# Justification of the KP-II approximation
in dynamics of two-dimensional FPU systems

Nikolay Hristov  Department of Mathematics and Statistics, McMaster
University, Hamilton, Ontario, Canada, L8S 4K1  and  Dmitry E. Pelinovsky
Department of Mathematics and Statistics, McMaster University, Hamilton,
Ontario, Canada, L8S 4K1<EMAIL_ADDRESS>
###### Abstract.

Dynamics of the Fermi–Pasta–Ulam (FPU) system on a two-dimensional square
lattice is considered in the limit of small-amplitude long-scale waves with
slow transverse modulations. In the absence of transverse modulations,
dynamics of such waves, even at an oblique angle with respect to the square
lattice, is known to be described by the Korteweg–de Vries (KdV) equation. For
the three basic directions (horizontal, vertical, and diagonal), we prove that
the modulated waves are well described by the Kadomtsev–Petviashvili (KP-II)
equation. The result was expected long ago but proving rigorous bounds on the
approximation error turns out to be complicated due to the nonlocal terms of
the KP-II equation and the vector structure of the FPU systems on two-
dimensional lattices. We have obtained these error bounds by extending the
local well-posedness result for the KP-II equation in Sobolev spaces and by
controlling the error terms with energy estimates. The bounds are useful in
the analysis of transverse stability of solitary and periodic waves in two-
dimensional FPU systems due to many results available for the KP-II equation.

##  1\. Introduction

A Fermi-Pasta-Ulam (FPU) system is formed by particles connected to their
nearest neighbours by nonlinear springs. If the particles are organized in a
one-dimensional chain, we can label the position variable of the $j$-th
particle by $q_{j}$. If we denote the corresponding velocity variable by
$p_{j}$, then the FPU system is generated by the Hamiltonian function in the
form

$H(p,q)=\sum_{j\in\mathbb{Z}}\frac{1}{2}p_{j}^{2}+V(q_{j+1}-q_{j}),$ (1.1)

where $V(q)$ is the potential energy. A useful approximation to dynamics of
small-amplitude long-scale waves in the FPU chain with smooth $V$ satisfying
$V^{\prime}(0)=0$, $V^{\prime\prime}(0)>0$, and $V^{\prime\prime\prime}(0)\neq
0$ is given by the Korteweg–de Vries (KdV) equation, which is a remarkable
model due to integrability, stability of periodic and solitary waves, and
global existence of solutions in Sobolev spaces (see [10] for review).

Bounds on the approximation error between solutions of the FPU system and the
KdV equation were obtained by G. Schneider and C.E. Wayne in a conference
proceeding [30] as an exercise related to the justification technique the
authors had developed for the water wave problem [31]. The same approximation
appeared to be very useful in the context of stability of FPU solitary waves
and was studied comprehensively in the series of papers by G. Friesecke and R.
L. Pego [13, 14, 15, 16]. It was also justified in the context of normal forms
and KAM theory for metastability and recurrency of the FPU systems [2, 29].

More recently, the same approximation but with other versions of the KdV
equation was justified for FPU systems with Hertzian potentials [8] and with
pure anharmonic powers [25]. In [25], the KdV approximation was extended to
logarithmically long time scales provided the global dynamics of the
generalized KdV equation is well defined in Sobolev spaces of higher
regularity. The KdV approximation was also used in the context of periodic
waves and for FPU systems with nonlocal interactions [12, 23]. Higher-order
corrections to the KdV approximation were studied in [20], where it was shown
that the second-order corrections are spanned by two members of the integrable
KdV hierarchy whereas the third-order corrections can only be spanned by three
members of the KdV hierarchy under a constraint on parameters of the FPU
system. Review of results on nonlinear waves in one-dimensional FPU chains can
be found in [36].

The purpose of this work is to consider dynamics of the two-dimensional FPU
systems and to justify the two-dimensional KdV approximation given by the
Kadomtsev–Petviashvili (KP-II) equation. Similar to the KdV equation, the KP-
II equation is remarkable due to integrability [39], stability of periodic and
solitary waves [22, 27], and global existence of solutions in Sobolev spaces
[21, 28].

The first formal derivation of the KP-II equation was performed in [9] for the
scalar extension of the two-dimensional FPU systems with the total energy of
the form

$H=\sum_{(j,k)\in\mathbb{Z}^{2}}\frac{1}{2}p_{j,k}^{2}+\frac{1}{2}(q_{j+1,k}-q_{j,k})^{2}+\frac{1}{2}\varepsilon^{2}(q_{j,k+1}-q_{j,k})^{2}+\frac{1}{3}\alpha(q_{j+1,k}-q_{j,k})^{3},$
(1.2)

where $\varepsilon^{2}$ is a small parameter of slow transverse modulations in
the $k$-direction of the dominant wave propagating in the $j$ direction and
$\alpha$ is the parameter for the cubic interaction potential. A similar
scalar model was considered in the recent work [19], where the KP-II equation
in the periodic domain was rigorously justified (among other integrable
models) as the normal form for metastability phenomenon in two-dimensional
rectangular lattices. Existence of two-dimensional solitary waves [6] and
breathers (space-localized and time-periodic solutions) [37, 38] was also
studied in the scalar two-dimensional FPU lattices. Extensions of the scalar
two-dimensional FPU models that include harmonic interactions between the
first and second nearest neighbors were considered in [1, 24]. Applications of
the scalar FPU models to the two-dimensional square-packed granular arrays
were studied numerically and experimentally in [26], where propagation of a
modulated one-dimensional solitary wave was observed. Periodic triangular
lattices were compared with square lattices in the numerical study in [3, 4],
where it was shown that non-square lattices do not have integrable
approximations leading to the energy localization phenomenon.

The scalar problem can not be realized mechanically in the dynamics of
particles organized in a two-dimensional square lattice and connected by the
nonlinear springs. On the other hand, in the context of mechanical models of
the elasticity theory, the long-wave reductions were used to derive the one-
dimensional KdV equation (rather than the two-dimensional KP-II equation) from
the vector two-dimensional FPU systems.

A two-dimensional square FPU lattice was considered in [11] with additional
springs along the diagonals and a pair of potential functions, one for the
horizontal and vertical displacements and the other one for the diagonal
displacements. Existence of small-amplitude supersonic longitudinal solitary
waves propagating along the horizontal direction was proven in [11].
Surprisingly, the result holds even when the potential functions are
quadratic, due to the geometric nonlinearity of the lattice. Nonlinear
couplings were included in the consideration of the same model in [7], where
the solitary wave propagated under an arbitrary angle with respect to the
square lattice.

Another work can be found in [32], where propagation of rings in two-
dimensional lattices was analyzed in the linear approximation and compared
rigorously with the approximation of the linearized KdV equation (rather than
with the linearized KP-II equation).

Compared to the previous works, we prove validity of the KP-II approximation
for dynamics of transversely modulated small-amplitude long-scale waves in a
vector FPU system on a two-dimensional square lattice.

Next we describe the mechanical model considered here. Each particle at the
$(j,k)$ site of the two-dimensional square lattice is characterized by the
vectors of relative displacements $(x_{j,k},y_{j,k})$ and relative velocities
$(\dot{x}_{j,k},\dot{y}_{j,k})$. The nonlinear springs connecting the
particles are shown on Figure 1.

j-1,k-1j,k-1j+1,k-1j-1,kj,kj+1,kj-1,k+1j,k+1j+1,k+1 Figure 1. A mass–spring
mechanical system arranged in a square lattice

The potential energy of a single spring between two particles in the
horizontal direction is defined by

$V(r,s)=\frac{1}{2}(c_{1}^{2}r^{2}+c_{2}^{2}s^{2})+\frac{1}{3}\alpha_{1}r^{3}+\frac{1}{2}\alpha_{2}rs^{2},$

where $(r,s)$ are the relative displacements of the two particles in the
$(x,y)$ coordinates, $(c_{1}^{2},c_{2}^{2})$ are some coefficients of the
quadratic interaction potential, and $(\alpha_{1},\alpha_{2})$ are some
coefficients of the cubic interaction potential. The cubic terms are chosen in
such a way that the potential energy of the horizontal spring is symmetric
with respect to the sign of vertical relative displacements of the particles.
Due to the symmetry between horizontal and vertical springs, the total energy
of the two-dimensional FPU lattice takes the form

$\displaystyle H$
$\displaystyle=\frac{1}{2}\sum_{(j,k)\in\mathbb{Z}^{2}}\dot{x}_{j,k}^{2}+\dot{y}_{j,k}^{2}$
$\displaystyle+\sum_{(j,k)\in\mathbb{Z}^{2}}V(x_{j+1,k}-x_{j,k},y_{j+1,k}-y_{j,k})+V(y_{j,k+1}-y_{j,k},x_{j,k+1}-x_{j,k}).$
(1.3)

As is discussed in [17], the model with horizontal and vertical springs may
not capture all properties of elastic materials and diagonal springs are
required to describe structural stability of some materials. The KP-II
approximation in the square lattices with additional diagonal springs should
be possible, but computations of coefficients will become more complicated.

Next we present the main result for the propagation of nonlinear waves in the
horizontal direction. We will seek a continuous approximating function of the
form

$\displaystyle x_{j,k}=\varepsilon X\left(\xi,\eta,\tau\right)+\text{error},$
(1.4)

with $\xi=\varepsilon(j-c_{1}t)$, $\eta=\varepsilon^{2}k$, and
$\tau=\varepsilon^{3}t$. We will show that $x_{j,k}$ satisfy the equations of
motion with small error controllable in $\ell^{2}(\mathbb{Z}^{2})$ if
$X\left(\xi,\eta,\tau\right)$ solves the following KP-II equation

$\displaystyle
2c_{1}\partial_{\xi}\partial_{\tau}X+\frac{c_{1}^{2}}{12}\partial_{\xi}^{4}X+2\alpha_{1}\left(\partial_{\xi}X\right)\left(\partial^{2}_{\xi}X\right)+c_{2}^{2}\partial_{\eta}^{2}X=0.$
(1.5)

For justification analysis, it is more convenient to solve the FPU system in
strain variables,

$\displaystyle\left\\{\begin{array}[]{l}u_{j,k}^{(1)}:=x_{j+1,k}-x_{j,k},\\\
u_{j,k}^{(2)}:=x_{j,k+1}-x_{j,k},\\\ v_{j,k}^{(1)}:=y_{j+1,k}-y_{j,k},\\\
v_{j,k}^{(2)}:=y_{j,k+1}-y_{j,k},\end{array}\right.$ (1.10)

which are defined by the relative displacements between adjacent particles,
and introduce the amplitude function $A\left(\xi,\eta,\tau\right)$ in the form

$\displaystyle
x_{j+1,k}-x_{j,k}=\varepsilon^{2}A\left(\xi,\eta,\tau\right)+\text{error}.$
(1.11)

The reason for the different scaling in (1.4) and (1.11) is that we can
formally consider the relationship between the function $A$ and $X$ through a
Taylor expansion

$A\left(\xi,\eta,\tau\right)=\partial_{\xi}X\left(\xi,\eta,\tau\right)+\mathcal{O}(\varepsilon),$
(1.12)

so that the KP-II equation (1.5) can be rewritten in the form

$\displaystyle
2c_{1}\partial_{\xi}\partial_{\tau}A+\frac{c_{1}^{2}}{12}\partial_{\xi}^{4}A+2\alpha_{1}\partial_{\xi}\left(A\partial_{\xi}A\right)+c_{2}^{2}\partial_{\eta}^{2}A=0,$
(1.13)

Associated with a smooth solution to the KP–II equation (1.13) at a given time
$\tau$, for which both $A\in H^{s}\left(\mathbb{R}^{2}\right)$ and
$\partial_{\xi}^{-1}A\in H^{s}\left(\mathbb{R}^{2}\right)$ with sufficiently
large $s$, we define the anti-derivative by

$\partial_{\xi}^{-1}A(\xi,\eta):=\int_{-\infty}^{\xi}A(\xi^{\prime},\eta)d\xi^{\prime}.$

The solution to the KP-II equation (1.13) is required to have enough
regularity so that $\partial_{\xi}^{-1}\partial_{\tau}^{2}A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s}\left(\mathbb{R}^{2}\right)\right)$
and $\partial_{\xi}^{-2}\partial_{\eta}^{2}\partial_{\tau}A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s}\left(\mathbb{R}^{2}\right)\right)$
with $s\geq 3$. Existence of such solutions is proven in Lemma 1 (Section 2).
Since $A(\varepsilon(j-c_{1}t),\varepsilon^{2}k,\varepsilon^{3}t)$ is
estimated in the $\ell^{2}(\mathbb{Z}^{2})$ norm over
$(j,k)\in\mathbb{Z}^{2}$, we will also need the bound on the
$\ell^{2}(\mathbb{Z}^{2})$ norm of these terms by their
$H^{s}(\mathbb{R}^{2})$ norm, this bound is obtained in Lemma 2 (Section 2).

The following theorem formulates the main result which justifies the KP-II
approximation (1.13) for the horizontal propagation of nonlinear waves.

######  Theorem 1.

Let $A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+9}\left(\mathbb{R}^{2}\right)\right)$
be a solution to the KP-II equation $\eqref{KPII}$ with fixed $s\geq 0$, whose
initial data $A(\xi,\eta,0)=A_{0}$ satisfies

$A_{0}\in
H^{s+9}\left(\mathbb{R}^{2}\right),\qquad\partial_{\xi}^{-2}\partial_{\eta}^{2}A_{0}\in
H^{s+9}\left(\mathbb{R}^{2}\right)$

and

$\partial_{\xi}^{-1}\partial_{\eta}^{2}\left[\partial_{\xi}^{-2}\partial_{\eta}^{2}A_{0}+A_{0}^{2}\right]\in
H^{s+3}\left(\mathbb{R}^{2}\right).$

Then there are constants $C_{0},C_{1},\varepsilon_{0}>0$ such that for
$\varepsilon\in(0,\varepsilon_{0})$ if the initial conditions of the two-
dimensional FPU system satisfies

$\displaystyle\begin{split}&\left\lVert
u^{(1)}_{\text{in}}-\varepsilon^{2}A_{0}\right\rVert_{\ell^{2}}+\left\lVert
u^{(2)}_{\text{in}}\right\rVert_{\ell^{2}}+\left\lVert\dot{x}_{\text{in}}+\varepsilon^{2}c_{1}A_{0}\right\rVert_{\ell^{2}}\\\
&+\left\lVert v^{(1)}_{\text{in}}\right\rVert_{\ell^{2}}+\left\lVert
v^{(2)}_{\text{in}}\right\rVert_{\ell^{2}}+\left\lVert\dot{y}_{\text{in}}\right\rVert_{\ell^{2}}\leq
C_{0}\varepsilon^{\frac{5}{2}}\end{split}$ (1.14)

then the solution to the two-dimensional FPU system satisfies

$\displaystyle\begin{split}&\left\lVert
u^{(1)}(t)-\varepsilon^{2}A\right\rVert_{\ell^{2}}+\left\lVert
u^{(2)}(t)\right\rVert_{\ell^{2}}+\left\lVert\dot{x}(t)+\varepsilon^{2}c_{1}A\right\rVert_{\ell^{2}}\\\
&+\left\lVert v^{(1)}(t)\right\rVert_{\ell^{2}}+\left\lVert
v^{(2)}(t)\right\rVert_{\ell^{2}}+\left\lVert\dot{y}(t)\right\rVert_{\ell^{2}}\leq
C_{1}\varepsilon^{\frac{5}{2}},\end{split}$ (1.15)

for $t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$.

######  Remark 1.

Extending this result to $\varepsilon^{\frac{7}{2}}$ is difficult as the next
order of the asymptotic expansion has terms which are not removed by seeking
solutions to the KP-II equation alone. They could be removed by seeking a
function of the form

$A\left(\xi,\eta,\tau\right)=A^{(0)}\left(\xi,\eta,\tau\right)+\varepsilon^{2}A^{(1)}\left(\xi,\eta,\tau\right),$

where $A^{(0)}$ solves the KP-II equation (1.13) and $A^{(1)}$ solves an
appropriately chosen linearized KP-II equation. However the linearized KP-II
equation is nonhomogeneous, where the nonhomogeneous piece contains higher
order antiderivative terms of $A^{(0)}$.

######  Remark 2.

Compared to the work [9], the slow transverse modulations in the expansion
(1.11) are not due to the external small parameter in the potential energy
$V(r,s)$. If we have the external small parameter $\varepsilon$ as in (1.2),
we can use $\eta=\varepsilon k$ so that the comparison between the $\ell^{2}$
norm and the Sobolev norm would only lose $\varepsilon^{-1}$ compared to
$\varepsilon^{-\frac{3}{2}}$ in Lemma 2. Performing the justification analysis
on a version of the FPU system as in [9] should also yield Theorem 1 but with
the errors bounds of $\varepsilon^{3}$ rather than
$\varepsilon^{\frac{5}{2}}$.

######  Remark 3.

Theorem 1 can be proven in the setting of periodic transverse modulations, for
which $A(\xi,\eta+P,\tau)=A(\xi,\eta,\tau)$ with fixed $P>0$. In view of the
scaling $\eta=\varepsilon^{2}k$, this would correspond to the periodic
transverse modulations with large, $\varepsilon$-dependent period
$P\varepsilon^{-2}$. Similarly, one can consider periodic waves of the KP-II
equation satisfying $A(\xi+L,\eta,\tau)=A(\xi,\eta,\tau)$, for which the
periodic waves of the FPU lattice has large, $\varepsilon$-dependent period
$L\varepsilon^{-1}$. The recent work [19] justifies the KP-II approximation
for periodic solutions in both spatial directions as the normal form for
metastable dynamics of the rectangular FPU system.

######  Remark 4.

By the obvious symmetry, the result of Theorem 1 can be formulated for the
vertical propagation of nonlinear waves in the $y$-coordinates.

$x_{m-1,n}$$\chi_{m-1,n}$$x_{m,n+1}$$\chi_{m,n+1}$$\chi_{m-1,n-1}$$x_{m,n}$$\chi_{m,n}$$x_{m+1,n+1}$$x_{m,n-1}$$\chi_{m,n-1}$$x_{m+1,n}$$\chi_{m+1,n}$$\chi_{m,n-2}$$x_{m+1,n-1}$$\chi_{m+1,n-1}$$x_{m+2,n}$
Figure 2. A diatomic mass spring system arranged in a square lattice

Finally, we consider the diagonal propagation of nonlinear waves in the FPU
lattice shown on Figure 1 and introduce a new coordinate system by

$m=\frac{j+k}{2},\quad n=\frac{j-k}{2},\quad(j,k)\in\mathbb{Z}^{2}.$

In the new coordinate system the particle experiences nearest-neighbour
interactions with neighbours located a half lattice site away. Due to this we
redefine $x_{j,k}$ as $x_{m,n}$ and introduce
$\chi_{m,n}:=x_{m+\frac{1}{2},n+\frac{1}{2}}$. The FPU system becomes a
diatomic system where $x_{m,n}$ particles communicate with four $\chi_{m,n}$
nearest-neighbour particles and vice versa, see Figure 2 for an illustration.

We will seek a continuous approximating function of the form

$x_{m,n}=\varepsilon
X\left(\varepsilon(m-c^{*}_{1}t),\varepsilon^{2}(n-c_{2}^{*}t),\varepsilon^{3}t\right)+\text{error},$

where $c_{1}^{*}=\frac{1}{2}\sqrt{c_{1}^{2}+c_{2}^{2}}$ and
$c_{2}^{*}=\frac{1}{2}\sqrt{c_{1}^{2}-c_{2}^{2}}$. It is hard to control the
error in a general case because nonlocal terms related to the solution of the
KP-II equation appear in lower orders of the asymptotic approximation.
However, if $c_{2}=c_{1}$ and $\alpha_{2}=2\alpha_{1}$, then
$c_{1}^{*}=\frac{c_{1}}{\sqrt{2}}$, $c_{2}^{*}=0$ and furthermore, the FPU
system is satisfied by the invariant reduction $x_{j,k}=y_{j,k}$. By using the
strain variables,

$\displaystyle
x_{m+1,n}-x_{m,n}=\varepsilon^{2}A\left(\varepsilon(m-c^{*}_{1}t),\varepsilon^{2}n,\varepsilon^{3}t\right)+\text{error},$
(1.16)

which corresponds to the displacement along the main diagonal, we derive the
following KP-II equation for $A(\xi,\eta,\tau)$ with the same relation (1.12)
between $A$ and $\partial_{\xi}X$:

$\displaystyle
2c_{1}^{*}\partial_{\xi}\partial_{\tau}A+\frac{(c_{1}^{*})^{2}}{48}\partial_{\xi}^{4}A+\alpha_{1}\partial_{\xi}\left(A\partial_{\xi}A\right)+(c_{1}^{*})^{2}\partial_{\eta}^{2}A=0.$
(1.17)

where $\xi=\varepsilon(m-c_{1}^{*}t)$, $\eta=\varepsilon^{2}n$, and
$\tau=\varepsilon^{3}t$. Similarly to the case of horizontal propagation, we
can redefine the strain variables for the diagonal propagation as follows:

$\displaystyle\left\\{\begin{array}[]{l}u_{m,n}^{(1)}:=x_{m+1,n}-x_{m,n},\\\
u_{m,n}^{(2)}:=x_{m,n+1}-x_{m,n},\\\
v_{m,n}^{(1)}:=\chi_{m+1,n}-\chi_{m,n},\\\
v_{m,n}^{(2)}:=\chi_{m,n+1}-\chi_{m,n}.\end{array}\right.$ (1.22)

The following theorem is similar to Theorem 1 but it justifies the KP-II
approximation (1.17) for the diagonal propagation of nonlinear waves.

######  Theorem 2.

Let $A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+9}\left(\mathbb{R}^{2}\right)\right)$
be a solution to the KP-II equation $\eqref{KPIIdiag}$ with fixed $s\geq 0$,
whose initial data $A(\xi,\eta,0)=A_{0}$ satisfies

$A_{0}\in
H^{s+9}\left(\mathbb{R}^{2}\right),\qquad\partial_{\xi}^{-2}\partial_{\eta}^{2}A_{0}\in
H^{s+9}\left(\mathbb{R}^{2}\right)$

and

$\partial_{\xi}^{-1}\partial_{\eta}^{2}\left[\partial_{\xi}^{-2}\partial_{\eta}^{2}A_{0}+A_{0}^{2}\right]\in
H^{s+3}\left(\mathbb{R}^{2}\right).$

Then there are constants $C_{0},C_{1},\varepsilon_{0}>0$ such that for
$\varepsilon\in(0,\varepsilon_{0})$ if the initial conditions of the two-
dimensional FPU system with $c_{2}=c_{1}$ and $\alpha_{2}=2\alpha_{1}$
satisfies

$\displaystyle\begin{split}&\left\lVert
u^{(1)}_{\text{in}}-\varepsilon^{2}A_{0}\right\rVert_{\ell^{2}}+\left\lVert
u^{(2)}_{\text{in}}\right\rVert_{\ell^{2}}+\left\lVert\dot{x}_{\text{in}}+\varepsilon^{2}c_{1}^{*}A_{0}\right\rVert_{\ell^{2}}\\\
&+\left\lVert
v^{(1)}_{\text{in}}-\varepsilon^{2}A_{0}\right\rVert_{\ell^{2}}+\left\lVert
v^{(2)}_{\text{in}}\right\rVert_{\ell^{2}}+\left\lVert\dot{\chi}_{\text{in}}+\varepsilon^{2}c_{1}^{*}A_{0}\right\rVert_{\ell^{2}}\leq
C_{0}\varepsilon^{\frac{5}{2}}\end{split}$ (1.23)

then the solution to the two-dimensional FPU system satisfies

$\displaystyle\begin{split}&\left\lVert
u^{(1)}(t)-\varepsilon^{2}A\right\rVert_{\ell^{2}}+\left\lVert
u^{(2)}(t)\right\rVert_{\ell^{2}}+\left\lVert\dot{x}(t)+\varepsilon^{2}c_{1}^{*}A\right\rVert_{\ell^{2}}\\\
&+\left\lVert v^{(1)}(t)-\varepsilon^{2}A\right\rVert_{\ell^{2}}+\left\lVert
v^{(2)}(t)\right\rVert_{\ell^{2}}+\left\lVert\dot{\chi}(t)+\varepsilon^{2}c_{1}^{*}A\right\rVert_{\ell^{2}}\leq
C_{1}\varepsilon^{\frac{5}{2}},\end{split}$ (1.24)

for $t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$.

######  Remark 5.

It is an open problem to justify the KP-II approximation for diagonal
propagation with $c_{2}\neq c_{1}$ and $\alpha_{2}\neq 2\alpha_{1}$ or for
other directions along the lattice. The main challenge arises in controlling
in Sobolev norm of the nonlocal terms computed at solutions of the KP-II
equation. Even for Theorem 2, if we use another equivalent choice for the
asymptotic approximation, e.g.

$\displaystyle\chi_{m,n}-x_{m,n}=\frac{1}{2}\varepsilon^{2}A\left(\varepsilon(m-c^{*}_{1}t),\varepsilon^{2}n,\varepsilon^{3}t\right)+\text{error},$
(1.25)

the asymptotic expansions contain some non-local terms. Although these
nonlocal terms can be transformed away by near-identity transformations, the
choice of (1.16) allows us to avoid the nonlocal terms, see Section 4.1.

The remainder of the paper is organized as follows. Section 2 contains
preliminary results needed for the justification analysis. In particular, we
extend [18] to obtain additional estimates on solutions of the KP-II equation
and extend [8, 30] to control $\ell^{2}(\mathbb{Z}^{2})$ norm at the slowly
varying solution of the KP-II equation. Section 3 gives the proof of Theorem 1
after equation of motion are set up in the strain variables and the near-
identity transformations are performed to reduce the residual. The error terms
are controlled from the energy estimates and Gronwall’s inequality. Section 4
gives relevant details for the very similar proof of Theorem 2. Summary and
discussion of further questions are contained in the concluding Section 5.

##  2\. Preliminary results

Consider the Cauchy problem for the normalized KP-II equation

$\left\\{\begin{array}[]{l}\partial_{\tau}A+\partial_{\xi}\left(A^{2}\right)+\partial_{\xi}^{3}A+\partial_{\xi}^{-1}\partial_{\eta}^{2}A=0,\quad
t>0,\\\ A|_{t=0}=A_{0}.\end{array}\right.$ (2.1)

The normalized KP-II equation differs slightly from (1.13) and (1.17), in the
choice of constants. However, the constants in the KP-II equation can be
changed, as long as each constant is positive, through a scaling of its
variables.

Global well-posedness of the Cauchy problem (2.1) was established in
$H^{s}\left(\mathbb{T}^{2}\right)$ or $H^{s}\left(\mathbb{R}^{2}\right)$ with
any $s\geq 0$ in [5], provided that the initial data satisfies the constraint

$\int_{\mathbb{R}}A_{0}\left(\xi,\eta\right)d\xi=0,\quad\mbox{\rm for every
}\eta.$ (2.2)

The result was proven by combining local well-posedness and conservation laws,
namely conservation of the $L^{2}$ norm along the solution.

Local well-posedness result was extended in [34] to Sobolev spaces of the type
$H^{s_{1},s_{2}}\left(\mathbb{R}^{2}\right)$, with $s_{1}>-\frac{1}{4}$ and
$s_{2}\geq 0$, the global result was also obtained by the conservation laws
provided that $s_{1}\geq 0$. It was further shown in [33] that the zero mean
constraint (2.2) can be dropped in the local-wellposedness result.

We will use the following local well-posedness result [18] (see [35] for
earlier work).

######  Proposition 1.

[18] For any $A_{0}\in H^{s+6}\left(\mathbb{R}^{2}\right)$ such that
$\partial_{\xi}^{-2}\partial_{\eta}^{2}A_{0}\in
H^{s+3}\left(\mathbb{R}^{2}\right)$ with fixed $s\geq 0$, there exists
$\tau_{0}>0$ such that the Cauchy problem (2.1) admits a unique solution

$\displaystyle A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+6}\right)\cap
C^{1}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+3}\right)\cap
C^{2}\left(\left[-\tau_{0},\tau_{0}\right],H^{s}\right)$

such that $\partial_{\xi}^{-1}\partial_{\eta}A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+5}\right)\cap
C^{1}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+2}\right)$.

However, for our work, we need to extend this result to
$C^{3}\left(\left[-\tau_{0},\tau_{0}\right],H^{s}\right)$ since we need the
antiderivative term $\partial_{\xi}^{-1}A$ as a function of $\tau$ to be twice
continuously differentiable in some Sobolev space. The following lemma
presents the corresponding extension of Proposition 1.

######  Lemma 1.

For any $A_{0}\in H^{s+9}\left(\mathbb{R}^{2}\right)$ such that
$\partial_{\xi}^{-2}\partial_{\eta}^{2}A_{0}\in
H^{s+6}\left(\mathbb{R}^{2}\right)$ and

$\partial_{\xi}^{-1}\partial_{\eta}^{2}\left[\partial_{\xi}^{-2}\partial_{\eta}^{2}A_{0}+A_{0}^{2}\right]\in
H^{s+3}\left(\mathbb{R}^{2}\right)$

with fixed $s\geq 0$, there exists $\tau_{0}>0$ such that the Cauchy problem
(2.1) admits a unique solution

$A\in C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+9}\right)\cap
C^{1}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+6}\right)\cap
C^{2}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+3}\right)\cap
C^{3}\left(\left[-\tau_{0},\tau_{0}\right],H^{s}\right)$

such that

$\partial_{\xi}^{-1}\partial_{\eta}A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+8}\right)\cap
C^{1}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+5}\right)\cap
C^{2}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+2}\right)$

and

$\partial_{\xi}^{-2}\partial_{\eta}^{2}A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+6}\right)\cap
C^{1}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+3}\right).$

###### Proof.

Assume that $A$ solves the Cauchy problem (2.1) and set
$D:=\partial_{\xi}^{-2}\partial_{\eta}^{2}A$. By Proposition 1, since the
initial data satisfies $A_{0}\in H^{s+9}\left(\mathbb{R}^{2}\right)$ and
$\partial_{\xi}^{-2}\partial_{\eta}^{2}A_{0}\in
H^{s+6}\left(\mathbb{R}^{2}\right)$ with $s\geq 0$, the KP-II equation has a
solution

$A\in C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+9}\right)\cap
C^{1}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+6}\right)\cap
C^{2}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+3}\right)$ (2.3)

such that $\partial_{\xi}^{-1}\partial_{\eta}A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+8}\right)\cap
C^{1}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+5}\right)$. Taking
$\partial_{\eta}^{2}\left(\cdot\right)$ of the KP-II equation yields

$\partial_{\xi}^{2}\partial_{\tau}D+\partial_{\xi}\partial_{\eta}^{2}\left(A^{2}\right)+\partial_{\xi}^{5}D+\partial_{\xi}\partial_{\eta}^{2}D=0.$

Setting $\tilde{D}:=D+A^{2}$ and taking
$\partial_{\xi}^{-2}\left(\cdot\right)$ yields the evolution equation

$\partial_{\tau}\tilde{D}+\partial_{\xi}^{3}\tilde{D}+\partial_{\xi}^{-1}\partial_{\eta}^{2}\tilde{D}=\partial_{\tau}\left(A^{2}\right)+\partial_{\xi}^{3}\left(A^{2}\right).$
(2.4)

Let us define
$\Omega:=\partial_{\xi}^{-1}\partial_{\eta}^{2}+\partial_{\xi}^{3}$ and
$S(\tau):=e^{\tau\Omega}$. Since $\Omega$ is skew-adjoint, the evolution
operator $S\left(\tau\right)$ is unitary in $L^{2}(\mathbb{R}^{2})$. Using
Duhamel’s principle we can write the evolution equation (2.4) in the integral
form:

$\tilde{D}\left(\tau\right)=S(\tau)\tilde{D}_{0}+\int_{0}^{\tau}S(\tau-s)\left[\partial_{s}\left(A^{2}\right)+\partial_{\xi}^{3}\left(A^{2}\right)\right](s)ds,$

where $\tilde{D}_{0}=\partial_{\xi}^{-2}\partial_{\eta}^{2}A_{0}+A_{0}^{2}$.
Expressing back $D=\tilde{D}-A$ gives us an integral equation for $D(\tau)$ of
the form

$D\left(\tau\right)=S(\tau)\tilde{D}_{0}-A(\tau)^{2}+\int_{0}^{\tau}S(\tau-s)\left[\partial_{s}\left(A^{2}\right)+\partial_{\xi}^{3}\left(A^{2}\right)\right](s)ds.$
(2.5)

Since $\tilde{D}_{0}\in H^{s+6}(\mathbb{R}^{2})$ and $S(\tau)$ is unitary in
$L^{2}(\mathbb{R}^{2})$, the local solution (2.3) yields $D\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+6}\right)$.

Taking the time derivative of the integral equation (2.5), using

$\frac{d}{d\tau}S(\tau-s)=-\frac{d}{ds}S(\tau-s),$

and integrating by parts, we obtain

$\displaystyle\partial_{\tau}D(\tau)=$ $\displaystyle
S(\tau)\left[\Omega\tilde{D}_{0}+2A_{0}\partial_{\tau}A_{0}+\partial_{\xi}^{3}\left(A^{2}_{0}\right)\right]-2A(\tau)\partial_{\tau}A(\tau)$
$\displaystyle+\int_{0}^{\tau}S\left(\tau-s\right)\partial_{s}\left[\partial_{s}\left(A^{2}\right)+\partial_{\xi}^{3}\left(A^{2}\right)\right](s)ds.$

Since $\Omega\tilde{D}_{0}\in H^{s+3}(\mathbb{R}^{2})$, then the solution
(2.3) satisfies $\partial_{\tau}D\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+3}\right)$, which gives
$\partial_{\xi}^{-2}\partial_{\eta}^{2}A\in
C^{1}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+3}\right)$.

It remains to control $\partial_{\tau}^{3}A$ and
$\partial_{\xi}^{-1}\partial_{\eta}\partial_{\tau}^{2}A$, which is achieved by
computing the time derivatives of the KP-II equation:

$\displaystyle\partial_{\tau}^{2}A$
$\displaystyle=-\left(2\partial_{\xi}\left(A\partial_{\tau}A\right)+\partial_{\xi}^{3}\partial_{\tau}A+\partial_{\xi}^{-1}\partial_{\eta}^{2}\partial_{\tau}A\right),$
$\displaystyle\partial_{\tau}^{3}A$
$\displaystyle=-\left(2\partial_{\xi}\left(\left(\partial_{\tau}A\right)^{2}+A\partial_{\tau}^{2}A\right)+\partial_{\xi}^{3}\partial_{\tau}^{2}A+\partial_{\xi}^{-1}\partial_{\eta}^{2}\partial_{\tau}^{2}A\right)$

It follows from (2.3) that all but the last term in $\partial_{\tau}^{3}A$ are
in $C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s}\right)$. By using the
expression for $\partial_{\tau}^{2}A$, we check that

$\displaystyle\partial_{\xi}^{-1}\partial_{\tau}^{2}A=-\left(2A\partial_{\tau}A+\partial_{\xi}^{2}\partial_{\tau}A+\partial_{\tau}D\right),$
(2.6)

so that $\partial_{\xi}^{-1}\partial_{\tau}^{2}A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+3}\right)$, which yields $A\in
C^{3}\left(\left[-\tau_{0},\tau_{0}\right],H^{s}\right)$ and
$\partial_{\xi}^{-1}\partial_{\eta}A\in
C^{2}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+2}\right)$. ∎

Next we derive a useful bound on the $\ell^{2}(\mathbb{Z}^{2})$ norm of a
function expressed in terms of the slowly varying solution of the KP-II
equation defined in $H^{s}(\mathbb{R}^{2})$. A similar result for one-
dimensional chains was obtained in [8] (see also [30] for earlier work).

######  Proposition 2.

[8] Let $u_{j}=U(\varepsilon j)$, where $U\in H^{1}(\mathbb{R})$. There is a
constant $C>0$ such that for every $\varepsilon\in(0,1]$ we have

$\left\lVert u\right\rVert_{\ell^{2}(\mathbb{Z})}\leq
C\varepsilon^{-\frac{1}{2}}\left\lVert
U\right\rVert_{H^{1}(\mathbb{R})},\qquad\forall U\in
H^{1}\left(\mathbb{R}\right).$ (2.7)

The following lemma generalizes Proposition 2.7 for two-dimensional square
lattices.

######  Lemma 2.

Let $u_{j,k}=U(\varepsilon j,\varepsilon^{2}k)$, where $U\in
H^{s}(\mathbb{R}^{2})$ with fixed $s>1$. There is a constant $C_{s}>0$ that
depends on $s$ such that for every $\varepsilon\in(0,1]$ we have

$\left\lVert u\right\rVert_{\ell^{2}(\mathbb{Z}^{2})}\leq
C_{s}\varepsilon^{-\frac{3}{2}}\left\lVert
U\right\rVert_{H^{s}(\mathbb{R}^{2})},\qquad\forall U\in
H^{s}\left(\mathbb{R}^{2}\right).$ (2.8)

###### Proof.

We use the discrete Fourier transform defined by

$\hat{u}\left(\theta,\phi\right)=\displaystyle\sum_{(j,k)\in\mathbb{Z}^{2}}u_{j,k}e^{-i(j\theta+k\phi)}$

with the inverse transform given by

$u_{j,k}=\frac{1}{(2\pi)^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\hat{u}\left(\theta,\phi\right)e^{i(j\theta+k\phi)}d\theta
d\phi.$ (2.9)

Since $U\in H^{s}(\mathbb{R}^{2})$ and $s>1$, the Fourier transform of $U$
denoted by $\hat{U}$ can be defined through the standard formula. We represent

$\displaystyle\begin{split}u_{j,k}&=U(\varepsilon
j,\varepsilon^{2}k)=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\hat{U}(\tilde{p},\tilde{q})e^{i(\varepsilon
j\tilde{p}+\varepsilon^{2}k\tilde{q})}d\tilde{p}d\tilde{q}\\\
&=\frac{1}{(2\pi)^{2}\varepsilon^{3}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\hat{U}\left(\frac{p}{\varepsilon},\frac{q}{\varepsilon^{2}}\right)e^{i(jp+kq)}dpdq\\\
&=\frac{1}{(2\pi)^{2}\varepsilon^{3}}\sum_{(n,m)\in\mathbb{Z}^{2}}\int_{(2n-1)\pi}^{(2n+1)\pi}\int_{(2m-1)\pi}^{(2m+1)\pi}\hat{U}\left(\frac{p}{\varepsilon},\frac{q}{\varepsilon^{2}}\right)e^{i(jp+kq)}dpdq\\\
&=\frac{1}{(2\pi)^{2}\varepsilon^{3}}\sum_{(n,m)\in\mathbb{Z}^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\hat{U}\left(\frac{\theta+2\pi
m}{\varepsilon},\frac{\phi+2\pi
n}{\varepsilon^{2}}\right)e^{i(j\theta+k\phi)}d\theta d\phi.\end{split}$
(2.10)

For any finite subset $\Lambda\subset\mathbb{Z}^{2}$ we have

$\displaystyle\sum_{(n,m)\in\Lambda}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\left|\hat{U}\left(\frac{\theta+2\pi
m}{\varepsilon},\frac{\phi+2\pi n}{\varepsilon^{2}}\right)\right|d\theta
d\phi\leq\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left|\hat{U}\left(\frac{p}{\varepsilon},\frac{q}{\varepsilon^{2}}\right)\right|dpdq,$

hence

$\displaystyle\sum_{(n,m)\in\mathbb{Z}^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\left|\hat{U}\left(\frac{\theta+2\pi
m}{\varepsilon},\frac{\phi+2\pi n}{\varepsilon^{2}}\right)\right|d\theta
d\phi\leq\|\hat{U}\|_{L^{1}(\mathbb{R}^{2})},$

where $\|U\|_{L^{1}(\mathbb{R}^{2})}<\infty$ if $U\in H^{s}(\mathbb{R}^{2})$
with $s>1$. Then we can interchange summation and integration in (2.10) by the
Fubini-Tonelli theorem. Comparing (2.9) and (2.10) yields

$\displaystyle\hat{u}\left(\theta,\phi\right)=\frac{1}{\varepsilon^{3}}\sum_{(n,m)\in\mathbb{Z}^{2}}\hat{U}\left(\frac{\theta+2\pi
m}{\varepsilon},\frac{\phi+2\pi n}{\varepsilon^{2}}\right).$

Parseval’s identity yields

$\displaystyle\|u\|_{\ell^{2}(\mathbb{Z}^{2})}^{2}$
$\displaystyle=\frac{1}{(2\pi)^{2}\varepsilon^{6}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\left|\sum_{(n,m)\in\mathbb{Z}^{2}}\hat{U}\left(\frac{\theta+2\pi
m}{\varepsilon},\frac{\phi+2\pi n}{\varepsilon^{2}}\right)\right|^{2}d\theta
d\phi$
$\displaystyle\leq\frac{1}{(2\pi)^{2}\varepsilon^{6}}\sum_{\begin{subarray}{c}(n_{1},m_{1})\in\mathbb{Z}^{2}\\\
(n_{2},m_{2})\in\mathbb{Z}^{2}\end{subarray}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\left|\hat{U}\left(\frac{\theta+2\pi
m_{1}}{\varepsilon},\frac{\phi+2\pi n_{1}}{\varepsilon^{2}}\right)\right|$
$\displaystyle\phantom{\leq\frac{1}{(2\pi)^{2}\varepsilon^{6}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\sum_{\begin{subarray}{c}(n_{1},m_{1})\in\mathbb{Z}^{2}\\\
(n_{2},m_{2})\in\mathbb{Z}^{2}\end{subarray}}}\times\left|\hat{U}\left(\frac{\theta+2\pi
m_{2}}{\varepsilon},\frac{\phi+2\pi
n_{2}}{\varepsilon^{2}}\right)\right|d\theta d\phi.$

Denote $\left\langle x,y\right\rangle_{2}=\sqrt{1+x^{2}+y^{2}}$. Inserting the
weights $\left\langle\frac{\pi m_{1}}{\varepsilon},\frac{\pi
n_{1}}{\varepsilon^{2}}\right\rangle^{-2s}_{2}$ and $\left\langle\frac{\pi
m_{2}}{\varepsilon},\frac{\pi n_{2}}{\varepsilon^{2}}\right\rangle^{-2s}_{2}$,
then applying Young’s inequality, $ab\leq\frac{1}{2}a^{2}+\frac{1}{2}b^{2}$,
yields

$\displaystyle\|u\|_{\ell^{2}(\mathbb{Z}^{2})}^{2}$
$\displaystyle\leq\frac{1}{(2\pi)^{2}\varepsilon^{6}}\sum_{\begin{subarray}{c}(n_{1},m_{1})\in\mathbb{Z}^{2}\\\
(n_{2},m_{2})\in\mathbb{Z}^{2}\end{subarray}}\left\langle\frac{\pi
m_{1}}{\varepsilon},\frac{\pi
n_{1}}{\varepsilon^{2}}\right\rangle^{-2s}_{2}\left\langle\frac{\pi
m_{2}}{\varepsilon},\frac{\pi n_{2}}{\varepsilon^{2}}\right\rangle^{-2s}_{2}$
$\displaystyle\qquad\times\left(\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\right.\frac{1}{2}\left\langle\frac{\pi
m_{1}}{\varepsilon},\frac{\pi
n_{1}}{\varepsilon^{2}}\right\rangle^{4s}_{2}\left|\hat{U}\left(\frac{\theta+2\pi
m_{1}}{\varepsilon},\frac{\phi+2\pi
n_{1}}{\varepsilon^{2}}\right)\right|^{2}d\theta d\phi$
$\displaystyle\qquad\qquad+\left.\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{1}{2}\left\langle\frac{\pi
m_{2}}{\varepsilon},\frac{\pi
n_{2}}{\varepsilon^{2}}\right\rangle^{4s}_{2}\left|\hat{U}\left(\frac{\theta+2\pi
m_{2}}{\varepsilon},\frac{\phi+2\pi
n_{2}}{\varepsilon^{2}}\right)\right|^{2}d\theta d\phi\right).$

Hence, by symmetry of coefficients, we obtain:

$\displaystyle\|u\|_{\ell^{2}(\mathbb{Z}^{2})}^{2}\leq\frac{1}{(2\pi)^{2}\varepsilon^{6}}$
$\displaystyle\left(\sum_{(n_{1},m_{1})\in\mathbb{Z}^{2}}\left\langle\frac{\pi
m_{1}}{\varepsilon},\frac{\pi
n_{1}}{\varepsilon^{2}}\right\rangle^{-2s}_{2}\right)$ $\displaystyle\times$
$\displaystyle\left(\sum_{(n_{2},m_{2})\in\mathbb{Z}^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\left\langle\frac{\pi
m_{2}}{\varepsilon},\frac{\pi
n_{2}}{\varepsilon^{2}}\right\rangle^{2s}_{2}\left|\hat{U}\left(\frac{\theta+2\pi
m_{2}}{\varepsilon},\frac{\phi+2\pi
n_{2}}{\varepsilon^{2}}\right)\right|^{2}d\theta d\phi\right).$

When $\varepsilon\in(0,1]$ the double series in first term in $\left\lVert
u\right\rVert^{2}_{\ell^{2}(\mathbb{Z})}$ converges for $s>1$ by the integral
test, hence $\exists C_{s}>0$ so that

$\sum_{(n_{1},m_{1})\in\mathbb{Z}^{2}}\left\langle\frac{\pi
m_{1}}{\varepsilon},\frac{\pi
n_{1}}{\varepsilon^{2}}\right\rangle^{-2s}_{2}<C_{s}^{2}.$

The second term in $\left\lVert u\right\rVert^{2}_{\ell^{2}(\mathbb{Z})}$ is
related to the $H^{s}$ norm of $U$ given by,

$\displaystyle\|U\|_{H^{s}}^{2}$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left\langle\tilde{p},\tilde{q}\right\rangle^{2s}_{2}\left|\hat{U}(\tilde{p},\tilde{q})\right|^{2}d\tilde{p}d\tilde{q}$
$\displaystyle=\frac{1}{(2\pi)^{2}\varepsilon^{3}}\sum_{(n_{2},m_{2})\in\mathbb{Z}^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\left\langle\frac{\theta+2\pi
m_{2}}{\varepsilon},\frac{\phi+2\pi
n_{2}}{\varepsilon^{2}}\right\rangle^{2s}_{2}$
$\displaystyle\phantom{=\frac{1}{(2\pi)^{2}\varepsilon^{3}}\sum_{(n_{2},m_{2})\in\mathbb{Z}^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}}\times\left|\hat{U}\left(\frac{\theta+2\pi
m_{2}}{\varepsilon},\frac{\phi+2\pi
n_{2}}{\varepsilon^{2}}\right)\right|^{2}d\theta d\phi.$

Since

$\left\langle\frac{\pi m_{2}}{\varepsilon},\frac{\pi
n_{2}}{\varepsilon^{2}}\right\rangle^{2s}_{2}\leq\left\langle\frac{\theta+2\pi
m_{2}}{\varepsilon},\frac{\phi+2\pi
n_{2}}{\varepsilon^{2}}\right\rangle^{2s}_{2},\quad\forall\theta,\phi\in[-\pi,\pi],$

the second term in $\left\lVert u\right\rVert^{2}_{\ell^{2}(\mathbb{Z})}$ is
bounded above by $(2\pi)^{2}\varepsilon^{3}\|U\|^{2}_{H^{s}}$. Hence, we
obtain
$\|u\|_{\ell^{2}(\mathbb{Z}^{2})}^{2}\leq\varepsilon^{-3}C_{s}^{2}\|U\|^{2}_{H^{s}}$
which yields (2.8). ∎

##  3\. Proof of Theorem 1

Here we use the results of Lemmas 1 and 2 in order to prove Theorem 1. We
start by writing equations of motions in terms of the strain variables (1.10):

$\displaystyle\dot{u}^{(1)}_{j,k}=$ $\displaystyle w_{j+1,k}-w_{j,k},$ (3.1)
$\displaystyle\dot{u}^{(2)}_{j,k}=$ $\displaystyle w_{j,k+1}-w_{j,k},$
$\displaystyle\dot{v}^{(1)}_{j,k}=$ $\displaystyle z_{j+1,k}-z_{j,k},$
$\displaystyle\dot{v}^{(2)}_{j,k}=$ $\displaystyle z_{j,k+1}-z_{j,k},$
$\displaystyle\dot{w}_{j,k}=$ $\displaystyle
c_{1}^{2}\left(u^{(1)}_{j,k}-u^{(1)}_{j-1,k}\right)+c_{2}^{2}\left(u^{(2)}_{j,k}-u^{(2)}_{j,k-1}\right)$
$\displaystyle+\alpha_{1}\left[\left(u^{(1)}_{j,k}\right)^{2}-\left(u^{(1)}_{j-1,k}\right)^{2}\right]$
$\displaystyle+\alpha_{2}\left[u^{(2)}_{j,k}v^{(2)}_{j,k}-u^{(2)}_{j,k-1}v^{(2)}_{j,k-1}+\frac{1}{2}\left(v^{(1)}_{j,k}\right)^{2}-\frac{1}{2}\left(v^{(1)}_{j-1,k}\right)^{2}\right]$
$\displaystyle\dot{z}_{j,k}=$ $\displaystyle
c_{1}^{2}\left(v^{(2)}_{j,k}-v^{(2)}_{j,k-1}\right)+c_{2}^{2}\left(v^{(1)}_{j,k}-v^{(1)}_{j-1,k}\right),$
$\displaystyle+\alpha_{1}\left[\left(v^{(2)}_{j,k}\right)^{2}-\left(v^{(2)}_{j,k-1}\right)^{2}\right]$
$\displaystyle+\alpha_{2}\left[u^{(1)}_{j,k}v^{(1)}_{j,k}-u^{(1)}_{j-1,k}v^{(1)}_{j-1,k}+\frac{1}{2}\left(u^{(2)}_{j,k}\right)^{2}-\frac{1}{2}\left(u^{(2)}_{j,k-1}\right)^{2}\right],$

where $w_{j,k}:=\dot{x}_{j,k}$, $z_{j,k}:=\dot{y}_{j,k}$, and
$(j,k)\in\mathbb{Z}^{2}$. The justification procedure is divided into the
following four steps.

###  3.1. Step 1. Decomposition

Let us use the following decomposition,

$\displaystyle u^{(1)}_{j,k}$
$\displaystyle=\varepsilon^{2}A\left(\xi,\eta,\tau\right)+$
$\displaystyle\varepsilon^{2}U^{(1)}_{j,k}$ (3.2) $\displaystyle
u^{(2)}_{j,k}$
$\displaystyle=\varepsilon^{2}U_{\varepsilon}\left(\xi,\eta,\tau\right)+$
$\displaystyle\varepsilon^{2}U^{(2)}_{j,k}$ $\displaystyle v^{(1)}_{j,k}$
$\displaystyle=$ $\displaystyle\varepsilon^{2}V^{(1)}_{j,k}$ $\displaystyle
v^{(2)}_{j,k}$ $\displaystyle=$ $\displaystyle\varepsilon^{2}V^{(2)}_{j,k}$
$\displaystyle w_{j,k}$
$\displaystyle=\varepsilon^{2}W_{\varepsilon}\left(\xi,\eta,\tau\right)+$
$\displaystyle\varepsilon^{2}W_{j,k}$ $\displaystyle z_{j,k}$ $\displaystyle=$
$\displaystyle\varepsilon^{2}Z_{j,k}$

where $\xi=\varepsilon\left(j-c_{1}t\right)$, $\eta=\varepsilon^{2}k$, and
$\tau=\varepsilon^{3}t$. The leading-order function $A$ is defined as a
suitable solution to the KP-II equation (1.13), whereas the
$\varepsilon$-dependent functions $U_{\varepsilon}$ and $W_{\varepsilon}$ are
introduced to eliminate the lower order terms in $\varepsilon$ arising from
time derivatives and finite differences of $A$ in the first and second
equations of system (3.1).

We denote the error terms of the formal power $\varepsilon^{5}$ by
$\mathcal{O}(\varepsilon^{5})$ and define $W_{\varepsilon}$ and
$U_{\varepsilon}$ from the following equations:

$\displaystyle
W_{\varepsilon}\left(\xi+\varepsilon,\eta\right)-W_{\varepsilon}\left(\xi,\eta\right)=-\varepsilon
c_{1}\partial_{\xi}A(\xi,\eta)+\varepsilon^{3}\partial_{\tau}A(\xi,\eta)+\mathcal{O}(\varepsilon^{5})$
(3.3)

and

$\displaystyle
W_{\varepsilon}\left(\xi,\eta+\varepsilon^{2}\right)-W_{\varepsilon}\left(\xi,\eta\right)=-\varepsilon
c_{1}\partial_{\xi}{U_{\varepsilon}}(\xi,\eta)+\varepsilon^{3}\partial_{\tau}{U_{\varepsilon}}(\xi,\eta)+\mathcal{O}(\varepsilon^{5}),$
(3.4)

where the time dependence is dropped from the list of arguments.

We look for an approximate solution to (3.3) in the form:

$W_{\varepsilon}=W^{(0)}+\varepsilon
W^{(1)}+\varepsilon^{2}W^{(2)}+\varepsilon^{3}W^{(3)},$ (3.5)

where the functions $W^{(j)}$ depend on $(\xi,\eta)$ and decay to zero at
infinity. Plugging (3.5) into (3.3) and expanding each $W^{(j)}$ in Taylor
series, we get

$\displaystyle\varepsilon\partial_{\xi}W^{(0)}+\frac{1}{2}\varepsilon^{2}\partial_{\xi}^{2}W^{(0)}+\frac{1}{6}\varepsilon^{3}\partial_{\xi}^{3}W^{(0)}+\frac{1}{24}\varepsilon^{4}\partial_{\xi}^{4}W^{(0)}$
$\displaystyle\qquad+\varepsilon^{2}\partial_{\xi}W^{(1)}+\frac{1}{2}\varepsilon^{3}\partial_{\xi}^{2}W^{(1)}+\frac{1}{6}\varepsilon^{4}\partial_{\xi}^{3}W^{(1)}+\varepsilon^{3}\partial_{\xi}W^{(2)}+\frac{1}{2}\varepsilon^{4}\partial_{\xi}^{2}W^{(2)}$
$\displaystyle\qquad\qquad+\varepsilon^{4}\partial_{\xi}W^{(3)}=-\varepsilon
c_{1}\partial_{\xi}A+\varepsilon^{3}\partial_{\tau}A+\mathcal{O}(\varepsilon^{5}).$

Grouping terms by their orders in powers of $\varepsilon$ yields a sequence of
equations with their relevant solutions:

$\displaystyle\begin{split}&\mathcal{O}(\varepsilon):\quad\partial_{\xi}W^{(0)}=-c_{1}\partial_{\xi}A\\\
&\phantom{O(\varepsilon):\partial_{\xi}W^{(0)}}\qquad\implies
W^{(0)}=-c_{1}A\\\
&\mathcal{O}(\varepsilon^{2}):\quad\frac{1}{2}\partial_{\xi}^{2}W^{(0)}+\partial_{\xi}W^{(1)}=0\\\
&\phantom{O(\varepsilon):\partial_{\xi}W^{(0)}}\qquad\implies
W^{(1)}=\frac{c_{1}}{2}\partial_{\xi}A\\\
&\mathcal{O}(\varepsilon^{3}):\quad\frac{1}{6}\partial_{\xi}^{3}W^{(0)}+\frac{1}{2}\partial_{\xi}^{2}W^{(1)}+\partial_{\xi}W^{(2)}=\partial_{\tau}A\\\
&\phantom{O(\varepsilon):\partial_{\xi}W^{(0)}}\qquad\implies
W^{(2)}=\partial_{\xi}^{-1}\partial_{\tau}A-\frac{c_{1}}{12}\partial_{\xi}^{2}A\\\
&\mathcal{O}(\varepsilon^{4}):\quad\frac{1}{24}\partial_{\xi}^{4}W^{(0)}+\frac{1}{6}\partial_{\xi}^{3}W^{(1)}+\frac{1}{2}\partial_{\xi}^{2}W^{(2)}+\partial_{\xi}W^{(3)}=0\\\
&\phantom{O(\varepsilon):\partial_{\xi}W^{(0)}}\qquad\implies
W^{(3)}=-\frac{1}{2}\partial_{\tau}A.\\\ \end{split}$ (3.6)

With the choice in (3.6), this construction ensures that equation (3.3) is
satisfied up to and including the order of $\mathcal{O}(\varepsilon^{4})$.
Substituting (3.6) into (3.5) yields

$W_{\varepsilon}=-c_{1}A+\varepsilon\left(\frac{c_{1}}{2}\partial_{\xi}A\right)+\varepsilon^{2}\left(\partial_{\xi}^{-1}\partial_{\tau}A-\frac{c_{1}}{12}\partial_{\xi}^{2}A\right)-\varepsilon^{3}\left(\frac{1}{2}\partial_{\tau}A\right).$
(3.7)

Similarly, we look for an approximate solution to (3.4) in the form:

$\displaystyle U_{\varepsilon}=\varepsilon
U^{(1)}+\varepsilon^{2}U^{(2)}+\varepsilon^{3}U^{(3)},$ (3.8)

where the functions $U^{(j)}$ depend on $(\xi,\eta)$ and decay to zero at
infinity. Plugging (3.5) and (3.8) into (3.4) and expanding each $W^{(j)}$ in
Taylor series, we get

$\displaystyle\begin{split}&\varepsilon^{2}\left(-c_{1}\partial_{\xi}U^{(1)}\right)+\varepsilon^{3}\left(-c_{1}\partial_{\xi}U^{(2)}\right)+\varepsilon^{4}\left(-c_{1}\partial_{\xi}U^{(3)}+\partial_{\tau}U^{(1)}\right)\\\
&\qquad=\varepsilon^{2}\partial_{\eta}W^{(0)}+\varepsilon^{3}\partial_{\eta}W^{(1)}+\varepsilon^{4}\left(\partial_{\eta}W^{(2)}+\frac{1}{2}\partial_{\eta}^{2}W^{(0)}\right)+\mathcal{O}(\varepsilon^{5}).\end{split}$
(3.9)

Grouping terms by their orders in powers of $\varepsilon$ and using the values
for $W_{\varepsilon}$ found in (3.6), we obtain a sequence of equations with
their relevant solutions:

$\displaystyle\begin{split}&\mathcal{O}(\varepsilon^{2}):\quad-
c_{1}\partial_{\xi}U^{(1)}=-c_{1}\partial_{\eta}A\\\
&\phantom{O(\varepsilon^{2}):-c_{1}\partial_{\eta}U=\partial_{\eta}W^{(0)}}\quad\implies
U^{(1)}=\partial_{\xi}^{-1}\partial_{\eta}A\\\
&\mathcal{O}(\varepsilon^{3}):\quad-
c_{1}\partial_{\xi}U^{(2)}=\frac{c_{1}}{2}\partial_{\xi}\partial_{\eta}A\\\
&\phantom{O(\varepsilon^{3}):\frac{c_{1}}{2}\partial_{\xi}\partial_{\eta}U=\partial_{\eta}W^{(1)}}\quad\implies
U^{(2)}=-\frac{1}{2}\partial_{\eta}A\\\ &\mathcal{O}(\varepsilon^{4}):\quad-
c_{1}\partial_{\xi}U^{(3)}+\partial_{\tau}U^{(1)}=\partial_{\xi}^{-1}\partial_{\eta}\partial_{\tau}A-\frac{c_{1}}{12}\partial_{\xi}^{2}\partial_{\eta}A-\frac{c_{1}}{2}\partial_{\eta}^{2}A\\\
&\phantom{O(\varepsilon^{4}):\partial_{\eta}W^{(2)}+\frac{1}{2}\partial_{\eta}^{2}W^{(0)}}\quad\implies
U^{(3)}=\frac{1}{2}\partial_{\xi}^{-1}\partial_{\eta}^{2}A+\frac{1}{12}\partial_{\xi}\partial_{\eta}A\end{split}$
(3.10)

Substituting (3.10) into (3.8) yields

$U_{\varepsilon}=\varepsilon\partial_{\xi}^{-1}\partial_{\eta}A-\varepsilon^{2}\left(\frac{1}{2}\partial_{\eta}A\right)+\varepsilon^{3}\left(\frac{1}{2}\partial_{\xi}^{-1}\partial_{\eta}^{2}A+\frac{1}{12}\partial_{\xi}\partial_{\eta}A\right).$
(3.11)

Substituting decomposition (3.2) into the first and second equations of system
(3.1) yield equations

$\displaystyle\begin{split}\dot{U}_{j,k}^{(1)}&=W_{j+1,k}-W_{j,k}+Res_{j,k}^{U^{(1)}},\\\
\dot{U}^{(2)}_{j,k}&=W_{j,k+1}-W_{j,k}+Res^{U^{(2)}}_{j,k},\end{split}$ (3.12)

where

$\displaystyle\begin{split}Res_{j,k}^{U^{(1)}}&:=c_{1}\varepsilon\partial_{\xi}A-\varepsilon^{3}\partial_{\tau}A+W_{\varepsilon}\left(\xi+\varepsilon,\eta\right)-W_{\varepsilon}\left(\xi,\eta\right),\\\
Res_{j,k}^{U^{(2)}}&:=c_{1}\varepsilon\partial_{\xi}{U_{\varepsilon}}-\varepsilon^{3}\partial_{\tau}{U_{\varepsilon}}+W_{\varepsilon}\left(\xi,\eta+\varepsilon^{2}\right)-W_{\varepsilon}\left(\xi,\eta\right).\end{split}$

If expansions (3.7) and (3.11) are used, the residual terms have the formal
order of $\mathcal{O}(\varepsilon^{5})$. The third and fourth equations of
system (3.1) have not been changed:

$\displaystyle\begin{split}\dot{V}_{j,k}^{(1)}=Z_{j+1,k}-Z_{j,k},\\\
\dot{V}^{(2)}_{j,k}=Z_{j,k+1}-Z_{j,k}.\end{split}$ (3.13)

Finally, the last two equations of system (3.1) can be rewritten explicitly as

$\displaystyle\begin{split}\dot{W}_{j,k}&=c_{1}^{2}\left[U^{(1)}_{j,k}-U^{(1)}_{j-1,k}\right]+c_{2}^{2}\left[U^{(2)}_{j,k}-U^{(2)}_{j,k-1}\right]\\\
&\quad+\alpha_{1}\varepsilon^{2}\left[2AU^{(1)}_{j,k}-2A\left(\xi-\varepsilon,\eta\right)U^{(1)}_{j-1,k}+\left(U_{j,k}^{(1)}\right)^{2}-\left(U_{j-1,k}^{(1)}\right)^{2}\right]\\\
&\quad+\alpha_{2}\varepsilon^{2}\left[U_{\varepsilon}\left(\xi,\eta\right)V^{(2)}_{j,k}-U_{\varepsilon}\left(\xi,\eta-\varepsilon^{2}\right)V^{(2)}_{j,k-1}+U^{(2)}_{j,k}V^{(2)}_{j,k}-U^{(2)}_{j,k-1}V^{(2)}_{j,k-1}\right]\\\
&\quad+\alpha_{2}\varepsilon^{2}\left[\frac{1}{2}\left(V^{(1)}_{j,k}\right)^{2}-\frac{1}{2}\left(V^{(1)}_{j-1,k}\right)^{2}\right]+Res^{W}_{j,k}\\\
\dot{Z}_{j,k}&=c_{2}^{2}\left[V_{j,k}^{(1)}-V_{j-1,k}^{(1)}\right]+c_{1}^{2}\left[V_{j,k}^{(2)}-V_{j,k-1}^{(2)}\right]\\\
&\quad+\alpha_{2}\varepsilon^{2}\left[U_{\varepsilon}\left(\xi,\eta\right)U_{j,k}^{(2)}-U_{\varepsilon}\left(\xi,\eta-\varepsilon^{2}\right)U_{j,k-1}^{(2)}+\frac{1}{2}\left(U_{j,k}^{(2)}\right)^{2}-\frac{1}{2}\left(U_{j,k-1}^{(2)}\right)^{2}\right]\\\
&\quad+\alpha_{2}\varepsilon^{2}\left[A\left(\xi,\eta\right)V_{j,k}^{(1)}-A\left(\xi-\varepsilon,\eta\right)V_{j-1,k}^{(1)}+V^{(1)}_{j,k}U^{(1)}_{j,k}-V^{(1)}_{j-1,k}U^{(1)}_{j-1,k}\right]\\\
&\quad+\alpha_{1}\varepsilon^{2}\left[\left(V^{(2)}_{j,k}\right)^{2}-\left(V^{(2)}_{j,k-1}\right)^{2}\right]+Res^{Z}_{j,k}\end{split}$
(3.14)

where

$\displaystyle\begin{split}Res^{W}_{j,k}:=&c_{1}\varepsilon\partial_{\xi}W_{\varepsilon}-\varepsilon^{3}\partial_{\tau}W_{\varepsilon}+c_{1}^{2}\left[A\left(\xi,\eta\right)-A\left(\xi-\varepsilon,\eta\right)\right]\\\
&+c_{2}^{2}\left[U_{\varepsilon}\left(\xi,\eta\right)-U_{\varepsilon}\left(\xi,\eta-\varepsilon^{2}\right)\right]+\alpha_{1}\varepsilon^{2}\left[A\left(\xi,\eta\right)^{2}-A\left(\xi-\varepsilon,\eta\right)^{2}\right],\\\
Res^{Z}_{j,k}:=&\frac{\alpha_{2}\varepsilon^{2}}{2}\left[U_{\varepsilon}\left(\xi,\eta\right)^{2}-U_{\varepsilon}\left(\xi,\eta-\varepsilon^{2}\right)^{2}\right].\end{split}$

Expanding each term in $Res^{W}_{j,k}$ by using expansions (3.7) and (3.11)
yields the following formal expansion

$\displaystyle Res^{W}_{j,k}=\quad$
$\displaystyle\varepsilon^{3}\left[2c_{1}\partial_{\tau}A+\frac{c_{1}^{2}}{12}\partial_{\xi}^{3}A+c_{2}^{2}\partial_{\xi}^{-1}\partial_{\eta}^{2}A+\alpha_{1}\partial_{\xi}\left(A^{2}\right)\right]$
$\displaystyle-$
$\displaystyle\varepsilon^{4}\left[c_{1}\partial_{\xi}\partial_{\tau}A+\frac{c_{1}^{2}}{24}\partial_{\xi}^{4}A+\frac{c_{2}^{2}}{2}\partial_{\eta}^{2}A+\frac{\alpha_{1}}{2}\partial_{\xi}^{2}\left(A^{2}\right)\right]+\mathcal{O}(\varepsilon^{5}).$

If the function $A$ is a solution of the KP-II equation (1.13), the residual
term $Res^{W}$ has the formal order of $\mathcal{O}(\varepsilon^{5})$. It is
also clear from expansion (3.11) that $Res^{Z}$ has the formal order of
$\mathcal{O}(\varepsilon^{6})$.

###  3.2. Step 2. Residual terms

The residual terms are handled by using Taylor’s theorem. If $A$ is defined in
Sobolev space $H^{s}(\mathbb{R}^{2})$ with sufficiently large $s$, then we can
estimate the residual terms in the $\ell^{2}(\mathbb{Z}^{2})$ norm by an
application of Lemma 2. Since all residual terms have the formal order of
$\mathcal{O}(\varepsilon^{5})$, we can obtain the bound of
$\mathcal{O}(\varepsilon^{\frac{7}{2}})$ on the residual terms in the
$\ell^{2}(\mathbb{Z})$ norm.

The following lemma gives us estimates of the $\ell^{2}$-norm for the residual
terms in equations (3.12) and (3.14). No residual terms appear in equations
(3.13).

######  Lemma 3.

Let $A\in C^{0}(\mathbb{R},H^{s})$ be a solution to the KP-II equation (1.13)
with $s\geq 9$. There is a positive constant $C$ that depend on $A$ such that
for all $\varepsilon\in\left(0,1\right]$, we have

$\displaystyle\left\lVert
Res^{U^{(1)}}_{j,k}\right\rVert_{\ell^{2}}+\left\lVert
Res^{U^{(2)}}_{j,k}\right\rVert_{\ell^{2}}+\left\lVert
Res^{W}_{j,k}\right\rVert_{\ell^{2}}+\left\lVert
Res^{Z}_{j,k}\right\rVert_{\ell^{2}}$ $\displaystyle\leq
C\varepsilon^{\frac{7}{2}}.$ (3.15)

###### Proof.

By construction, all terms in $Res_{j,k}^{U^{(1)}}$ below the formal order of
$\mathcal{O}(\varepsilon^{5})$ vanish. From the Taylor’s theorem for
$W_{\varepsilon}\left(\xi+\varepsilon,\eta\right)-W_{\varepsilon}\left(\xi,\eta\right)$,
the nonzero terms at $\mathcal{O}(\varepsilon^{5})$ are given by the
integrals:

$\displaystyle\varepsilon^{5}\int_{0}^{1}\partial_{\xi}^{5-l}W^{(l)}\left(\varepsilon\left(j+r\right),\varepsilon^{2}k\right)\frac{\left(1-r\right)^{5-l-1}}{\left(5-l-1\right)!}dr,\quad
0\leq l\leq 3.$

In view of corrections for $W^{(l)}$ in (3.7), the error is given by a linear
combination of the following two terms:

$\varepsilon^{5}\sup_{r\in[0,1]}\left|\partial_{\xi}^{5}A\left(\varepsilon\left(j+r\right),\varepsilon^{2}k\right)\right|,\quad\varepsilon^{5}\sup_{r\in[0,1]}\left|\partial_{\xi}^{2}\partial_{\tau}A\left(\varepsilon\left(j+r\right),\varepsilon^{2}k\right)\right|.$

Using Lemma 2, there is a constant $C_{s}>0$ such that the $\ell^{2}$ norm of
the residual term $Res^{U^{(1)}}$ is bounded by

$\left\lVert Res^{U^{(1)}}\right\rVert_{\ell^{2}}\leq
C_{s}\varepsilon^{\frac{7}{2}}\left(\|A\|_{H^{s+5}}+\|\partial_{\tau}A\|_{H^{s+2}}\right),$

for $s>1$.

Similarly, all terms in $Res_{j,k}^{U^{(2)}}$ below the formal order of
$\mathcal{O}(\varepsilon^{5})$ vanish. From the Taylor’s theorem for
$W_{\varepsilon}\left(\xi,\eta+\varepsilon^{2}\right)-W_{\varepsilon}\left(\xi,\eta\right)$
and the corrections for $W_{\varepsilon}$ and $U_{\varepsilon}$ given by (3.7)
and (3.11), the error is given by a linear combination of the following six
terms:

$\displaystyle\varepsilon^{5}\sup_{r\in[0,1]}\left|\partial_{\xi}\partial_{\eta}^{2}A\left(\varepsilon
j,\varepsilon^{2}(k+r)\right)\right|,\quad$
$\displaystyle\varepsilon^{5}\sup_{r\in[0,1]}\left|\partial_{\eta}\partial_{\tau}A\left(\varepsilon
j,\varepsilon^{2}(k+r)\right)\right|,$
$\displaystyle\varepsilon^{6}\sup_{r\in[0,1]}\left|\partial_{\eta}^{3}A\left(\varepsilon
j,\varepsilon^{2}(k+r)\right)\right|,\quad$
$\displaystyle\varepsilon^{6}\sup_{r\in[0,1]}\left|\partial_{\xi}^{2}\partial_{\eta}^{2}A\left(\varepsilon
j,\varepsilon^{2}(k+r)\right)\right|,$
$\displaystyle\varepsilon^{6}\sup_{r\in[0,1]}\left|\partial_{\xi}^{-1}\partial_{\eta}^{2}\partial_{\tau}A\left(\varepsilon
j,\varepsilon^{2}(k+r)\right)\right|,\quad$
$\displaystyle\varepsilon^{6}\left|\partial_{\xi}\partial_{\eta}\partial_{\tau}A\left(\varepsilon
j,\varepsilon^{2}k\right)\right|.$

Using Lemma 2, there is a constant $C_{s}>0$ such that the $\ell^{2}$ norm of
the residual term $Res^{U^{(2)}}$ is bounded for $\varepsilon\in(0,1]$ by

$\left\lVert Res^{U^{(2)}}\right\rVert_{\ell^{2}}\leq
C_{s}\varepsilon^{\frac{7}{2}}\left(\|A\|_{H^{s+4}}+\|\partial_{\tau}A\|_{H^{s+2}}+\|\partial_{\xi}^{-1}\partial_{\eta}\partial_{\tau}A\|_{H^{s+1}}\right),$

for $s>1$.

If $A\in C^{0}(\mathbb{R},H^{s})$ is a solution to the KPII equation (1.13)
with $s\geq 9$, all terms in $Res_{j,k}^{W}$ below the formal order of
$\mathcal{O}(\varepsilon^{5})$ vanish. Expanding all terms in $Res_{j,k}^{W}$
shows that the error is given by a linear combination of the following eight
terms:

$\displaystyle\varepsilon^{5}\left|\partial_{\xi}^{-1}\partial_{\tau}^{2}A\left(\varepsilon
j,\varepsilon^{2}k\right)\right|,\quad\varepsilon^{5}\left|\partial_{\xi}^{2}\partial_{\tau}A\left(\varepsilon
j,\varepsilon^{2}k\right)\right|,\quad\varepsilon^{6}\left|\partial_{\tau}^{2}A\left(\varepsilon
j,\varepsilon^{2}k\right)\right|,$
$\displaystyle\varepsilon^{5}\sup_{r\in[0,1]}\left|\partial_{\xi}^{5}A\left(\varepsilon(j+r),\varepsilon^{2}k\right)\right|,\quad\varepsilon^{5}\sup_{r_{1},r_{2}\in[0,1]}\left|A\left(\varepsilon(j+r_{1}),\varepsilon^{2}k\right)\partial_{\xi}^{3}A\left(\varepsilon(j+r_{2}),\varepsilon^{2}k\right)\right|,$
$\displaystyle\varepsilon^{5}\sup_{r\in[0,1]}\left|\partial_{\xi}^{-1}\partial_{\eta}^{3}A\left(\varepsilon
j,\varepsilon^{2}(k+r)\right)\right|,\quad\varepsilon^{5}\sup_{r\in[0,1]}\left|\partial_{\xi}\partial_{\eta}^{2}A\left(\varepsilon
j,\varepsilon^{2}(k+r)\right)\right|,$

and

$\displaystyle\varepsilon^{6}\sup_{r\in[0,1]}\left|\partial_{\eta}^{3}A\left(\varepsilon
j,\varepsilon^{2}(k+r)\right)\right|.$

Using Lemma 2 and the relation (2.6) for
$\partial_{\xi}^{-1}\partial_{\tau}^{2}A$, there is a constant $C_{s}>0$ such
that the $\ell^{2}$ norm of the residual term $Res^{W}$ is bounded for
$\varepsilon\in(0,1]$ by

$\displaystyle\left\lVert Res^{W}\right\rVert_{\ell^{2}}$ $\displaystyle\leq
C_{s}\varepsilon^{\frac{7}{2}}\left(\|\partial_{\tau}^{2}A\|_{H^{s}}+\|\partial_{\tau}A\|_{H^{s+2}}+\|\partial_{\xi}^{-2}\partial_{\eta}^{2}\partial_{\tau}A\|_{H^{s}}+\|A\|_{H^{s+5}}+\|\partial_{\xi}^{-1}\partial_{\eta}A\|_{H^{s+2}}\right.$
$\displaystyle\qquad\qquad\left.+\|A\|_{H^{s}}\|A\|_{H^{s+3}}+\|A\|_{H^{s}}\|\partial_{\tau}A\|_{H^{s}}\right),$

for $s>1$.

The final residual term $Res^{Z}_{j,k}$ is estimated from the expansion
(3.11). The error is given by

$\displaystyle\varepsilon^{6}\sup_{r_{1},r_{2},r_{3},r_{4}\in[0,1]}\left(|\partial_{\xi}^{-1}\partial_{\eta}A(\varepsilon
j,\varepsilon^{2}(k+r_{1}))|+\varepsilon|\partial_{\eta}A(\varepsilon(j+r_{2}),\varepsilon^{2}k)|\right)$
$\displaystyle\qquad\qquad\left(|\partial_{\xi}^{-1}\partial_{\eta}^{2}A(\varepsilon
j,\varepsilon^{2}(k+r_{3}))|+\varepsilon|\partial_{\eta}^{2}A(\varepsilon(j+r_{4}),\varepsilon^{2}k)|\right)$

and is controlled in the $\ell^{2}$ norm by using the bound

$\left\lVert Res^{Z}\right\rVert_{\ell^{2}}\leq
C_{s}\varepsilon^{\frac{9}{2}}\left(\left\lVert\partial_{\xi}^{-1}\partial_{\eta}A\right\rVert^{2}_{H^{s+1}}+\left\lVert
A\right\rVert^{2}_{H^{s+2}}\right),$

where $C_{s}>0$, $s>1$, and $\varepsilon\in(0,1]$.

Combining all four bounds together and using the fact that a solution $A\in
C^{0}(\mathbb{R},H^{s})$ to the KP-II equation (1.13) with $s\geq 9$ enjoys
the estimates of Lemma 1, we obtain the bound (3.15). ∎

###  3.3. Step 3. Energy estimates

In order to control the growth of the approximation error from solutions to
system (3.12), (3.13), and (3.14), we will introduce the following energy
function,

$\displaystyle\begin{split}E(t)=&\frac{1}{2}\sum_{j,k\in\mathbb{Z}^{2}}W_{j,k}^{2}+Z_{j,k}^{2}+c_{1}^{2}\left(U_{j,k}^{(1)}\right)^{2}+c_{2}^{2}\left(U_{j,k}^{(2)}\right)^{2}+c_{1}^{2}\left(V_{j,k}^{(1)}\right)^{2}+c_{2}^{2}\left(V_{j,k}^{(2)}\right)^{2}\\\
&+\alpha_{1}\varepsilon^{2}\left[2A\left(U_{j,k}^{(1)}\right)^{2}+\frac{2}{3}\left(U_{j,k}^{(1)}\right)^{3}+\frac{2}{3}\left(V_{j,k}^{(2)}\right)^{3}\right]\\\
&+\alpha_{2}\varepsilon^{2}\left[A\left(V_{j,k}^{(1)}\right)^{2}+U_{j,k}^{(1)}\left(V_{j,k}^{(1)}\right)^{2}+\left(U_{j,k}^{(2)}\right)^{2}V_{j,k}^{(2)}+2U_{\varepsilon}U_{j,k}^{(2)}V_{j,k}^{(2)}\right].\end{split}$
(3.16)

The $\varepsilon$-dependent terms of $E(t)$ are chosen from the condition that
the growth rate $E^{\prime}(t)$ along the solution of system (3.12), (3.13),
and (3.14) does not contain terms of the formal orders
$\mathcal{O}(\varepsilon)$ and $\mathcal{O}(\varepsilon^{2})$ (see Lemma 5
below).

The following lemma establishes coercivity of the energy $E(t)$ with respect
to the $\ell^{2}$ norm of the perturbations as long as the perturbations are
not large in the $\ell^{2}$ norm.

######  Lemma 4.

Let $A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s}\left(\mathbb{R}^{2}\right)\right)$
and $\partial_{\xi}^{-1}\partial_{\eta}A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s-1}\left(\mathbb{R}^{2}\right)\right)$
with $s>3$ and assume that $E(t)\leq E_{0}$ for some $\varepsilon$-independent
constant $E_{0}>0$ for every
$t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$. There exists some
constants $\varepsilon_{0}>0$ and $K_{0}>0$ that depend on $A$ such that

$\left\lVert W\right\rVert_{\ell^{2}}^{2}+\left\lVert
Z\right\rVert_{\ell^{2}}^{2}+\left\lVert
U^{(1)}\right\rVert_{\ell^{2}}^{2}+\left\lVert
U^{(2)}\right\rVert_{\ell^{2}}^{2}+\left\lVert
V^{(1)}\right\rVert_{\ell^{2}}^{2}+\left\lVert
V^{(2)}\right\rVert_{\ell^{2}}^{2}\leq 2K_{0}E(t),$ (3.17)

for each $\varepsilon\in\left(0,\varepsilon_{0}\right)$ and
$t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$.

###### Proof.

It follows from the decomposition (3.11) and Sobolev’s embedding of
$H^{s}(\mathbb{R}^{2})$ into $L^{\infty}(\mathbb{R}^{2})$ for $s>1$ that if
$A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s}\left(\mathbb{R}^{2}\right)\right)$
and $\partial_{\xi}^{-1}\partial_{\eta}A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s-1}\left(\mathbb{R}^{2}\right)\right)$
with $s>3$, then there exists a constant $C_{0}>0$ that depends on $A$ such
that

$\displaystyle\sup_{t\in[-\tau_{0},\tau_{0}]}\left(\|A(\cdot,\tau)\|_{L^{\infty}(\mathbb{R}^{2})}+\|U_{\varepsilon}(\cdot,\tau)\|_{L^{\infty}(\mathbb{R}^{2})}\right)\leq
C_{0}.$

By using (3.16) and the Cauchy–Schwarz inequality for
$U^{(2)}_{j,k}V^{(2)}_{j,k}$, we derive the following lower bound for $E(t)$:

$\displaystyle 2E(t)\geq$ $\displaystyle\left\lVert
W\right\rVert_{\ell^{2}}^{2}+\left\lVert Z\right\rVert_{\ell^{2}}^{2}$
$\displaystyle+\left(c_{1}^{2}-2\alpha_{1}\varepsilon^{2}C_{0}-\frac{2}{3}\alpha_{1}\varepsilon^{2}\|U^{(1)}\|_{\ell^{\infty}}\right)\left\lVert
U^{(1)}\right\rVert_{\ell^{2}}^{2}$
$\displaystyle+\left(c_{2}^{2}-\alpha_{2}\varepsilon^{2}C_{0}-\alpha_{2}\varepsilon^{2}\|V^{(2)}\|_{\ell^{\infty}}\right)\left\lVert
U^{(2)}\right\rVert_{\ell^{2}}^{2}$
$\displaystyle+\left(c_{1}^{2}-\alpha_{2}\varepsilon^{2}C_{0}-\alpha_{2}\varepsilon^{2}\|U^{(1)}\|_{\ell^{\infty}}\right)\left\lVert
V^{(1)}\right\rVert_{\ell^{2}}^{2}$
$\displaystyle+\left(c_{2}^{2}-\alpha_{2}\varepsilon^{2}C_{0}-\frac{2}{3}\alpha_{1}\varepsilon^{2}\|V^{(2)}\|_{\ell^{\infty}}\right)\left\lVert
V^{(2)}\right\rVert_{\ell^{2}}^{2}$

For fixed $c_{1},c_{2}>0$, we use the bound
$\|U\|_{\ell^{\infty}}\leq\|U\|_{\ell^{2}}$ and choose $\varepsilon_{0}>0$ and
$K_{0}>0$ such that

$\displaystyle
c_{1}^{2}-2\alpha_{1}\varepsilon_{0}^{2}C_{0}-\frac{2}{3}\alpha_{1}\varepsilon_{0}^{2}(2K_{0}E_{0})^{1/2}\geq{\rm
min}(1,K_{0}^{-1}),$ $\displaystyle
c_{2}^{2}-\alpha_{2}\varepsilon_{0}^{2}C_{0}-\alpha_{2}\varepsilon_{0}^{2}(2K_{0}E_{0})^{1/2}\geq{\rm
min}(1,K_{0}^{-1}),$ $\displaystyle
c_{1}^{2}-\alpha_{2}\varepsilon_{0}^{2}C_{0}-\alpha_{2}\varepsilon_{0}^{2}(2K_{0}E_{0})^{1/2}\geq{\rm
min}(1,K_{0}^{-1}),$ $\displaystyle
c_{2}^{2}-\alpha_{2}\varepsilon_{0}^{2}C_{0}-\frac{2}{3}\alpha_{1}\varepsilon_{0}^{2}(2K_{0}E_{0})^{1/2}\geq{\rm
min}(1,K_{0}^{-1}),$

which is always possible if $\varepsilon_{0}$ is sufficiently small. This
gives the bound (3.17) with $K_{0}$ redefined as $\max(1,K_{0})$. ∎

The following lemma uses the coercivity of the energy $E(t)$ in Lemma 4 to
establish the rate at which it may grow in time along the solutions of system
(3.12), (3.13), and (3.14). We will be able to use this in Step 4, along with
a Gronwall lemma argument of Lemma 6 below, in order to get a bound on the
size of the energy quantity. This will in turn gives a bound on how far
solutions of the KP-II equation (1.13) drift away from solutions of the FPU
system (3.1) and hence will complete the proof of Theorem 1.

######  Lemma 5.

Let $A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+9}\left(\mathbb{R}^{2}\right)\right)$
be a solution to the KP-II equation (1.13) with $s\geq 0$ in the class of
functions of Lemma 1 and assume that $E(t)\leq E_{0}$ for some
$\varepsilon$-independent constant $E_{0}>0$ for every
$t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$. There exist some
constants $\varepsilon_{0}>0$ and $K_{0}>0$ that depend on $A$ such that

$\left|E^{\prime}(t)\right|\leq
K_{0}\left(\varepsilon^{\frac{7}{2}}E(t)^{\frac{1}{2}}+\varepsilon^{3}E(t)\right),$
(3.18)

for each $\varepsilon\in(0,\varepsilon_{0})$ and
$t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$.

###### Proof.

By differentiating $E(t)$, defined by (3.16), in time $t$, we obtain

$\displaystyle E^{\prime}(t)=$
$\displaystyle\sum_{j,k\in\mathbb{Z}^{2}}\dot{W}_{j,k}W_{j,k}+\dot{Z}_{j,k}Z_{j,k}+c_{1}^{2}\dot{U}_{j,k}^{(1)}U_{j,k}^{(1)}+c_{2}^{2}\dot{U}_{j,k}^{(2)}U_{j,k}^{(2)}+c_{1}^{2}\dot{V}_{j,k}^{(1)}V_{j,k}^{(1)}+c_{2}^{2}\dot{V}_{j,k}^{(2)}V_{j,k}^{(2)}$
$\displaystyle+\alpha_{1}\varepsilon^{2}\left[\dot{A}\left(U_{j,k}^{(1)}\right)^{2}+2A\dot{U}_{j,k}^{(1)}U_{j,k}^{(1)}+\dot{U}_{j,k}^{(1)}\left(U_{j,k}^{(1)}\right)^{2}+\dot{V}_{j,k}^{(2)}\left(V_{j,k}^{(2)}\right)^{2}\right]$
$\displaystyle+\alpha_{2}\varepsilon^{2}\left[\dot{U_{\varepsilon}}U_{j,k}^{(2)}V_{j,k}^{(2)}+\frac{1}{2}\dot{U}_{j,k}^{(1)}\left(V_{j,k}^{(1)}\right)^{2}+U_{j,k}^{(1)}\dot{V}_{j,k}^{(1)}V_{j,k}^{(1)}+\frac{1}{2}\left(U_{j,k}^{(2)}\right)^{2}\dot{V}_{j,k}^{(2)}\right.$
$\displaystyle\left.+\dot{U}_{j,k}^{(2)}U_{j,k}^{(2)}V_{j,k}^{(2)}+U_{\varepsilon}\dot{U}_{j,k}^{(2)}V_{j,k}^{(2)}+U_{\varepsilon}U_{j,k}^{(2)}\dot{V}_{j,k}^{(2)}+\frac{1}{2}\dot{A}\left(V_{j,k}^{(1)}\right)^{2}+A\dot{V}_{j,k}^{(1)}V_{j,k}^{(1)}\right],$

where the dot denotes derivative in $t$ and is applied with the chain rule to
$A$ and $U_{\varepsilon}$ that depends on
$(\varepsilon(j-c_{1}t),\varepsilon^{2}k,\varepsilon^{3}t)$. Substituting
equations of motion (3.12), (3.13), and (3.14) and summing across
$(j,k)\in\mathbb{Z}^{2}$ simplifies $E^{\prime}(t)$ to the form:

$\displaystyle E^{\prime}(t)=$
$\displaystyle\sum_{j,k\in\mathbb{Z}^{2}}W_{j,k}Res^{W}_{j,k}+Z_{j,k}Res^{Z}_{j,k}$
$\displaystyle+\left(c_{1}^{2}U^{(1)}_{j,k}+2\alpha_{1}\varepsilon^{2}AU^{(1)}_{j,k}+\alpha_{1}\varepsilon^{2}\left(U_{j,k}^{(1)}\right)^{2}+\frac{1}{2}\alpha_{2}\varepsilon^{2}\left(V_{j,k}^{(1)}\right)^{2}\right)Res^{U^{(1)}}$
$\displaystyle+\left(c_{2}^{2}U^{(2)}_{j,k}+\alpha_{2}\varepsilon^{2}U_{\varepsilon}V^{(2)}_{j,k}+\alpha_{2}\varepsilon^{2}U_{j,k}^{(2)}V_{j,k}^{(2)}\right)Res^{U^{(2)}}$
$\displaystyle+\alpha_{1}\varepsilon^{2}\left(-c_{1}\varepsilon\partial_{\xi}A+\varepsilon^{3}\partial_{\tau}A\right)\left(U^{(1)}_{j,k}\right)^{2}$
$\displaystyle+\alpha_{2}\varepsilon^{2}\left(-c_{1}\varepsilon\partial_{\xi}{U_{\varepsilon}}+\varepsilon^{3}\partial_{\tau}{U_{\varepsilon}}\right)U^{(2)}_{j,k}V^{(2)}_{j,k}$
$\displaystyle+\frac{\alpha_{2}}{2}\varepsilon^{2}\left(-c_{1}\varepsilon\partial_{\xi}A+\varepsilon^{3}\partial_{\tau}A\right)\left(V^{(1)}_{j,k}\right)^{2}.$

Applying the Cauchy-Schwartz inequality and the bound
$\|U\|_{\ell^{\infty}}\leq\|U\|_{\ell^{2}}$ yields

$\displaystyle\left|E^{\prime}(t)\right|\leq$ $\displaystyle\left\lVert
W\right\rVert_{\ell^{2}}\left\lVert Res^{W}\right\rVert_{\ell^{2}}+\left\lVert
Z\right\rVert_{\ell^{2}}\left\lVert Res^{Z}\right\rVert_{\ell^{2}}$
$\displaystyle+\left(c_{1}^{2}\left\lVert
U^{(1)}\right\rVert_{\ell^{2}}+2\alpha_{1}\varepsilon^{2}\left\lVert
A\right\rVert_{L^{\infty}}\left\lVert
U^{(1)}\right\rVert_{\ell^{2}}+\alpha_{1}\varepsilon^{2}\left\lVert
U^{(1)}\right\rVert_{\ell^{2}}^{2}+\frac{1}{2}\alpha_{2}\varepsilon^{2}\left\lVert
V^{(1)}\right\rVert_{\ell^{2}}^{2}\right)\left\lVert
Res^{U^{(1)}}\right\rVert_{\ell^{2}}$
$\displaystyle+\left(c_{2}^{2}\left\lVert
U^{(2)}\right\rVert_{\ell^{2}}+\alpha_{2}\varepsilon^{2}\left\lVert
U_{\varepsilon}\right\rVert_{L^{\infty}}\left\lVert
V^{(2)}\right\rVert_{\ell^{2}}+\alpha_{2}\varepsilon^{2}\|U^{(2)}\|_{\ell^{2}}\|V^{(2)}\|_{\ell^{2}}\right)\left\lVert
Res^{U^{(2)}}\right\rVert_{\ell^{2}}$
$\displaystyle+\varepsilon^{2}\left(c_{1}\varepsilon\left\lVert\partial_{\xi}A\right\rVert_{L^{\infty}}+\varepsilon^{3}\left\lVert\partial_{\tau}A\right\rVert_{L^{\infty}}\right)\left(\alpha_{1}\left\lVert
U^{(1)}\right\rVert_{\ell^{2}}^{2}+\frac{1}{2}\alpha_{2}\left\lVert
V^{(1)}\right\rVert_{\ell^{2}}^{2}\right)$
$\displaystyle+\varepsilon^{2}\left(c_{1}\varepsilon\left\lVert\partial_{\xi}U_{\varepsilon}\right\rVert_{L^{\infty}}+\varepsilon^{3}\left\lVert\partial_{\tau}U_{\varepsilon}\right\rVert_{L^{\infty}}\right)\left\lVert
U^{(2)}\right\rVert_{\ell^{2}}\left\lVert V^{(2)}\right\rVert_{\ell^{2}}.$

Estimating the residual terms and the perturbation terms with the help of
Lemmas 3 and 4, respectively, yields (3.18). ∎

###  3.4. Step 4. Bound on the approximation error

By making the substitution $E(t):=\frac{1}{2}Q(t)^{2}$, we obtain from (3.18):

$\left|Q^{\prime}(t)\right|\leq
K_{0}\left(\varepsilon^{\frac{7}{2}}+\varepsilon^{3}Q\right),$ (3.19)

where the constant $K_{0}$ may change from one line to another line. The norm
of the perturbation terms controlled by $Q(t)$ is further estimated by using
the Gronwall’s inequality.

######  Lemma 6.

Assume that $Q(t)$ satisfies (3.19) for
$t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$ and $Q(0)\leq
C_{0}\varepsilon^{\frac{1}{2}}$ for some $\varepsilon$-independent constant
$C_{0}$. There exists $\varepsilon_{0}>0$ such that

$Q(t)\leq\varepsilon^{\frac{1}{2}}(1+C_{0})\exp\left(K_{0}\tau_{0}\right)$
(3.20)

for each $\varepsilon\in\left(0,\varepsilon_{0}\right)$ and
$t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$.

###### Proof.

By using the integrating factor, we can rewrite (3.19) in the form:

$\frac{d}{dt}\left[\exp\left(-\varepsilon^{3}K_{0}t\right)Q\right]\leq
K_{0}\varepsilon^{\frac{7}{2}}\exp\left(-\varepsilon^{3}K_{0}t\right).$ (3.21)

Integrating (3.21) yields the Gronwall’s inequality

$Q(t)\leq\left(Q(0)+\varepsilon^{\frac{1}{2}}\right)\exp\left(\varepsilon^{3}K_{0}t\right).$
(3.22)

Since $Q(0)\leq C\varepsilon^{\frac{1}{2}}$ and
$t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$, the inequality
(3.22) yields (3.20). ∎

We can finish the proof of Theorem 1 by using Lemmas 4 and 6. Since

$\displaystyle Q(0)\leq C_{A}\left(\left\lVert
U^{(1)}_{in}\right\rVert_{\ell^{2}}+\left\lVert
U^{(2)}_{in}\right\rVert_{\ell^{2}}+\left\lVert
W_{in}\right\rVert_{\ell^{2}}+\left\lVert
V^{(1)}_{in}\right\rVert_{\ell^{2}}+\left\lVert
V^{(2)}_{in}\right\rVert_{\ell^{2}}+\left\lVert
Z_{in}\right\rVert_{\ell^{2}}\right),$

where the subscript $in$ stands for the initial condition, the decomposition
(3.2) and the hypothesis (1.14) yield that $Q(0)\leq
C\varepsilon^{\frac{1}{2}}$, so that Lemma 6 gives the bound (3.20). With
decomposition (3.2) and Lemma 4 we have that

$\displaystyle\left\lVert
u^{(1)}(t)-\varepsilon^{2}A\left(\varepsilon(j-c_{1}t),\eta^{2}k,\varepsilon^{3}t\right)\right\rVert^{2}_{\ell^{2}}+\left\lVert
u^{(2)}(t)-\varepsilon^{2}U_{\varepsilon}\left(\varepsilon(j-c_{1}t),\eta^{2}k,\varepsilon^{3}t\right)\right\rVert^{2}_{\ell^{2}}$
$\displaystyle+\left\lVert
w(t)-\varepsilon^{2}W_{\varepsilon}\left(\varepsilon(j-c_{1}t),\eta^{2}k,\varepsilon^{3}t\right)\right\rVert^{2}_{\ell^{2}}+\left\lVert
v^{(1)}(t)\right\rVert^{2}_{\ell^{2}}+\left\lVert
v^{(2)}(t)\right\rVert^{2}_{\ell^{2}}+\left\lVert
z(t)\right\rVert^{2}_{\ell^{2}}\leq K_{0}\varepsilon^{4}Q(t)^{2},$

where the $\varepsilon$-dependent functions $W_{\varepsilon}$ and
$U_{\varepsilon}$ are given in terms of $A$ by (3.7) and (3.11), respectively.
Due to the bound (3.20) and the triangle inequality, we obtain the bound
(1.15) and the result of Theorem 1 follows.

##  4\. Proof of Theorem 2

Here we give relevant details of the proof of Theorem 2. As is explained in
the introduction, we will only consider the reduction of the general FPU
system if $c_{1}=c_{2}$ and $\alpha_{2}=2\alpha_{1}$, when the vertical and
horizontal displacements on the square two-dimensional lattice coincide with
$x_{j,k}=y_{j,k}$.

Instead of working with the strain variables in (1.22), we introduce the
following strain variables (see Fig. 2):

$\displaystyle\left\\{\begin{array}[]{l}a^{l}_{m,n}=\chi_{m,n}-x_{m,n},\\\
a^{d}_{m,n}=x_{m+1,n+1}-\chi_{m,n},\\\ a^{x}_{m,n}=x_{m+1,n}-\chi_{m,n},\\\
a^{y}_{m,n}=x_{m,n+1}-\chi_{m,n},\end{array}\right.$ (4.5)

in order to write equations of motion in the form:

$\displaystyle\begin{split}\dot{a}^{l}_{m,n}=&v_{m,n}-u_{m,n},\\\
\dot{a}^{d}_{m,n}=&u_{m+1,n+1}-v_{m,n},\\\
\dot{a}^{x}_{m,n}=&u_{m+1,n}-v_{m,n},\\\
\dot{a}^{y}_{m,n}=&u_{m,n+1}-v_{m,n},\\\
\dot{u}_{m,n}=&c_{1}^{2}\left(a^{l}_{m,n}-a^{d}_{m-1,n-1}-a^{x}_{m-1,n}-a^{y}_{m,n-1}\right)\\\
&+2\alpha_{1}\left[\left(a^{l}_{m,n}\right)^{2}-\left(a^{d}_{m-1,n-1}\right)^{2}-\left(a^{x}_{m-1,n}\right)^{2}+\left(a^{y}_{m,n-1}\right)^{2}\right],\\\
\dot{v}_{m,n}=&c_{1}^{2}\left(a^{d}_{m,n}-a^{l}_{m,n}+a^{x}_{m,n}+a^{y}_{m,n}\right)\\\
&+2\alpha_{1}\left[\left(a^{d}_{m,n}\right)^{2}-\left(a^{l}_{m,n}\right)^{2}+\left(a^{x}_{m,n}\right)^{2}-\left(a^{y}_{m,n}\right)^{2}\right],\end{split}$
(4.6)

where $u_{m,n}:=\dot{x}_{m,n}$, $v_{m,n}:=\dot{\chi}_{m,n}$, and
$(m,n)\in\mathbb{Z}^{2}$. The justification procedure is divided into the same
four steps as in the case of the horizontal propagation.

###  4.1. Step 1. Decomposition

We will use the following decomposition,

$\displaystyle\begin{split}a^{l}_{m,n}&=\varepsilon^{2}L_{\varepsilon}\left(\xi,\eta,\tau\right)+\varepsilon^{2}L_{m,n},\\\
a^{d}_{m,n}&=\varepsilon^{2}D_{\varepsilon}\left(\xi,\eta,\tau\right)+\varepsilon^{2}D_{m,n},\\\
a^{x}_{m,n}&=\varepsilon^{2}X_{\varepsilon}\left(\xi,\eta,\tau\right)+\varepsilon^{2}X_{m,n},\\\
a^{y}_{m,n}&=\varepsilon^{2}Y_{\varepsilon}\left(\xi,\eta,\tau\right)+\varepsilon^{2}Y_{m,n},\\\
u_{m,n}&=\varepsilon^{2}U_{\varepsilon}\left(\xi,\eta,\tau\right)+\varepsilon^{2}U_{m,n},\\\
v_{m,n}&=\varepsilon^{2}V_{\varepsilon}\left(\xi,\eta,\tau\right)+\varepsilon^{2}V_{m,n},\end{split}$
(4.7)

where $\xi=\varepsilon(m-c_{1}^{*}t)$, $\eta=\varepsilon^{2}n$, and
$\tau=\varepsilon^{3}t$ with $c_{1}^{*}:=\frac{c_{1}}{\sqrt{2}}$. By ignoring
the error terms and the residual terms of the formal order of
$\mathcal{O}(\varepsilon^{5})$ for the time being, we shall use the equations
of motion (4.6) and define the expansions of the functions
$L_{\varepsilon},\dots,V_{\varepsilon}$ in $\varepsilon$ from the condition
that all residual terms of the formal order below
$\mathcal{O}(\varepsilon^{5})$ are removed.

The first equation in system (4.6) gives us the relationship:

$-\varepsilon
c_{1}^{*}\partial_{\xi}L_{\varepsilon}+\varepsilon^{3}\partial_{\tau}L_{\varepsilon}=V_{\varepsilon}-U_{\varepsilon},$
(4.8)

which is used to eliminate $V_{\varepsilon}$ from all other relations.

Adding the first and third equations in system (4.6) implies

$-\varepsilon
c_{1}^{*}\partial_{\xi}(L_{\varepsilon}+X_{\varepsilon})+\varepsilon^{3}\partial_{\tau}(L_{\varepsilon}+X_{\varepsilon})=U_{\varepsilon}\left(\xi+\varepsilon,\eta\right)-U_{\varepsilon}\left(\xi,\eta\right),$
(4.9)

which coincides with equation (3.3) up to notations. As follows from (1.16),
we set

$X_{\varepsilon}+L_{\varepsilon}=A,$ (4.10)

where $A$ is a suitable solution of the KP-II equation (1.17). Since (4.9)
coincides with (3.3), we rewrite expansions (3.5) and (3.7) in new notations:

$\displaystyle
U_{\varepsilon}=-c_{1}^{*}A+\varepsilon\left(\frac{c^{*}_{1}}{2}\partial_{\xi}A\right)+\varepsilon^{2}\left(\partial_{\xi}^{-1}\partial_{\tau}A-\frac{c^{*}_{1}}{12}\partial_{\xi}^{2}A\right)-\varepsilon^{3}\left(\frac{1}{2}\partial_{\tau}A\right).$
(4.11)

Adding the first and fourth equations in system (4.6) implies

$-\varepsilon
c_{1}^{*}\partial_{\xi}(L_{\varepsilon}+Y_{\varepsilon})+\varepsilon^{3}\partial_{\tau}(L_{\varepsilon}+Y_{\varepsilon})=U_{\varepsilon}\left(\xi,\eta+\varepsilon^{2}\right)-U_{\varepsilon}\left(\xi,\eta\right)$
(4.12)

which coincides with equation (3.4) up to notations, Again, we rewrite
expansions (3.8) and (3.11) in new notations:

$\displaystyle
Y_{\varepsilon}+L_{\varepsilon}=\varepsilon\partial_{\xi}^{-1}\partial_{\eta}A-\varepsilon^{2}\left(\frac{1}{2}\partial_{\eta}A\right)+\varepsilon^{3}\left(\frac{1}{2}\partial_{\xi}^{-1}\partial_{\eta}^{2}A+\frac{1}{12}\partial_{\eta}\partial_{\xi}A\right).$
(4.13)

Finally, adding the first and second equations in system (4.6) implies

$-\varepsilon
c_{1}^{*}\partial_{\xi}(L_{\varepsilon}+D_{\varepsilon})+\varepsilon^{3}\partial_{\tau}(L_{\varepsilon}+D_{\varepsilon})=U_{\varepsilon}\left(\xi+\varepsilon,\eta+\varepsilon^{2}\right)-U_{\varepsilon}\left(\xi,\eta\right).$
(4.14)

We derive by using Taylor series and expansion (4.11) up to the formal order
of $\mathcal{O}(\varepsilon^{5})$:

$\displaystyle
U_{\varepsilon}\left(\xi+\varepsilon,\eta+\varepsilon^{2}\right)-U_{\varepsilon}\left(\xi,\eta\right)$
$\displaystyle=\varepsilon\partial_{\xi}U_{\varepsilon}+\varepsilon^{2}\left(\
\frac{1}{2}\partial_{\xi}^{2}U_{\varepsilon}+\partial_{\eta}U_{\varepsilon}\right)$
$\displaystyle\quad+\varepsilon^{3}\left(\frac{1}{6}\partial_{\xi}^{3}U_{\varepsilon}+\partial_{\xi}\partial_{\eta}U_{\varepsilon}\right)+\varepsilon^{4}\left(\frac{1}{24}\partial_{\xi}^{4}U_{\varepsilon}+\frac{1}{2}\partial_{\xi}^{2}\partial_{\eta}U_{\varepsilon}+\frac{1}{2}\partial_{\eta}^{2}U_{\varepsilon}\right)$
$\displaystyle=-c_{1}^{*}\varepsilon\partial_{\xi}A-c_{1}^{*}\varepsilon^{2}\partial_{\eta}A+\varepsilon^{3}\left(\partial_{\tau}A-\frac{c_{1}^{*}}{2}\partial_{\xi}\partial_{\eta}A\right)$
$\displaystyle\quad+\varepsilon^{4}\left(\partial_{\xi}^{-1}\partial_{\eta}\partial_{\tau}A-\frac{c_{1}^{*}}{12}\partial_{\xi}^{2}\partial_{\eta}A-\frac{c_{1}^{*}}{2}\partial_{\eta}^{2}A\right).$

Expanding the left hand side of equation (4.14) in orders of $\varepsilon$ and
comparing with the previous expansions yields

$\displaystyle
D_{\varepsilon}+L_{\varepsilon}=A+\varepsilon\partial_{\xi}^{-1}\partial_{\eta}A+\varepsilon^{2}\left(\frac{1}{2}\partial_{\eta}A\right)+\varepsilon^{3}\left(\frac{1}{2}\partial_{\xi}^{-1}\partial_{\eta}^{2}A+\frac{1}{12}\partial_{\xi}\partial_{\eta}A\right).$
(4.15)

All quantities of the decomposition (4.7) are now defined in terms of $A$ and
$L_{\varepsilon}$. We can now use the fifth and sixth equations in system
(4.6) in order to define $L_{\varepsilon}$ and to verify the validity of the
KP-II equation (1.17) for $A$ up to truncation at the formal order of
$\mathcal{O}(\varepsilon^{5})$. The fifth and sixth equations in system (4.6)
yield

$\begin{split}&-\varepsilon
c_{1}^{*}\partial_{\xi}U_{\varepsilon}+\varepsilon^{3}\partial_{\tau}U_{\varepsilon}=c_{1}^{2}\left(L_{\varepsilon}\left(\xi,\eta\right)-D_{\varepsilon}\left(\xi-\varepsilon,\eta-\varepsilon^{2}\right)-X_{\varepsilon}\left(\xi-\varepsilon,\eta\right)-Y_{\varepsilon}\left(\xi,\eta-\varepsilon^{2}\right)\right)\\\
&+2\alpha_{1}\varepsilon^{2}\left[L_{\varepsilon}\left(\xi,\eta\right)^{2}-D_{\varepsilon}\left(\xi-\varepsilon,\eta-\varepsilon^{2}\right)^{2}-X_{\varepsilon}\left(\xi-\varepsilon,\eta\right)^{2}+Y_{\varepsilon}\left(\xi,\eta-\varepsilon^{2}\right)^{2}\right]\end{split}$
(4.16)

and

$\begin{split}&-\varepsilon
c_{1}^{*}\partial_{\xi}V_{\varepsilon}+\varepsilon^{3}\partial_{\tau}V_{\varepsilon}=c_{1}^{2}\left(D_{\varepsilon}\left(\xi,\eta\right)-L_{\varepsilon}\left(\xi,\eta\right)+X_{\varepsilon}\left(\xi,\eta\right)+Y_{\varepsilon}\left(\xi,\eta\right)\right)\\\
&+2\alpha_{1}\varepsilon^{2}\left[D_{\varepsilon}\left(\xi,\eta\right)^{2}-L_{\varepsilon}\left(\xi,\eta\right)^{2}+X_{\varepsilon}\left(\xi,\eta\right)^{2}-Y_{\varepsilon}\left(\xi,\eta\right)^{2}\right]\end{split}$
(4.17)

In view of equation (4.8), the left-hand side of equation (4.17) is expanded
as

$\displaystyle-\varepsilon
c_{1}^{*}\partial_{\xi}V_{\varepsilon}+\varepsilon^{3}\partial_{\tau}V_{\varepsilon}=-\varepsilon
c_{1}^{*}\partial_{\xi}U_{\varepsilon}+\varepsilon^{2}(c_{1}^{*})^{2}\partial_{\xi}^{2}L_{\varepsilon}+\varepsilon^{3}\partial_{\tau}U_{\varepsilon}-2\varepsilon^{4}c_{1}^{*}\partial_{\xi}\partial_{\tau}L_{\varepsilon}+\varepsilon^{6}\partial_{\tau}^{2}L_{\varepsilon},$

whereas the right-hand side of equation (4.17) can be written as

$\displaystyle
c_{1}^{2}\left(D_{\varepsilon}+L_{\varepsilon}+X_{\varepsilon}+L_{\varepsilon}+Y_{\varepsilon}+L_{\varepsilon}-4L_{\varepsilon}\right)$
$\displaystyle+2\alpha_{1}\varepsilon^{2}\left[(D_{\varepsilon}+L_{\varepsilon})^{2}+(X_{\varepsilon}+L_{\varepsilon})^{2}-(Y_{\varepsilon}+L_{\varepsilon})^{2}-2L_{\varepsilon}\left(D_{\varepsilon}+L_{\varepsilon}+X_{\varepsilon}+L_{\varepsilon}-Y_{\varepsilon}-L_{\varepsilon}\right)\right]$

We expand $L_{\varepsilon}$ in powers of $\varepsilon$ as

$L_{\varepsilon}=L^{(0)}+\varepsilon
L^{(1)}+\varepsilon^{2}L^{(2)}+\varepsilon^{3}L^{(3)},$ (4.18)

where the functions $L^{(j)}$ depend on $(\xi,\eta)$ and decay to zero at
infinity. Substituting (4.10), (4.11), (4.13), (4.15), and (4.18) into the
left-hand and right-hand sides of equation (4.17) yields the following
equations in different powers of $\varepsilon$ with their corresponding
solutions:

$\displaystyle\mathcal{O}(1):\quad 0=2(c_{1}^{*})^{2}(2A-4L^{(0)})$
$\displaystyle\phantom{O(1):\partial_{\xi}W^{(0)}}\qquad\implies
L^{(0)}=\frac{1}{2}A$
$\displaystyle\mathcal{O}(\varepsilon):\quad(c_{1}^{*})^{2}\partial_{\xi}A=2(c_{1}^{*})^{2}(2\partial_{\xi}^{-1}\partial_{\eta}A-4L^{(1)})$
$\displaystyle\phantom{O(\varepsilon):\partial_{\xi}W^{(0)}}\qquad\implies
L^{(1)}=\frac{1}{2}\partial_{\xi}^{-1}\partial_{\eta}A-\frac{1}{8}\partial_{\xi}A$
$\displaystyle\mathcal{O}(\varepsilon^{2}):\quad
0=2(c_{1}^{*})^{2}(-4L^{(2)})$
$\displaystyle\phantom{O(\varepsilon^{2}):\partial_{\xi}W^{(0)}}\qquad\implies
L^{(2)}=0$
$\displaystyle\mathcal{O}(\varepsilon^{3}):\quad-2c_{1}^{*}\partial_{\tau}A-\frac{(c_{1}^{*})^{2}}{24}\partial_{\xi}^{3}A+\frac{(c_{1}^{*})^{2}}{2}\partial_{\xi}\partial_{\eta}A=2(c_{1}^{*})^{2}(\partial_{\xi}^{-1}\partial_{\eta}^{2}A+\frac{1}{6}\partial_{\xi}\partial_{\eta}A-4L^{(3)})+\alpha_{1}A\partial_{\xi}A.$

By using the KPII equation (1.17), we eliminate

$\displaystyle
2c_{1}^{*}\partial_{\tau}A+\alpha_{1}A\partial_{\xi}A=-\frac{1}{48}(c_{1}^{*})^{2}\partial_{\xi}^{3}A-(c_{1}^{*})^{2}\partial_{\xi}^{-1}\partial_{\eta}^{2}A$

and obtain from the equation at the order of $\mathcal{O}(\varepsilon^{3})$
that

$L^{(3)}=\frac{1}{8}\partial_{\xi}^{-1}\partial_{\eta}^{2}A+\frac{1}{384}\partial_{\xi}^{3}A-\frac{1}{48}\partial_{\xi}\partial_{\eta}A.$

Substituting this expansion into (4.18) yields yields the expansion

$L_{\varepsilon}=\frac{1}{2}A+\varepsilon\left(\frac{1}{2}\partial_{\xi}^{-1}\partial_{\eta}A-\frac{1}{8}\partial_{\xi}A\right)+\varepsilon^{3}\left(\frac{1}{8}\partial_{\xi}^{-1}\partial_{\eta}^{2}A+\frac{1}{384}\partial_{\xi}^{3}A-\frac{1}{48}\partial_{\xi}\partial_{\eta}A\right).$
(4.19)

By using (4.10), (4.13), and (4.15), we also obtain

$\displaystyle X_{\varepsilon}$
$\displaystyle=\frac{1}{2}A-\varepsilon\left(\frac{1}{2}\partial_{\xi}^{-1}\partial_{\eta}A-\frac{1}{8}\partial_{\xi}A\right)-\varepsilon^{3}\left(\frac{1}{8}\partial_{\xi}^{-1}\partial_{\eta}^{2}A+\frac{1}{384}\partial_{\xi}^{3}A-\frac{1}{48}\partial_{\xi}\partial_{\eta}A\right),$
(4.20) $\displaystyle Y_{\varepsilon}$
$\displaystyle=-\frac{1}{2}A+\varepsilon\left(\frac{1}{2}\partial_{\xi}^{-1}\partial_{\eta}A+\frac{1}{8}\partial_{\xi}A\right)-\varepsilon^{2}\left(\frac{1}{2}\partial_{\eta}A\right)$
$\displaystyle\qquad\qquad+\varepsilon^{3}\left(\frac{3}{8}\partial_{\xi}^{-1}\partial_{\eta}^{2}A-\frac{1}{384}\partial_{\xi}^{3}A+\frac{5}{48}\partial_{\xi}\partial_{\eta}A\right),$
(4.21) $\displaystyle D_{\varepsilon}$
$\displaystyle=\frac{1}{2}A+\varepsilon\left(\frac{1}{2}\partial_{\xi}^{-1}\partial_{\eta}A+\frac{1}{8}\partial_{\xi}A\right)+\varepsilon^{2}\left(\frac{1}{2}\partial_{\eta}A\right)$
$\displaystyle\qquad\qquad+\varepsilon^{3}\left(\frac{3}{8}\partial_{\xi}^{-1}\partial_{\eta}^{2}A-\frac{1}{384}\partial_{\xi}^{3}A+\frac{5}{48}\partial_{\xi}\partial_{\eta}A\right),$
(4.22)

Finally, substituting decompositions (4.11), (4.19), (4.20), (4.21), and
(4.22) into (4.16) gives the expansion:

$\displaystyle\varepsilon(c_{1}^{*})^{2}\partial_{\xi}A-\frac{1}{2}\varepsilon^{2}(c_{1}^{*})^{2}\partial_{\xi}^{2}A+\varepsilon^{3}\left(-2c_{1}^{*}\partial_{\tau}A+\frac{1}{12}(c_{1}^{*})^{2}\partial_{\xi}^{3}A\right)+\varepsilon^{4}c_{1}^{*}\partial_{\xi}\partial_{\tau}A$
$\displaystyle=\varepsilon(c_{1}^{*})^{2}\partial_{\xi}A-\frac{1}{2}\varepsilon^{2}(c_{1}^{*})^{2}\partial_{\xi}^{2}A+\varepsilon^{3}\left(\frac{5}{48}(c_{1}^{*})^{2}\partial_{\xi}^{3}A+(c_{1}^{*})^{2}\partial_{\xi}^{-1}\partial_{\eta}^{2}A+\alpha_{1}A\partial_{\xi}A\right)$
$\displaystyle\qquad+\varepsilon^{4}\left(-\frac{1}{96}(c_{1}^{*})^{2}\partial_{\xi}^{4}A-\frac{1}{2}(c_{1}^{*})^{2}\partial_{\eta}^{2}A-\frac{1}{2}\alpha_{1}\partial_{\xi}(A\partial_{\xi}A)\right),$

which is satisfied up to the formal order of $\mathcal{O}(\varepsilon^{5})$ if
$A$ is a suitable solution of the KP-II equation (1.17).

###  4.2. Step 2. Residual terms

Plugging the decomposition (4.7) into equations of motion (4.6) gives the
following equations for the error terms:

$\displaystyle\begin{split}\dot{L}_{m,n}&=V_{m,n}-U_{m,n},\\\
\dot{D}_{m,n}&=U_{m+1,n+1}-V_{m,n}+Res^{D}_{m,n},\\\
\dot{X}_{m,n}&=U_{m+1,n}-V_{m,n}+Res^{X}_{m,n},\\\
\dot{Y}_{m,n}&=U_{m,n+1}-V_{m,n}+Res^{Y}_{m,n},\\\
\dot{U}_{m,n}&=c_{1}^{2}\left(L_{m,n}-D_{m-1,n-1}-X_{m-1,n}-Y_{m,n-1}\right)\\\
&\qquad+2\alpha_{1}\varepsilon^{2}\left[L_{m,n}^{2}-D_{m-1,n-1}^{2}-X_{m-1,n}^{2}+Y_{m,n-1}^{2}\right]\\\
&\qquad+4\alpha_{1}\varepsilon^{2}\left[L_{m,n}L_{\varepsilon}\left(\xi,\eta\right)-D_{m-1,n-1}D_{\varepsilon}\left(\xi-\varepsilon,\eta-\varepsilon^{2}\right)\right]\\\
&\qquad+4\alpha_{1}\varepsilon^{2}\left[Y_{m,n-1}Y_{\varepsilon}\left(\xi,\eta-\varepsilon^{2}\right)-X_{m-1,n}X_{\varepsilon}\left(\xi-\varepsilon,\eta\right)\right]\\\
&\qquad+Res^{U}_{m,n},\\\
\dot{V}_{m,n}&=c_{1}^{2}\left(D_{m,n}-L_{m,n}+X_{m,n}+Y_{m,n}\right)\\\
&\qquad+2\alpha_{1}\varepsilon^{2}\left[D_{m,n}^{2}-L_{m,n}^{2}+X_{m,n}^{2}-Y_{m,n}^{2}\right]\\\
&\qquad+4\alpha_{1}\varepsilon^{2}\left[D_{m,n}D_{\varepsilon}-L_{m,n}L_{\varepsilon}+X_{m,n}X_{\varepsilon}-Y_{m,n}Y_{\varepsilon}\right]+Res^{V}_{m,n},\end{split}$
(4.23)

where

$\displaystyle Res^{D}_{m,n}$
$\displaystyle:=U_{\varepsilon}(\xi+\varepsilon,\eta+\varepsilon^{2})-V_{\varepsilon}(\xi,\eta)+\varepsilon
c_{1}^{*}\partial_{\xi}D_{\varepsilon}-\varepsilon^{3}\partial_{\tau}D_{\varepsilon},$
$\displaystyle Res^{X}_{m,n}$
$\displaystyle:=U_{\varepsilon}(\xi+\varepsilon,\eta)-V_{\varepsilon}(\xi,\eta)+\varepsilon
c_{1}^{*}\partial_{\xi}X_{\varepsilon}-\varepsilon^{3}\partial_{\tau}X_{\varepsilon},$
$\displaystyle Res^{Y}_{m,n}$
$\displaystyle:=U_{\varepsilon}(\xi,\eta+\varepsilon^{2})-V_{\varepsilon}(\xi,\eta)+\varepsilon
c_{1}^{*}\partial_{\xi}Y_{\varepsilon}-\varepsilon^{3}\partial_{\tau}Y_{\varepsilon},$

and the residuals $Res^{U}$ and $Res^{V}$ are computed from the residual terms
of equations (4.16) and (4.17). Similarly to Lemma 3, the residual terms are
controlled in the $\ell^{2}(\mathbb{Z}^{2})$ norm if $A$ is a smooth solution
of the KP-II equation (1.17). This estimate is summarized in the following
lemma, which we give without proof.

######  Lemma 7.

Let $A\in C^{0}(\mathbb{R},H^{s})$ be a solution to the KP-II equation (1.17)
with $s\geq 9$. There is a positive constant $C$ that depend on $A$ such that
for all $\varepsilon\in(0,1]$, we have

$\displaystyle\begin{split}\left\lVert
Res^{D}\right\rVert_{\ell^{2}}+\left\lVert
Res^{X}\right\rVert_{\ell^{2}}+\left\lVert
Res^{Y}\right\rVert_{\ell^{2}}+\left\lVert
Res^{U}\right\rVert_{\ell^{2}}+\left\lVert Res^{V}\right\rVert_{\ell^{2}}\leq
C\varepsilon^{\frac{7}{2}}.\end{split}$ (4.24)

###  4.3. Step 3. Energy estimates

In order to control the growth of the approximation error from solutions to
system (4.23), we introduce the following energy function,

$\displaystyle\begin{split}E(t)=&\sum_{m,n}\frac{1}{2}\left(U^{2}_{m,n}+V^{2}_{m,n}\right)+\frac{1}{2}c_{1}^{2}\left(L^{2}_{m,n}+D^{2}_{m,n}+X^{2}_{m,n}+Y^{2}_{m,n}\right)\\\
&+\sum_{m,n}2\alpha_{1}\varepsilon^{2}\left(L^{2}_{m,n}L_{\varepsilon}+D^{2}_{m,n}D_{\varepsilon}+X^{2}_{m,n}X_{\varepsilon}-Y^{2}_{m,n}Y_{\varepsilon}\right)\\\
&+\sum_{m,n}\frac{2}{3}\alpha_{1}\varepsilon^{2}\left(L^{3}_{m,n}+D^{3}_{m,n}+X^{3}_{m,n}-Y^{3}_{m,n}\right).\end{split}$
(4.25)

Similarly to the proof of Lemma 4, the energy is coercive with respect to the
$\ell^{2}$ norm of the perturbations if $\varepsilon$ is sufficiently small
and the perturbations are not large in the $\ell^{2}$ norm. The only
difference between the expansions (4.19), (4.20), (4.21), and (4.22) from the
expansion (3.11) is that the former involve three derivatives of $A$ and one
derivative of $\partial_{\xi}^{-1}\partial_{\eta}A$, whereas the latter
involves two derivatives of $A$ and one derivative of
$\partial_{\xi}^{-1}\partial_{\eta}A$. This modifies the statement of the
following lemma, which we give without proof.

######  Lemma 8.

Let $A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+1}\left(\mathbb{R}^{2}\right)\right)$
and $\partial_{\xi}^{-1}A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s}\left(\mathbb{R}^{2}\right)\right)$
with $s>3$ and assume that $E(t)\leq E_{0}$ for some $\varepsilon$-independent
constant $E_{0}>0$ for every
$t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$. There exists some
constants $\varepsilon_{0}>0$ and $K_{0}>0$ that depend on $A$ such that

$\left\lVert U\right\rVert_{\ell^{2}}^{2}+\left\lVert
V\right\rVert_{\ell^{2}}^{2}+\left\lVert
L\right\rVert_{\ell^{2}}^{2}+\left\lVert
D\right\rVert_{\ell^{2}}^{2}+\left\lVert
X\right\rVert_{\ell^{2}}^{2}+\left\lVert Y\right\rVert_{\ell^{2}}^{2}\leq
2K_{0}E(t),$ (4.26)

for each $\varepsilon\in\left(0,\varepsilon_{0}\right)$ and
$t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$.

Finally, the growth of the energy (4.25) is estimated from the balance
equation:

$\displaystyle\begin{split}E^{\prime}(t)=&\sum_{m,n}U_{m,n}Res^{U}_{m,n}+V_{m,n}Res^{V}_{m,n}\\\
&+c_{1}^{2}\sum_{m,n}D_{m,n}Res^{D}_{m,n}+X_{m,n}Res^{X}_{m,n}+Y_{m,n}Res^{Y}_{m,n}\\\
&+4\alpha_{1}\varepsilon^{2}\sum_{m,n}D_{m,n}^{2}Res^{D}_{m,n}+X_{m,n}^{2}Res^{X}_{m,n}-Y_{m,n}^{2}Res^{Y}_{m,n}\\\
&+4\alpha_{1}\varepsilon^{2}\sum_{m,n}D_{m,n}D_{\varepsilon}Res^{D}_{m,n}+X_{m,n}X_{\varepsilon}Res^{X}_{m,n}-Y_{m,n}Y_{\varepsilon}Res^{Y}_{m,n}\\\
&+2\alpha_{1}\varepsilon^{2}\sum_{m,n}L_{m,n}^{2}\dot{L}_{\varepsilon}+D_{m,n}^{2}\dot{D}_{\varepsilon}+X_{m,n}^{2}\dot{X}_{\varepsilon}-Y_{m,n}^{2}\dot{Y}_{\varepsilon},\end{split}$
(4.27)

where the dot denotes the derivative in $t$ of the function of
$\xi=\varepsilon(m-c_{1}^{*}t)$, $\eta=\varepsilon^{2}n$, and
$\tau=\varepsilon^{3}t$. In view of Lemmas 7 and 8, similar to the proof of
Lemma 5, we can obtain a bound on the growth of the energy. This bound is
summarized in the following lemma, which we give without proof.

######  Lemma 9.

Let $A\in
C^{0}\left(\left[-\tau_{0},\tau_{0}\right],H^{s+9}\left(\mathbb{R}^{2}\right)\right)$
be a solution to the KP-II equation (1.17) with $s\geq 0$ in the class of
functions of Lemma 1 and assume that $E(t)\leq E_{0}$ for some
$\varepsilon$-independent constant $E_{0}>0$ for every
$t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$. There exist some
constants $\varepsilon_{0}>0$ and $K_{0}>0$ that depend on $A$ such that

$\left|E^{\prime}(t)\right|\leq
K_{0}\left(\varepsilon^{\frac{7}{2}}E(t)^{\frac{1}{2}}+\varepsilon^{3}E(t)\right),$
(4.28)

for each $\varepsilon\in(0,\varepsilon_{0})$ and
$t\in[-\tau_{0}\varepsilon^{-3},\tau_{0}\varepsilon^{-3}]$.

###  4.4. Step 4. Bound on the approximation error

Since the bounds (3.18) and (4.28) coincide, application of the Gronwall’s
inequality in Lemma 6 gives the desired result of Theorem 2 exactly like in
Step 4 of the proof of Theorem 1.

##  5\. Conclusion

We have proved here the validity of the KP–II approximation for dynamics of
transversely modulatied small-amplitude long-scale waves in a vector FPU
system on a two-dimensional square lattice. The justification was performed
for horizontal and vertical propagations of the waves and, under some
restrictions on parameters of the FPU system, for the diagonal propagation.
While the general algorithm of the justification analysis is well understood
by now, the technical details of the justification analysis have been
developed for the first time in the vector FPU systems, to the best of our
knowledge.

This research opens up new directions. First, it is interesting to see if the
justification analysis can be generalized for the vector FPU mass–spring
systems with diagonal springs and for the wave propagation under an arbitrary
angle with respect to the square lattice. Second, in terms of applications of
the FPU models to the graphene materials, it is important to consider other
two-dimensional models such as hexagonal lattices. Finally, known properties
of the KP-II equation can be applied to study other problems of the nonlinear
dynamics of small-amplitude waves in the two-dimensional FPU lattices such as
the linear and nonlinear stability of periodic and solitary waves with respect
to transverse modulations.

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