq 0,\;\;\;A,B,C>0.$ (2.25)

For further treatment of (2.25) we distinguish between the following cases:

  1. 1.

$D=A^{2}-4BC<0$, which due to the definition of the constants $A,B,C$ in
(2.24) implies the condition (2.16). In this case, the quantity
$A\chi-B\chi^{2}-C>0$ for any $\chi\in\mathbb{R}$, and (2.25) becomes
$\dot{\chi}(t)\leq 0$, which gives the uniform bound (2.17).

  2. 2.

$D=A^{2}-4BC>0$ which gives the condition (2.18). Then the quadratic equation
$A\chi-B\chi^{2}-C=0$ has two positive roots

$\displaystyle
R_{1,2}=\frac{(\delta-1)\pm\sqrt{(\delta-1)^{2}-\frac{8\sqrt{1+\beta^{2}}||g||^{2}_{l^{2}}}{(\delta-1)}}}{4\sqrt{1+\beta^{2}}}=\frac{A\pm\sqrt{D}}{2B}.$
(2.26)

Let now

$\chi(t)=z(t)+R_{i},$ (2.27)

where $R_{i}$ is one of the roots $R_{1}$ or $R_{2}$. Since $R_{i}$ is a
solution of the quadratic equation $A\chi-B\chi^{2}-C=0$, the $R_{i}$ satisfy
inequality (2.25) as an equality, that is

$\dot{R}_{i}+AR_{i}-BR_{i}^{2}-C=0,$ (2.28)

($\dot{R}_{i}=0$). Then, subtraction of (2.28) from (2.25), yields that $z(t)$
satisfies the inequality (2.25). Thus, the change of variable defined by
(2.27) can be substituted into (2.25), and little algebra results in the
Bernoulli inequality for $z(t)$:

$\dot{z}(t)+(A-2BR_{i})z(t)-Bz(t)^{2}\leq 0.$ (2.29)

To derive a uniform in time upper bound for the solutions, we need that the
coefficient

$A-2BR_{i}>0\Longrightarrow R_{i}<\frac{A}{2B}.$ (2.30)

Clearly, (2.30) is satisfied only by the root $R_{2}$. Then, we set

$\displaystyle K=A-2BR_{2}=\sqrt{D}>0.$ (2.31)

Since (2.31) holds for $K$, we treat the Bernoulli differential inequality as
follows: We use the change of variable $\psi=z^{-1}$. As
$\dot{\psi}=-z^{-2}\dot{z}$, we have that $\dot{z}=-\dot{\psi}z^{2}$.
Substituting in (2.30), we carefully integrating the latter as follows:

$\displaystyle-\dot{\psi}z^{2}+Kz-Bz^{2}$ $\displaystyle\leq$ $\displaystyle
0\Rightarrow$ $\displaystyle\dot{\psi}-K\psi+B$ $\displaystyle\geq$
$\displaystyle 0\Rightarrow$ $\displaystyle\psi(t)$ $\displaystyle\geq$
$\displaystyle\bigg{(}\psi(0)-\frac{B}{K}\bigg{)}e^{Kt}+\frac{B}{K}.$ (2.32)

We may ensure positivity of $\psi(t)$ for all $t\geq 0$, and thus of $z(t)$.
This follows by assuming that

$\displaystyle\psi(0)\geq\frac{B}{K}\Longrightarrow
z(0)\leq\frac{K}{B}\Longrightarrow\chi(0)\leq\frac{K}{B}+R_{2}=\frac{\sqrt{D}}{B}+R_{2}=R_{1},$

giving the assumption on the initial data (2.19). With this assumption on the
initial data, we get from (II) (when inverting for $z$),

$z(t)\leq\bigg{[}\bigg{(}\psi(0)-\frac{B}{K}\bigg{)}e^{Kt}+\frac{B}{K}\bigg{]}^{-1}.$
(2.33)

Eventually from (2.33) we observe that

$\displaystyle 0<z(t),\;\;\mbox{for
all}\;\;t>0,\;\;\mbox{and}\lim_{t\rightarrow\infty}z(t)=0.$ (2.34)

We use (2.34) to conclude with the description of the dynamics. Since $z(t)>0$
for all $t>0$, we have by its definition (2.27), that
$\chi(t)=||v(t)||_{l^{2}}>R_{2}$. Thus, the solution belongs to the regime of
positivity of the quadratic $A\chi-B\chi^{2}-C$, for all $t>0$. With this
being the case, we derive again from (2.25) that $\dot{x}(t)\leq 0$, for all
$t>0$, implying the boundedness
$||v(t)||^{2}_{l^{2}}\leq||v(0)||^{2}_{l^{2}}$. Finally, the definition of $z$
implies the limit for $||v(t)||_{l^{2}}^{2}$. $\square$

In conclusion, for the L-DGL initial value problem (1.1)-(1.2) solutions exist
globally unconditionally with respect to the initial data, the forcing and the
parameters of the system: for any $u^{0}\in l^{2}$, the corresponding solution
$u(t)$ of (1.1)-(1.2) is bounded for all $t\in[0,\infty)$ and the solution
operator determined by

$S_{u}(t):u^{0}\in l^{2}\rightarrow u(t)=S_{u}(t)u^{0}\in l^{2},\,\,\,t\geq
0,$ (2.35)

forms a continuous semigroup $\left\\{S_{u}(t)\right\\}_{t\geq 0}$. This is
not the case for the solutions $v(t)$ of the initial-value problem of the NL-
DGL (1.3)-(1.4), as they exist globally for each of the in Lemma II.3
classified regimes for the parameters, the forcing and the initial data.
Nevertheless, in these regimes the solution operator of the NL-DGL problem
(1.3)-(1.4)

$S_{v}(t):v^{0}\in l^{2}\rightarrow v(t)=S_{v}(t)v^{0}\in l^{2},\,\,\,t\geq
0,$ (2.36)

generates a continuous semigroup $\left\\{S_{v}(t)\right\\}_{t\geq 0}$ on
$l^{2}$.

##  III Attractors and invariant sets

In this section we establish the existence of restricted attractors, global
attractors and other invariant sets for the dynamical systems associated to
the discrete Ginzburg-Landau equations (1.1) and (1.3).

###  III.1 Existence of a restricted attractor for the semigroup
$\left\\{S_{v}(t)\right\\}_{t\geq 0}$ in $l^{2}$

This section is devoted to the study of the semigroup
$\left\\{S_{v}(t)\right\\}_{t\geq 0}$ of the NL-DGL system (1.3)-(1.4). The
richer scenarios for invariant sets of the semigroup
$\left\\{S_{v}(t)\right\\}_{t\geq 0}$ of the NL-DGL system is already
indicated by Lemma II.3. First, we discuss the implications of the above
possibilities on the domain of attraction and then we pay attention to the
existence of a restricted attractor.

####  III.1.1 Invariant sets in $l^{2}$ for the NL-DGL (1.3)-(1.4)

There exist at least two possibilities for attracting balls capturing the
orbits of the semigroup $\left\\{S_{v}(t)\right\\}_{t\geq 0}$ of the NL-DGL
system (1.3)-(1.4), as well as the possibility of an invariant annulus. These
scenarios correspond to distinct parametric regimes and have distinct domain
of attractions as explained in the following lemma.

---
Figure 1: The possibilities of Lemma III.1. Case 1: The existence of a non-
escaping ball $\mathcal{B}(0,R_{0})$ with
$R_{0}(\delta,\beta,||g||_{l^{2}},R)$ for the orbit $S_{v}(t)v^{0}$ of the
initial condition $v^{0}\in\overline{\mathcal{B}}(0,R)$. Case 2: The existence
of a restricted absorbing ball $\mathcal{B}(0,\tilde{\rho})$ with a domain of
attraction the closed ball $\overline{\mathcal{B}}(0,R_{v})$. Case 3: The
existence of the invariant annulus $\mathcal{A}_{1,2}$. For any initial
condition $R_{2}<||v^{0}||_{l^{2}}^{2}\leq R_{1}$, the orbit
$v(t)=S_{v}(t)v^{0}$ satisfies $R_{2}<||v(t)||_{l^{2}}^{2}\leq R_{1}$, and
$lim_{t\rightarrow\infty}||v(t)||_{l^{2}}^{2}=R_{2}$.

###### Lemma III.1

For the NL-DGL (1.3) we assume that $\delta>1$, $\beta>0$ and consider the
following cases for $\beta$, $||g||_{l^{2}}$ and the initial data
$v(0)=v^{0}\in l^{2}$:

  1. 1.

(Non-escaping ball). Assume that $\delta$, $\beta$ and $||g||_{l^{2}}$ are in
the regime described by the condition (2.16) and that the initial data $v^{0}$
are in a closed ball $\overline{\mathcal{B}}(0,R)$ for some $R>0$. Then, there
exists a closed ball $\overline{\mathcal{B}}(0,R_{0})\subset l^{2}$ with
$R_{0}(\delta,\beta,||g||_{l^{2}},R)$ for the semigroup
$\left\\{S_{v}(t)\right\\}_{t\geq 0}$ of the system (1.3)-(1.4), with the
property $\overline{\mathcal{B}}(0,R)\subset\overline{\mathcal{B}}(0,R_{0})$,
and

$\displaystyle
S_{v}(t)\overline{\mathcal{B}}(0,R)\subset\overline{\mathcal{B}}(0,R_{0})\;\;\mbox{for
all}\;\;t\geq 0.$ (3.1)

  2. 2.

(Absorbing ball for restricted initial data). Assume that $\delta$, $\beta$
and $||g||_{l^{2}}$ are in the regime described by the condition (2.16) and
that the initial data $v^{0}$ further satisfy

$||v^{0}||_{l^{2}}^{2}\leq R_{v,0}^{2}:=\frac{\delta-1}{2\sqrt{1+\beta^{2}}},$
(3.2)

that is
$v^{0}\in\mathcal{B}_{R_{v,0}}:=\overline{\mathcal{B}}(0,{R_{v,0}})\subset
l^{2}$. Then, for the semigroup $\left\\{S_{v}(t)\right\\}_{t\geq 0}$ there
exists a bounded absorbing set $\mathcal{B}_{v,0}$ in $l^{2}$ for the above
restricted initial data, viz. for every bounded set
$\mathcal{B}\subset\overline{\mathcal{B}}(0,{R_{v,0}})$ of $l^{2}$, there is a
$t_{0}(\mathcal{B},\mathcal{B}_{v,0})$ s.t.
$S_{v}(t)\mathcal{B}\subset\mathcal{B}_{v,0}\,\,\,\forall t\geq
t_{0}(\mathcal{B},\mathcal{B}_{v,0})$.

  3. 3.

(Positively invariant annulus in $l^{2}$). Assume that $\delta$, $\beta$ and
$||g||_{l^{2}}$ are in the regime described by the condition (2.18). Then the
annulus

$\displaystyle\mathcal{A}_{1,2}=\bigg{\\{}v\in
l^{2}\;\;:\;\;R_{2}<||v||_{l^{2}}^{2}\leq R_{1}\bigg{\\}},$ (3.3)

is positively invariant for the semigroup $\left\\{S_{v}(t)\right\\}_{t\geq
0}$, viz. for every bounded set $\mathcal{B}\subset\mathcal{A}_{1,2}$ it holds
that $S_{v}(t)\mathcal{B}\subset\mathcal{A}_{1,2}$ for all $t\geq 0$. Moreover
the limit $\lim_{t\rightarrow\infty}S_{v}(t)v^{0}$ exists for any
$v^{0}\in\mathcal{A}_{1,2}$, and
$\lim_{t\rightarrow\infty}S_{v}(t)v^{0}=R_{2}$.

Proof: 1\. (Non-escaping ball): We turn again to the energy equation (2.21).
By assumption, $||v^{0}||^{2}\leq R^{2}$. Then the right-hand side can be
estimated as follows:

$\begin{split}\bigg{|}\mathrm{Re}(1+i\beta)|\sum_{n\in\mathbb{Z}}|v_{n}|^{2}(v_{n+1}+v_{n})\overline{v}_{n}\bigg{|}&+2\bigg{|}\mathrm{Re}\sum_{n\in\mathbb{Z}}g_{n}\overline{v}_{n}\bigg{|}\\\
&\leq
2\sqrt{1+\beta^{2}}||v||^{4}_{l^{4}}+(\delta-1)||v||^{2}_{l^{2}}+\frac{1}{\delta-1}||g||_{l^{2}}^{2}\\\
&\leq
2\sqrt{1+\beta^{2}}||v^{0}||^{4}_{l^{2}}+(\delta-1)||v||^{2}_{l^{2}}+\frac{1}{\delta-1}||g||_{l^{2}}^{2}.\end{split}$
(3.4)

We set

$\begin{split}\rho_{0}&=2\sqrt{1+\beta^{2}}||v^{0}||^{4}_{l^{2}}+\frac{1}{\delta-1}||g||_{l^{2}}^{2},\\\
\rho_{0}&\leq
2\sqrt{1+\beta^{2}}R^{4}+\frac{1}{\delta-1}||g||_{l^{2}}^{2}:=\rho_{1}(\beta,\delta,||g||_{l^{2}},R).\end{split}$
(3.5)

Then, from (3.4) and (3.5), the accordingly modified differential inequality
obtained from equation (2.21) implies the estimate

$\displaystyle||v(t)||_{l^{2}}^{2}\leq||v^{0}||_{l^{2}}^{2}\exp[-(\delta-1)t]+\frac{\rho_{0}}{\delta-1}||g||_{l^{2}}^{2}\left(1-\exp[-(\delta-1)t]\right).$
(3.6)

Letting $t\rightarrow\infty$ in (3.6), we get

${\lim\sup}_{t\rightarrow\infty}||v(t)||_{l^{2}}^{2}\leq\frac{\rho_{0}}{\delta-1}||g||_{l^{2}}^{2}\leq\frac{\rho_{1}}{\delta-1}||g||_{l^{2}}^{2}:=R_{0}^{2},$
(3.7)

where $R_{0}$ is determined by the dependence of $\rho_{1}$ on the parameters
$\beta$, $\delta$ and the initial radius $R$. Any orbit starting from the
closed ball $\overline{\mathcal{B}}(0,R)$ cannot escape from the closed ball
$\overline{\mathcal{B}}(0,R_{0})$ and each $\overline{\mathcal{B}}(0,R)$
defines a non-escaping ball $\overline{\mathcal{B}}(0,R_{0})$. The latter is
“trivially attracting” since (3.1) holds, and the time of entry is $t_{0}=0$.
2\. (Absorbing ball for restricted data): The existence of the restricted
absorbing ball follows by an alternative manipulation of the energy equation
(2.21) and the estimate (3.4), when we focus on the damping term on the right-
hand side of the former. More precisely, instead of (3.4) we may have the
following estimate:

$\begin{split}\bigg{|}\mathrm{Re}(1+i\beta)|\sum_{n\in\mathbb{Z}}|v_{n}|^{2}(v_{n+1}+v_{n})\overline{v}_{n}\bigg{|}&+2\bigg{|}\mathrm{Re}\sum_{n\in\mathbb{Z}}g_{n}\overline{v}_{n}\bigg{|}\\\
&\leq
2\sqrt{1+\beta^{2}}||v||^{2}_{l^{\infty}}\sum_{n\in\mathbb{Z}}|v_{n}|^{2}+(\delta-1)||v||^{2}_{l^{2}}+\frac{1}{\delta-1}||g||_{l^{2}}^{2}\\\
&\leq
2\sqrt{1+\beta^{2}}||v^{0}||^{2}_{l^{2}}\sum_{n\in\mathbb{Z}}|v_{n}|^{2}+(\delta-1)||v||^{2}_{l^{2}}+\frac{1}{\delta-1}||g||_{l^{2}}^{2}.\end{split}$
(3.8)

Now we assume that

$\delta_{0}=(\delta-1)-2\sqrt{1+\beta^{2}}||v^{0}||^{2}_{l^{2}}>0,$

which is actually condition (3.2). Inserting the modified estimate (3.8) in
(2.21) we get the differential inequality

$\frac{d}{dt}||v||_{l^{2}}^{2}+\delta_{0}||v||_{l^{2}}^{2}\leq\frac{1}{\delta-1}||g||_{l^{2}}^{2},$

which implies the estimate

$||v(t)||_{l^{2}}^{2}\leq||v(0)||_{l^{2}}^{2}\exp[-\delta_{0}t]+\frac{1}{\delta_{0}(\delta-1)}||g||_{l^{2}}^{2}\left(1-\exp[-\delta_{0}t]\right).$

Letting $t\rightarrow\infty$, yields that

${\lim\sup}_{t\rightarrow\infty}||v(t)||_{l^{2}}^{2}\leq\frac{1}{\delta_{0}(\delta-1)}||g||_{l^{2}}^{2}=\rho^{2}.$
(3.9)

Now, due to (3.9), we observe that for any number $\tilde{\rho}$, satisfying
$\tilde{\rho}>\rho$, the ball $\mathcal{B}_{v,0}$ of $l^{2}$ centered at $0$
of radius $\tilde{\rho}$ is an absorbing set for the semigroup $S_{v}(t):$
That is, for a bounded set $\mathcal{B}\in l^{2}$ included in
$\overline{\mathcal{B}}(0,{R_{v,0}})$, it follows that, for $t\geq
t_{0}(\mathcal{B},\mathcal{B}_{v,0})$, where

$t_{0}=\frac{1}{\tilde{\delta}}\log\left(\frac{R^{2}-\rho^{2}}{\tilde{\rho}^{2}-\rho^{2}}\right),$
(3.10)

one has $||v(t)||_{l^{2}}^{2}\leq\tilde{\rho}^{2}$, i.e.
$S_{v}(t)\mathcal{B}\subset\mathcal{B}_{v,0}$.
3\. (Positively invariant annulus in $l^{2}$). The proof is given in Lemma
II.3, case 2. $\square$

###### Remark III.1

Regarding the scenarios of Lemma III.1, note that scenario 1 may include
scenario 2, since 1 holds under the most generic condition (2.17) on the
boundedness of the initial data and both scenarios hold under the same
conditions on the parameters (2.16). However, scenario 2 establishes under a
further smallness condition on the initial data, namely (3.2), the existence
of the restricted absorbing ball $\mathcal{B}(0,\tilde{\rho})$ with an
explicitly defined domain of attraction, the closed ball
$\overline{\mathcal{B}}(0,R_{v})$. On the other hand, scenario 3, is purely
distinct since it holds when the parametric condition (2.16) is violated, that
is, when (2.18) is satisfied and the initial data belong to the identified
annulus $\mathcal{A}_{1,2}$.

####  III.1.2 Asymptotic compactness of the semigroup
$\left\\{S_{v}(t)\right\\}_{t\geq 0}$

Here, we verify that the semigroup $\left\\{S_{v}(t)\right\\}_{t\geq 0}$
associated with the NL-DGL (1.3)-(1.4) possesses the asymptotic tail-end
property, if the hypotheses of Lemma III.1.2 for the existence of the
absorbing ball for the restricted initial data are fulfilled. We present an
alternative approach for the proof instead of the tails-estimates method (see
Bates ; Zhou0 ; Zhou ; Zhou1 ; Zhou3 ; Wang ; Zhou2 ; Nikos ; NAE2007 ; Du ).
The approach is based on a contradiction argument and suitable implementation
of the generalized comparison principle of ODE’s McNabb1986 . We recall that
the asymptotic tail property is essential in establishing the asymptotic
compactness of the semigroup.

###### Lemma III.2

Assume the regime for the parameters, forcing and initial data stated in Lemma
III.1.2, that is, conditions (2.16) and (3.2). Let
$(v_{n}(0))_{n\in{\mathbb{Z}}}=v^{0}\in\mathcal{B}$ a bounded set of $l^{2}$,
where $\mathcal{B}\subset\mathcal{B}_{R_{v}}$. Then, for any $\xi>0$, there
exist $T(\xi)$ and $M(\xi)$ such that the solution $v(t)$ of (1.3)-(1.4)
satisfies for all $t\geq T(\xi)$:

$\sum_{|n|>2K}|v_{n}(t)|^{2}\leq\xi\,\,\,\,{\rm for\,any\,\,\,\,}K>M(\xi).$
(3.11)

Proof: For a contradiction, let us suppose that this assertion is not true,
i.e. there is an $\epsilon_{0}>0$ and a subsequence
$(n_{k})_{k\in{\mathbb{Z}}}$ of $\mathbb{Z}$ s.t.

$\sum_{|k|>m}|v_{n_{k}}(t)|^{2}\geq\epsilon_{0},\,\,\,\forall t\geq
0,\,\,\,{\rm for\,\,any}\,\,m\in{\mathbb{N}}.$ (3.12)

Next, we elaborate the following argument based on the comparison principle
McNabb1986 : Consider the equation

$\dot{w}(t)+Aw(t)-Bw^{2}(t)-C=0,$ (3.13)

with $A,B,C$ determined in (2.24) and $w(t)\geq 0$ for all $t\geq 0$. Then, if
$w(0)\geq\chi(0)$, by the comparison principle the solution $\chi(t)$ of
(2.25) is bounded from above by the solution $w(t)$ of (3.13) for all $t\geq
0$. Performing a Laplace transformation ${\cal{L}}\\{w(t)\\}=\tilde{w}(s)$,
yields

$s\tilde{w}(s)-w(0)+A\tilde{w}(s)=\frac{C}{s}+BF(s),$ (3.14)

where $F(s)={\cal{L}}\\{w^{2}(t)\\}$. A rearrangement of the terms of (3.14),
gives the equation

$\tilde{w}(s)=\frac{1}{s+A}\left(w(0)+\frac{C}{s}+BF(s)\right).$ (3.15)

Applying the inverse Laplace transformation results in

$w(t)=w(0)\exp(-At)+\frac{C}{A}\left(1-\exp(-At)\right)+B\int_{0}^{t}\exp(-A\theta)w^{2}(t-\theta)d\theta.$
(3.16)

Hence, we obtain the upper bound

$\displaystyle\chi(t)$ $\displaystyle\leq$
$\displaystyle\chi(0)\exp(-At)+\frac{C}{A}\left(1-\exp(-At)\right)+B\int_{0}^{t}\exp(-A\theta)\chi^{2}(t-\theta)ds$
(3.17) $\displaystyle\leq$
$\displaystyle\chi(0)\exp(-At)+\frac{C}{A}\left(1-\exp(-At)\right)+B\max_{0\leq\theta\leq
t}\chi^{2}(\theta)\int_{0}^{t}\exp(-A\theta)d\theta$ $\displaystyle\leq$
$\displaystyle\chi(0)\exp(-At)+\frac{1}{A}(C+B\chi^{2}(0))(1-\exp(-At),$

where we used the relation (2.17). Asymptotically for $t\rightarrow\infty$,
this gives the upper bound:

$\chi(t)\leq\frac{1}{A}(C+B\chi^{2}(0)).$ (3.18)

Now, let us note that since the NL-DGL equation is satisfied by the solution
$v_{n}(t)$ for all $n\in\mathbb{Z}$, it is also satisfied by the subsequence
$v_{n_{k}}(t)$, for all $n_{k}\in\mathbb{Z}$. We multiply equation (1.3) by
$\overline{v}_{n_{k}}$, sum over $|k|>m$, keeping the real parts to get the
inequality

$\frac{d}{dt}\sum_{|k|>m}|v_{n_{k}}(t)|^{2}+(\delta-1)\sum_{|k|>m}|v_{n_{k}}(t)|^{2}-2\sqrt{1+\beta^{2}}\left(\sum_{|k|>m}|v_{n_{k}}(t)|^{2}\right)^{2}-\frac{\sum_{|k|>m}|g_{n_{k}}|^{2}}{\delta-1}\leq
0,$ (3.19)

which is a differential inequality of the form (2.25) with constants

$A=(\delta-1),\;\;B=2\sqrt{1+\beta^{2}},\;\;C=\frac{\sum_{|k|>m}|g_{n_{k}}|^{2}}{\delta-1}.$

Since inequality (3.18) is valid, one obtains

$\displaystyle\sum_{|k|>m}|v_{n_{k}}(t)|^{2}$ $\displaystyle\leq$
$\displaystyle\sum_{|k|>m}|v_{n_{k}}(0)|^{2}\exp(-(\delta-1)t)$ (3.20)
$\displaystyle+$
$\displaystyle\frac{1}{\delta-1}\left(\frac{\sum_{|k|>m}|g_{n_{k}}|^{2}}{\delta-1}+2\sqrt{1+\beta^{2}}\left(\sum_{|k|>m}|v_{n_{k}}(0)|^{2}\right)^{2}\right)\left(1-\exp(-(\delta-1)t)\right).$

¿From (3.12) and (3.20), we deduce that

$\sup_{t\in[0,\infty]}\sum_{|k|>m}|v_{n_{k}}(t)|^{2}=\frac{1}{(\delta-1)^{2}}\sum_{|k|>m}|g_{n_{k}}|^{2}+2\frac{\sqrt{1+\beta^{2}}}{\delta-1}\left(\sum_{|k|>m}|v_{n_{k}}(0)|^{2}\right)^{2}\geq\epsilon_{0},\,\,\,{\rm
for\,\,any}\,\,m\in{\mathbb{N}}.$ (3.21)

Since $g\in l^{2}$, by choosing $m$ sufficiently large, this is equivalent to

$\sum_{|k|>m}|v_{n_{k}}(0)|^{2}\geq C(\beta,\delta)\tilde{\epsilon}>0,$ (3.22)

where

$\tilde{\epsilon}=\left(\epsilon_{0}-\frac{1}{(\delta-1)^{2}}\sum_{|k|>m}|g_{n_{k}}|^{2}\right)^{1/2},\,\,\,\,\,C(\beta,\delta)=\left(\frac{\delta-1}{2\sqrt{1+\beta^{2}}}\right)^{1/2}.$
(3.23)

Thus, inequality (3.22) contradicts the fact that for $v^{0}\in l^{2}$ every
subsequence $(v_{n_{k}}(0))_{k\in{\mathbb{N}}}$ must converge. $\square$

We also recall for the convenience of the reader the notion of the
asymptotically compact semigroup.

###### Definition III.1

The semigroup $\left\\{S_{v}(t)\right\\}_{t\geq 0}$ is said to be
asymptotically compact in $l^{2}$ if, for any bounded
$\mathcal{B}\subset\mathcal{B}_{R_{v}}\subset l^{2}$, and any sequence
$\left\\{t_{n}\right\\}$, $\left\\{v_{n}\right\\}$ with $t_{n}\geq 0$,
$t_{n}\rightarrow\infty$ as $n\rightarrow\infty$, and $v_{n}\in\mathcal{B}$,
the sequence $\left\\{S_{v}(t_{n})v_{n}\right\\}$ is relatively compact in
$l^{2}$.

With Lemma III.2 in hand, we establish the asymptotic compactness of the
semigroup associated with the system (1.3)-(1.4) in the next proposition.

###### Proposition III.1

Under the same conditions of Lemma III.2 the semigroup
$\left\\{S_{v}(t)\right\\}_{t\geq 0}$ is asymptotically compact.

Proof: By contradiction: Suppose that a subsequence of
$v^{n}(t_{n})=S_{v}(t_{n})v^{0,n}\in\mathcal{B}$ diverges in $l^{2}$ as
$t_{n}\rightarrow\infty$ (equivalently, the sequence
$v^{n}(t_{n})=S_{v}(t_{n})v^{0,n}$ has not a convergent subsequence in $l^{2}$
as $t_{n}\rightarrow\infty$). Take any subsequence
$v^{n_{k}}(t_{n_{k}})=S_{v}(t_{n_{k}})v^{0,n_{k}}\in\mathcal{B}$,
$k\in{\mathbb{N}}$. Suppose
$||v^{n_{k}}(t_{n_{k}})||_{l^{2}}^{2}\rightarrow\infty$ as
$t_{n_{k}}\rightarrow\infty$, viz. $k\rightarrow\infty$.

Then for any $M\in{\mathbb{R}}>0$ there are only finitely many $k$ s.t.

$||v^{n_{k}}(t_{n_{k}})||_{l^{2}}^{2}\leq M.$ (3.24)

Denote all values of $k$ for which (3.24) is satisfied by $k_{1},...,k_{m}$.
Then, setting $N_{M}=\max\left\\{k_{1},...,k_{m}\right\\}+1$, for any
$k>N_{M}$ it holds that

$||v^{n_{k}}(t_{n_{k}})||_{l^{2}}^{2}=\sum_{i\in{\mathbb{Z}}}|v^{n_{k}}_{i}(t_{n_{k}})|^{2}>M,\,\,\,\forall
k>N_{M}.$ (3.25)

We split the infinite sum in (3.25) as

$\sum_{i\in{\mathbb{Z}}}|v^{n_{k}}_{i}(t_{n_{k}})|^{2}=\sum_{|i|>2L}|v^{n_{k}}_{i}(t_{n_{k}})|^{2}+\sum_{|i|\leq
2L}|v^{n_{k}}_{i}(t_{n_{k}})|^{2}>M,\,\,\,\forall k>N_{M},$ (3.26)

for any fixed $0<L<\infty$. For the finite sum, we get $\sum_{|i|\leq
2L}|v^{n_{k}}_{i}(t_{n_{k}})|^{2}<M_{L}$ with $M>M_{L}$, implying that

$\sum_{|i|\geq 2L}|v^{n_{k}}_{i}(t_{n_{k}})|^{2}>M-M_{L}>0,\,\,\,\forall
k>N_{M}.$ (3.27)

Since the relation (3.27) holds for all $t\geq t_{n_{k>N_{M}}}$, for all $M>0$
and any $L>0$, it contradicts the asymptotic tail end property of
$\left\\{S_{v}(t)\right\\}_{t\geq 0}$ as established by Lemma III.2.

$\square$

Finally, facilitating Proposition III.1 and Theorem 1.1 in Temam , we are now
able to state the main result of this section.

###### Theorem III.1

The semigroup $\left\\{S_{v}(t)\right\\}_{t\geq 0}$ associated with the NL-DGL
(1.3)-(1.4), possesses a unique restricted global attractor ${\cal
A}_{v}\subset\mathcal{B}_{v,0}\subset l^{2}$.

###  III.2 Existence of a global attractor for the semigroup
$\left\\{S_{u}(t)\right\\}_{t\geq 0}$

Here, we prove the existence of an absorbing set in $l^{2}$ for the dynamical
system belonging to the L-DGL (1.1)-(1.2) in the asymptotic regime
$t\rightarrow\infty$.

###### Lemma III.3

Assume that $\delta>1$, $\beta>0$ and let $(g_{n})_{n\in{\mathbb{Z}}}=g\in
l^{2}$ and $(u_{n}(0))_{n\in{\mathbb{Z}}}=u^{0}$. For the dynamical system
determined by (1.1)-(1.2)

$S_{u}(t):\,u^{0}\in l^{2}\rightarrow u(t)\in l^{2}$ (3.28)

there exists a bounded absorbing set $\mathcal{B}_{u,0}$ in $l^{2}$, viz. for
every bounded set $\mathcal{B}$ of $l^{2}$, there is a
$t_{0}(\mathcal{B},\mathcal{B}_{u,0})$ s.t.
$S_{u}(t)\mathcal{B}\subset\mathcal{B}_{u,0}\,\,\,\forall t\geq
t_{0}(\mathcal{B},\mathcal{B}_{u,0})$.

Proof: Defining

$\rho^{2}=\frac{1}{(\delta-1)^{2}}||g||_{l^{2}}^{2},$ (3.29)

we observe from Lemma II.2, that for any number $\tilde{\rho}>\rho$ the ball
$\mathcal{B}_{u,0}\in l^{2}$ centered at $0$ of radius $\tilde{\rho}$ is an
absorbing set for the semigroup $S_{u}(t).$ That is, if $\mathcal{B}$ is a
bounded set of $l^{2}$ included in a ball centered at $0$ of radius $R$, it
follows that for $t\geq t_{0}(\mathcal{B},\mathcal{B}_{u,0})$ where

$t_{0}=\frac{1}{\delta-1}\log\left(\frac{R^{2}-\rho^{2}}{\tilde{\rho}^{2}-\rho^{2}}\right),$
(3.30)

one has $||u(t)||_{l^{2}}^{2}\leq\tilde{\rho}^{2}$, viz.
$S_{u}(t)\mathcal{B}\subset\mathcal{B}_{u,0}$. $\square$

Concerning the asymptotic tail-end property, for the L-DGL (1.1)-(1.2) we have
the following Lemma.

###### Lemma III.4

Let $(u_{n}(0))_{n\in{\mathbb{Z}}}=u^{0}\in\mathcal{B}$, where $\mathcal{B}$
is a bounded set of $l^{2}$ and $(g_{n})_{n\in{\mathbb{Z}}}=g\in l^{2}$. For
any $\xi>0$ there exist $T(\xi)$ and $M(\xi)$ s.t. the solution $u(t)$ of
(1.1),(1.2) satisfies for all $t\geq T(\xi)$:

$\sum_{|n|\geq 2K}|u_{n}(t)|^{2}\leq\xi\,\,\,\,{\rm
for\,any\,\,\,\,}K>M(\xi).$ (3.31)

Proof: For a contradiction, let us suppose that this assertion is is not true,
i.e. there is an $\epsilon_{0}>0$ and a subsequence
$(n_{k})_{k\in{\mathbb{Z}}}$ of $\mathbb{Z}$ s.t.

$\sum_{|k|>m}|u_{n_{k}}(t)|^{2}\geq\epsilon_{0},\,\,\,\forall t\geq
0,\,\,\,{\rm for\,\,any}\,\,m\in{\mathbb{N}}.$ (3.32)

Working as for the derivation of (3.19), we get the differential inequality
for $u_{n_{k}}$,

$\frac{d}{dt}\sum_{|k|>m}|u_{n_{k}}(t)|^{2}+(\delta-1)\sum_{|k|>m}|u_{n_{k}}(t)|^{2}\leq\frac{1}{\delta-1}\sum_{|k|>m}|g_{n_{k}}|^{2},$
(3.33)

and Gronwall’s inequality gives

$\sum_{|k|>m}|u_{n_{k}}(t)|^{2}\leq\sum_{|k|>m}|u_{n_{k}}(0)|^{2}\exp(-(\delta-1)t)+\frac{\sum_{|k|>m}|g_{n_{k}}|^{2}}{(\delta-1)^{2}}\left(1-\exp(-(\delta-1)t)\right).$
(3.34)

Conclusively,

$\inf_{t\in[0,\infty]}\sum_{|k|>m}|u_{n_{k}}(t)|^{2}=\frac{\sum_{|k|>m}|g_{n_{k}}|^{2}}{(\delta-1)^{2}}\geq\epsilon_{0}>0,\,\,\,{\rm
for\,\,any}\,\,m\in{\mathbb{N}}.$ (3.35)

Therefore, for every $m\in{\mathbb{N}}$,
$\sum_{|k|>m}|g_{n_{k}}|^{2}>C(\delta)\epsilon_{0}>0$, which contradicts the
fact that for $(g_{n})_{n\in{\mathbb{Z}}}=g$ with $||g||_{l^{2}}<\infty$,
every subsequence $(g_{n_{k}})_{k\in{\mathbb{N}}}$ must converge.

$\square$

###### Proposition III.2

Under the same conditions of Lemma 3.31, the semigroup
$\left\\{S_{u}(t)\right\\}_{t\geq 0}$ is asymptotically compact.

The proof of Proposition III.2 proceeds in a similar manner to the
corresponding proof for the NL-DGL and is omitted here.

In conclusion, by virtue of Proposition III.1 and Theorem 1.1 in Temam , we
have:

###### Theorem III.2

The semigroup $\left\\{S_{u}(t)\right\\}_{t\geq 0}$ attributed to the L-DGL
(1.1)-(1.2) has a unique global attractor ${\cal
A}_{u}\subset\mathcal{B}_{u,0}\subset l^{2}$.

##  IV Closeness of the NL-DGL and L-DGL solutions and congruence of the
attractors

###  IV.1 Closeness of the solutions

In this section, we give the proof of Theorem I.1. The result gives in a sense
of “continuous dependence” the closeness of the solutions of (1.1) and (1.3)
for small enough initial data. A first application can be found in
establishing the persistence of small amplitude dissipative solitons for the
L-DGL (1.1), close to those identified for the NL-DGL (1.3), when the suitable
parametric regimes for the models described in AGL3 ; AGL4 ; UGL6  are
considered.

Proof of Theorem I.1: 1\. Case $\delta>1$. One derives for $t>0$, the
following differential inequality:

$\displaystyle\frac{d}{dt}||\Delta||_{l^{2}}^{2}$ $\displaystyle=$
$\displaystyle 2||\Delta||_{l^{2}}\frac{d}{dt}||\Delta||_{l^{2}}$ (4.1)
$\displaystyle=$
$\displaystyle\sum_{n\in{\mathbb{Z}}}\bigg{\\{}2(1-\delta)|\Delta_{n}|^{2}+[(1+i\alpha)(\Delta_{n+1}+\Delta_{n-1}-2\Delta_{n})\overline{\Delta}_{n}+c.c.]$
$\displaystyle-$
$\displaystyle[(1+i\beta)|u_{n}|^{2}u_{n}\overline{\Delta}_{n}+c.c.]$
$\displaystyle-$
$\displaystyle\left[\frac{1}{2}(1+i\beta)|v_{n}|^{2}(v_{n+1}+v_{n-1})\overline{\Delta}_{n}+c.c.\right]\bigg{\\}}$
$\displaystyle\leq$ $\displaystyle
2(1-\delta)||\Delta||_{l^{2}}^{2}+2(1+\beta)(\sup_{n\in{\mathbb{Z}}}|u_{n}|^{2}||u||_{l^{2}}+\sup_{n\in{\mathbb{Z}}}|v_{n}|^{2}||v||_{l^{2}})||\Delta||_{l^{2}}$
$\displaystyle\leq$ $\displaystyle
2(1-\delta)||\Delta||_{l^{2}}^{2}+2(1+\beta)(||u||_{l^{2}}^{3}+||v||_{l^{2}}^{3})||\Delta||_{l^{2}}$

For the derivation of (4.1), we exploited the continuous embeddings (2.6). For
$t>0$, it holds that

$\frac{d}{dt}||\Delta||_{l^{2}}+2(\delta-1)||\Delta||_{l^{2}}\leq
2(1+\beta)(R_{u}^{3}+R_{v}^{3}).$ (4.2)

Applying Gronwall’s inequality to (4.2), one gets

$||\Delta(t)||_{l^{2}}\leq||\Delta(0)||\exp[-2(\delta-1)t]+\frac{(1+\beta)}{\delta-1}(R^{3}_{u}+R^{3}_{v})\big{[}1-\exp[-2(\delta-1)t]\big{]}.$
(4.3)

From (4.3), we see that the estimate (1.8) is satisfied with the constant

$C=C_{0}+\bigg{[}\frac{(1+\beta)}{\delta-1}(C_{u,0}^{3}+C_{v,0}^{3})\bigg{]}.$
(4.4)

Taking the limit as $t\rightarrow\infty$, in (4.3), we deduce that

$\displaystyle\limsup_{t\rightarrow\infty}||\Delta(t)||_{l^{2}}\leq\bigg{[}\frac{(1+\beta)}{\delta-1}(C_{u,0}^{3}+C_{v,0}^{3})\bigg{]}\epsilon^{3}.$

justifying (1.9) with

$\displaystyle
C_{1}=\bigg{[}\frac{(1+\beta)}{\delta-1}(C_{u,0}^{3}+C_{v,0}^{3})\bigg{]}\epsilon^{3}.$
(4.5)

2\. Case $\delta\leq 1$. When $\delta=1$, we derive the claimed estimate from
(4.2), with the constant

$C_{2}=C_{0}+2(1+\beta)(C_{u,0}^{3}+C_{v,0}^{3})T_{f}.$

Furthermore, when $\delta<1$, we get from (4.2) the estimate

$\begin{split}||\Delta(t)||_{l^{2}}&\leq||\Delta(0)||\exp[2(1-\delta)t]+\frac{(1+\beta)}{1-\delta}(R^{3}_{u}+R^{3}_{v})\big{[}\exp[-2(1-\delta)t]-1\big{]},\end{split}$
(4.6)

which holds for any $t\in(0,\infty]$. Hence, for any $0<T_{f}<\infty$, by
setting this time

$C_{2}=C_{0}\exp[2(1-\delta)T_{f}]+\frac{(1+\beta)}{1-\delta}(C_{u,0}^{3}+C_{v,0}^{3})\big{[}\exp[2(1-\delta)T_{f}]-1\big{]},$
(4.7)

we conclude with the claimed estimate (1.10). $\square$

After having shown that the distance between the solutions of the NL-DGL and
the L-DGL measured in terms of the $l^{2}-$metric remains small (bounded above
by ${\cal{O}}(\epsilon^{3})$), compared to the $l^{2}-$norm of the solutions
themselves (bounded above by ${\cal{O}}(\epsilon$), an analogous feature
directly follows for the $l^{\infty}-$norm (sup norm) determining the maximal
distance between individual units, which is a consequence of the embeddings
(2.6). We have the following

###### Corollary IV.1

Assume the conditions of Theorem I.1. Then the closeness estimates (1.8)-(1.9)
for $\delta>1$, and (1.10) for $\delta\leq 1$, are satisfied by the
$l^{\infty}$-distance of solutions
$||\Delta(t)||_{l^{\infty}}=||u(t)-v(t)||_{l^{\infty}}$.

###  IV.2 Congruence of the attractors ${\cal A}_{v}$ and ${\cal A}_{u}$

A second application of Theorem I.1 is that it can be used in the dissipative
regime $\delta>1$ to establish the congruence of the attractors ${\cal A}_{u}$
and ${\cal A}_{v}$ associated with the dynamical systems $S_{u}(t)$ and
$S_{v}(t)$, that is the proof of Theorem 1.12.

Proof of Theorem 1.12: For any bounded subset $\mathcal{B}\subset l^{2}$, it
holds that

$\displaystyle{\rm dist}({\cal A}_{v},{\cal A}_{u})$ $\displaystyle\leq$
$\displaystyle{\rm dist}({\cal A}_{v},S_{v}(t)\mathcal{B})+{\rm
dist}(S_{v}(t)\mathcal{B},S_{u}(t)\mathcal{B})+{\rm
dist}(S_{u}(t)\mathcal{B},{\cal A}_{u}).$ (4.8)

As ${\cal A}_{v}$ attracts any bounded set
$\mathcal{B}\subseteq\mathcal{B}_{R_{v}}\subset l^{2}$, for any $\xi>0$, there
is some $T_{v}(\xi)>0$ s.t.

${\rm dist}\left({\cal A}_{v},S_{v}(t)\mathcal{B}\right)=\sup_{a\in{\cal
A}_{v}}\,\inf_{v^{0}\in\mathcal{B}}{\rm
dist}\left(a,S_{v}(t)v^{0}\right)_{l^{2}}<\frac{\xi}{3},\,\,\,\forall t\geq
T_{v}.$ (4.9)

Analogously, as ${\cal A}_{u}$ attracts any bounded set $\mathcal{B}\subset
l^{2}$ (${\cal A}_{u}$ actually attracts any bounded set in $l^{2}$ anyway),
for any $\xi>0$, there is some $T_{u}(\xi)>0$ s.t.

${\rm dist}\left(S_{u}(t)\mathcal{B},{\cal A}_{u}\right)={\rm dist}\left({\cal
A}_{u},S_{u}(t)\mathcal{B}\right)=\sup_{a\in{\cal
A}_{u}}\,\inf_{u^{0}\in\mathcal{B}}{\rm
dist}\left(a,S_{u}(t)u^{0}\right)_{l^{2}}<\frac{\xi}{3},\,\,\,\forall t\geq
T_{u}.$ (4.10)

Let $\overline{T}=\max\\{T_{v}(\xi),T_{u}(\xi)\\}$ and consider the time
interval $[0,t_{0}]$ with $t_{0}\geq\overline{T}$. In light of Theorem
I.1-case 1, for $\delta>1$, we have that for every $t_{0}>\overline{T}$, there
exists a constant $C>0$ s.t. for all
$v^{0},\,u^{0}\in\mathcal{B}\subseteq\mathcal{B}_{R_{v}}$, fulfilling
hypotheses (1.7), it holds that for every $t\in[0,t_{0}]$,

${\rm
dist}(S_{v}(t)\mathcal{B},S_{u}(t)\mathcal{B})=\inf_{v^{0}\in\mathcal{B}}{\rm
dist}_{l^{2}}\left(S_{v}(t)v^{0},S_{u}(t)v^{0}\right)<C{\epsilon^{3}},\,\,\,\forall\epsilon>0.$
(4.11)

Combining (4.9),(4.10) and (4.11) we get

$\displaystyle 0\leq{\rm dist}\left({\cal A}_{v},{\cal A}_{u}\right)$
$\displaystyle\leq$
$\displaystyle\frac{\xi}{3}+C\epsilon^{3}+\frac{\xi}{3}=\frac{2\xi}{3}+C\epsilon^{3},\,\,\,\forall\xi>0.$
(4.12)

By setting $\epsilon_{0}^{3}=\xi/(3C)$, one gets

${\rm dist}\left({\cal A}_{v},{\cal
A}_{u}\right)<\xi,\,\,\,\forall\epsilon\in[0,\epsilon_{0}],$ (4.13)

from which, by the arbitrariness of $\xi$ it follows that ${\rm
dist}\left({\cal A}_{v},{\cal A}_{u}\right)=0$, and the proof is finished.
$\square$

##  V Conclusions

While the integrable, non-local Ablowitz-Ladik (AL) lattice Ablowitz  is less
significant in physical applications than its local variant-the celebrated
Discrete Nonlinear Schrödinger equation (DNLS), its counterpart incorporating
linear and nonlinear damping (or even forcing), namely the non-local Discrete
Ginzburg-Landau equation, has proven to be more relevant in various physical
contexts particularly in nonlinear optics AchBook  and in the mathematical
description of arrays of vortices in fluids AGL1 -AGL2 . For this important
model, we have identified distinct scenarios for the asymptotic behavior of
its globally existing solutions, depending on the involved parameters and
external driving. Crucially, we have proven closeness of the solutions of the
non-local Discrete Ginzburg-Landau equation to the ones of its local
counterpart; the latter is another physically relevant nonlinear lattice. As a
consequence, the compact attractors governing the asymptotic dynamics coincide
when the initial data are selected within a suitable domain of attraction
defined by the restricted absorbing ball of the non-local system. Our results
on the closeness of globally existing solutions and the congruence of
attractors may inspire further studies in the context of dissipative localized
structures and their dynamics taking into account stochastic and other random
effects Stoch1 -Stoch4  or space and time discretisations GLord ,Han  relevant
for the numerical analysis of the (continuum) Ginzburg-Landau partial
differential equation. One direction of research shall be the investigation of
the inviscid limit Zhao  of the dissipative lattices (such as the discrete
GLEs considered in this paper) to the Hamiltonian ones (represented by the
integrable Ablowitz-Ladik lattice with its variety of analytical solutions)
motivated also by current closeness results for the conservative AL and DNLS
lattices DNJ2021 . Corresponding investigations are in progress and will be
reported in future publications.

## Acknowledgment

We would like to thank the referee for his/her constructive comments and
suggestions.

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