Stability of Benney-Luke line solitary waves in 2D

TL;DR

This paper proves the nonlinear stability of line solitary waves in the 2D Benney-Luke equation, extending previous linear stability results and employing methods from KP-II soliton stability analysis.

Contribution

It establishes the nonlinear stability of line solitary waves in the 2D Benney-Luke equation, a significant advancement over prior linear stability results.

Findings

Nonlinear stability of line solitary waves proven.

Method adapted from KP-II soliton stability analysis.

Extends understanding of water wave models.

Abstract

The $2$D Benney-Luke equation is an isotropic model which describes long water waves of small amplitude in $3$D whereas the KP-II equation is a unidirectional model for long waves with slow variation in the transverse direction. In the case where the surface tension is weak or negligible, linearly stability of small line solitary waves of the $2$D Benney-Luke equation was proved by Mizumachi and Shimabukuro [Nonlinearity, 30 (2017), 3419--3465]. In this paper, we prove nonlinear stability of the line solitary waves by adopting the argument by Mizumachi ([Mem. Amer. Math. Soc. no. 1125], [Proc. Roy. Soc. Edinburgh Sect. A., 148 (2018), 149--198] and [arXiv:1808.00809]) which prove nonlinear stability of $1$-line solitons for the KP-II equation.

Equations

$${\partial }_t^2\varphi -\Delta \varphi +a{\Delta }^2\varphi -b\Delta {\partial }_t^2\varphi +({\partial }_t\varphi )(\Delta \varphi )+{\partial }_t(|\nabla \varphi {|}^2)=0\text{on R×R2.}$$

$$2f_{\tilde{x}\tilde{t}}+(b-a)f_{\tilde{x}\tilde{x}\tilde{x}\tilde{x}}+3f_{\tilde{x}}{f}_{\tilde{x}\tilde{x}}+f_{\tilde{y}\tilde{y}}=0$$

$$E(\varphi (t),{\partial }_t\varphi (t))=E({\varphi }_0,{\varphi }_1)\text{for t∈R,}$$

$$E(f,g):=\frac{1}{2}{\int }_{{\mathbb{R}}^2}\left\{|\nabla f{|}^2+a(\Delta f{)}^2+g^2+b|\nabla g{|}^2\right\}dxdy,$$

$$\varphi (t,x,y)={\phi }_c(x\cos \theta +y\sin \theta -ct+\gamma ),\pm c>1,\gamma \in \mathbb{R},\theta \in [0,2\pi ),$$

$$\varphi_{c}(x)=\beta(c)\left\{\tanh\Bigl{(}\frac{\alpha_{c}}{2}x\Bigr{)}-1\right\}\,,\quad\alpha_{c}=\sqrt{\frac{c^{2}-1}{bc^{2}-a}}\,,\quad\beta(c)=\frac{2(c^{2}-1)}{c\alpha_{c}}\,,$$

$$q_{c}(x):=\varphi_{c}^{\prime}(x)=\frac{c^{2}-1}{c}\operatorname{sech}^{2}\bigl{(}\frac{\alpha_{c}x}{2}\bigr{)}$$

$$(b{c}^2-a)q_c^{\prime \prime }-(c^2-1)q_c+\frac{3c}{2}{q}_c^2=0.$$

$${\omega }^2=({\xi }^2+{\eta }^2)\frac{1+a({\xi }^2+{\eta }^2)}{1+b({\xi }^2+{\eta }^2)}$$

$$\{\begin{array}{rl}& {\gamma }_t\simeq {\lambda }_{2,c_0}{\gamma }_{yy}+c-c_0,\\ & c_t\simeq {\lambda }_{1,c_0}^2{\gamma }_{yy}+{\lambda }_{2,c_0}{c}_{yy}.\end{array}$$

$$(\mathcal{F}f)(\xi )=\hat{f}(\xi )=(2\pi {)}^{-n/2}{\int }_{{\mathbb{R}}^n}f(x)e^{-ix\xi }dx,$$

$$({\mathcal{F}}^{-1}f)(x)=\check{f}(x)=\hat{f}(-x),$$

$$⟨f,g⟩=\sum _{j=1}^m{\int }_{\mathbb{R}}{f}_j(x)\overline{g_j(x)}dx$$

$$\Vert u{\Vert }_{H_{\alpha }^k({\mathbb{R}}^2)}={\left(\Vert {\partial }_x^{k}u{\Vert }_{L_{\alpha }^2({\mathbb{R}}^2)}^2+\Vert {\partial }_y^{k}u{\Vert }_{L_{\alpha }^2({\mathbb{R}}^2)}^2+\Vert u{\Vert }_{L_{\alpha }^2({\mathbb{R}}^2)}^2\right)}^{1/2},$$

$$\Vert u{\Vert }_{H_{\alpha }^k(\mathbb{R})}={\left(\Vert {\partial }_x^{k}u{\Vert }_{L_{\alpha }^2(\mathbb{R})}^2+\Vert u{\Vert }_{L_{\alpha }^2(\mathbb{R})}^2\right)}^{1/2},$$

$$\Vert (u_1,u_2){\Vert }_{{\mathbf{X}}_k}=\sqrt{\Vert \nabla u_1{\Vert }_{H_{\alpha }^k({\mathbb{R}}^2)}^2+\Vert u_2{\Vert }_{H_{\alpha }^k({\mathbb{R}}^2)}^2},$$

$${\partial }_t\Phi =L\Phi +N(\Phi ),$$

$$\Phi =\left(\begin{array}{c}{\varphi }_1\\ {\varphi }_2\end{array}\right),L=\left(\begin{array}{cc}0& I\\ B^{-1}A\Delta & 0\end{array}\right),N(\Phi )=-B^{-1}\left(\begin{array}{c}0\\ {\varphi }_2\Delta {\varphi }_1+2\nabla {\varphi }_1\cdot \nabla {\varphi }_2\end{array}\right).$$

$$(c{\partial }_x+L){\Phi }_c+N({\Phi }_c)=0,$$

$${\partial }_t\Phi ={\mathcal{L}}_c\Phi ,$$

$$V_c=-B^{-1}\left(\begin{array}{cc}0& 0\\ v_{1,c}& v_{2,c}\end{array}\right),v_{1,c}=2r_c^{\prime }(z){\partial }_z+r_c(z)\Delta ,v_{2,c}=2q_c(z){\partial }_z+q_c^{\prime }(z).$$

$$L(\eta )=\left(\begin{array}{cc}0& 1\\ B(\eta {)}^{-1}A(\eta )({\partial }_x^2-{\eta }^2)& 0\end{array}\right),V_c(\eta )=-B(\eta {)}^{-1}\left(\begin{array}{cc}0& 0\\ v_{1,c}(\eta )& v_{2,c}(\eta )\end{array}\right),$$

$$v_{1,c}(\eta )=2r_c^{\prime }{\partial }_x+r_c({\partial }_x^2-{\eta }^2),v_{2,c}(\eta )=2q_c{\partial }_z+q_c^{\prime }.$$

$$L(\eta )=L_0+{\eta }^2{L}_1(\eta ),L_1(0)=B_0^{-1}(I-A_0-B_0^{-1}{A}_0)E_{21},$$

$${\mathcal{L}}_c(\eta )={\mathcal{L}}_c(0)+{\eta }^2{\mathcal{L}}_{1,c}(\eta ),$$

$${\mathcal{L}}_{1,c}(0)=L_1(0)+B_0^{-1}{r}_c{E}_{21}+b{B}_0^{-2}\{v_{1,c}(0)E_{21}+v_{2,c}(0)E_2\},$$

$${\zeta }_c(\cdot ,\eta )\in C^{\infty }([-{\eta }_0,{\eta }_0];H_{\alpha }^k(\mathbb{R})\times H_{\alpha }^{k-1}(\mathbb{R})),$$

$${\zeta }_c^{\ast }(\cdot ,\eta )\in C^{\infty }([-{\eta }_0,{\eta }_0];H_{-\alpha }^k(\mathbb{R})\times H_{-\alpha }^{k-1}(\mathbb{R}))$$

$${\mathcal{L}}_c(\eta ){\zeta }_c(z,\eta )={\lambda }_c(\eta ){\zeta }_c(z,\eta ),{\mathcal{L}}_c(\eta {)}^{\ast }{\zeta }_c^{\ast }(z,\eta )={\lambda }_c(-\eta ){\zeta }_c^{\ast }(z,\eta ),$$

$${\lambda }_c(\eta )=i{\lambda }_{1,c}\eta -{\lambda }_{2,c}{\eta }^2+O({\eta }^3),$$

$${\zeta }_c(\cdot ,\eta )={\zeta }_{1,c}+\{i{\lambda }_{1,c}\eta +O({\eta }^2)\}{\zeta }_{2,c}+z_c(\eta ),z_c(\eta )\perp {\zeta }_{1,c}^{\ast },{\zeta }_{2,c}^{\ast },$$

$${\zeta }_c^{\ast }(\cdot ,\eta )={\zeta }_{2,c}^{\ast }-\{i{\lambda }_{1,c}\eta +O({\eta }^2)\}{\zeta }_{1,c}^{\ast }+z_c^{\ast }(\eta ),z_c^{\ast }(\eta )\perp {\zeta }_{1,c},{\zeta }_{2,c},$$

$$\Vert z_c(\eta ){\Vert }_{H_{\alpha }^k(\mathbb{R})\times H_{\alpha }^{k-1}(\mathbb{R})}+\Vert z_c^{\ast }(\eta ){\Vert }_{H_{-\alpha }^k(\mathbb{R})\times H_{-\alpha }^{k-1}(\mathbb{R})}=O({\eta }^2),$$

$$\overline{{\lambda }_c(\eta )}={\lambda }_c(-\eta ),\overline{{\zeta }_c(z,\eta )}={\zeta }_c(z,-\eta ),\overline{{\zeta }_c^{\ast }(z,\eta )}={\zeta }_c^{\ast }(z,-\eta ),$$

$${\zeta }_{1,c}={\partial }_x{\Phi }_c=\left(\begin{array}{c}{q}_c\\ r_c^{\prime }\end{array}\right),{\zeta }_{2,c}=-{\partial }_c{\Phi }_c=-\left(\begin{array}{c}{\partial }_c{\phi }_c\\ {\partial }_c{r}_c\end{array}\right),$$

$${\zeta }_{1,c}^{\ast }=c\left(\begin{array}{c}-B_0{\partial }_c{r}_c-2q_c{\partial }_c{q}_c-q_c^{\prime }{\partial }_c{\phi }_c^{\ast }\\ B_0{\partial }_c{\phi }_c^{\ast }\end{array}\right),{\zeta }_{2,c}^{\ast }=\left(\begin{array}{c}{A}_0{q}_c^{\prime }\\ -B_0{r}_c\end{array}\right),$$

Full text

Secondary 35Q35

1. Introduction

In this paper, we study nonlinear transverse stability of line solitary waves for the Benney-Luke equation

[TABLE]

The Benney-Luke equation is an approximation model of small amplitude long water waves with finite depth originally derived by Benney and Luke [3] as a model for 3D water waves and its mathematically rigorous derivation from the water wave equation was given by [15]. Here $\phi=\phi(t,x,y)$ corresponds to a velocity potential of water waves. We remark that (1.1) is an isotropic model for the propagation of water waves whereas KdV, BBM and KP equations are unidirectional models. See e.g. [5, 6, 7] for the other bidirectional models of 2D and 3D water waves. Since the Benney-Luke equation is isotropic as the water wave equation, it could be more useful to describe nonlinear interactions of waves at a high angle than the KP equations.

The parameters $a$, $b$ are positive and satisfy $a-b=\hat{\tau}-1/3$, where $\hat{\tau}$ is the inverse Bond number. In this paper, we will assume $0<a<b$, which corresponds to the case where the surface tension is weak or negligible.

If we think of waves propagating in one direction, slowly evolving in time and having weak transverse variation, then the Benney-Luke equation can be formally reduced to the KP-II equation if $0<a<b$ and to the KP-I equation if $a>b>0$. More precisely, the Benney-Luke equation (1.1) is reduced to

[TABLE]

in the coordinate $\tilde{t}=\epsilon^{3}t$, $\tilde{x}=\epsilon(x-t)$ and $\tilde{y}=\epsilon^{2}y$ by taking terms only of order $\epsilon^{5}$, where $\phi(t,x,y)=\epsilon f(\tilde{t},\tilde{x},\tilde{y})$. See e.g. [20] for the details. On the other hand, the Benney-Luke equation (1.1) can realize the finite time dynamics of the KP-II equation as in the case of $2$-dimensional Boussinesq equations (see [12] and [28]).

The solution $\phi(t)$ of the Benney-Luke equation (1.1) formally satisfies the energy conservation law

[TABLE]

where

[TABLE]

and (1.1) is globally well-posed in the energy class $\mathbf{E}:=(\dot{H}^{2}(\mathbb{R}^{2})\cap\dot{H}^{1}(\mathbb{R}^{2}))\times H^{1}(\mathbb{R}^{2})$ (see [33]). The Benney Luke equation (1.1) has a 3-parameter family of line solitary wave solutions

[TABLE]

where

[TABLE]

and

[TABLE]

is a solution of

[TABLE]

Stability of solitary waves to the $1$-dimensional Benney-Luke equation was studied by [31] for the strong surface tension case $a>b>0$ by using the variational argument ([10, 13]) which was originated by [2, 4] and by [25] for the weak surface tension case $b>a>0$ by adopting the semigroup approach of [30].

If $a>b>0$, then (1.1) has a stable ground state for $c$ satisfying $0<c^{2}<1$ ([29, 32]). Note that for the water wave equation with strong surface tension, orbital stability of solitary waves conditional on global solvability has been proved by Mielke [19] and Buffoni [8] by the variational argument. See also [17] for the algebraic decay property of the ground state. In view of [34, 35], line solitary waves for the $2$-dimensional Benney-Luke equation are expected to be unstable in this parameter regime.

On the other hand if $0<a<b$ and $c:=\sqrt{1+\epsilon^{2}}$ is close to $1$ (the sonic speed), then $\varphi_{c}(x-ct)$ is expected to be transversally stable because $q_{c}(x)$ is similar to a KdV $1$-soliton and line solitons of the KP-II equation is transversally stable ([14, 22, 23, 27]).

The dispersion relation for the linearized equation of (1.1) around [math] is

[TABLE]

for a plane wave solution $\phi(t,x,y)=e^{i(x\xi+y\eta-\omega t)}$. If $b>a>0$, then $|\nabla\omega|\leq 1$, $\nabla\omega(\xi,\eta)\parallel(\xi,\eta)$ and line solitary waves travel faster than the maximum group velocity of linear waves. Using this property and transverse linear stability of $1$-line solitons for the KP-II equation ([9, 22]), transverse linear stability of small line solitary waves of (1.1) was proved by [26]. The difference between the linear stability result for solitary waves of the $1$-dimensional Benney-Luke equation ([25, Lemma 2.1 and Theorems 2.2 and 2.3]) is that in the $1$-dimensional case $\lambda=0$ is an isolated eigenvalue in exponentially weighted spaces whereas $\lambda=0$ is not an isolated eigenvalue of the linearized operator around line solitary waves because line solitary waves do not decay in the transverse direction. In [26], we investigate the spectrum of the linearized operator in a weighted space $L^{2}(\mathbb{R}^{2};e^{2\alpha x}dxdy)$ with $\alpha>0$ and find a curve of continuous spectrum $\{\lambda\mid\pm i\lambda_{1,c}+\lambda_{2,c}\eta^{2}+O(\eta^{3})\,,\eta\in[-\eta_{*},\eta_{*}]\}$, where $\lambda_{1,c}$, $\lambda_{2,c}$, $\eta_{*}$ are positive constants. We remark that the continuous eigenmodes found in [26] grow exponentially as $x\to-\infty$ and cannot be recognized as continuous eigenmodes in the $L^{2}$-framework. These resonant continuous eigenmodes have to do with modulations of line solitary waves. Indeed, we find in [26] that linear evolution of those resonant continuous eigenmodes can be approximately described by solutions of $1$-dimensional dissipative linear wave equation on the time variable $t$ and the transverse variable $y$ and that it illustrates phase shifts of modulating line solitary waves.

In this paper, we show that motion of the local amplitude $c(t,y)$ and the local phase shift $\gamma(t,y)$ of a modulating line solitary wave $\varphi_{c(t,y)}(x-\gamma(t,y))$ is described by $1$-dimensional a system of nonlinear dissipative wave equations and prove nonlinear transverse stability of line solitary waves. The nonlinear stability of line solitary waves in the entire domain has been proved only for the KP-II equation ([22, 23, 24]) and it is interesting to give its mathematical proof for non-integrable system such as the Benny-Luke equations. Since the arguments in [22, 23, 24] are based on PDE methods and integrability of the KP-II equation is used only to prove linear stability of line solitary waves, we are able to extend the arguments for the Benney-Luke equation.

Our plan of the present paper is as follows. In Section 2.1, we recall the linear stability property of line solitary waves in [26] (Theorem 2.1) and introduce our main result (Theorem 2.2) in Section 2.2. In Section 3, we introduce $L^{2}-\mathcal{F}^{-1}L^{\infty}$ estimates of solutions for the linearized modulation equations as well as a substitute of d’Alembert’s formula to prove the $L^{\infty}$-bound of the phase shifts of line solitary waves. We also investigate the large time behavior of the solutions to the linearized equation. In Section 4, we decompose a solution around line solitary waves into a sum of the modulating line solitary wave, a small freely propagating solution of (1.1), exponentially localized remainder part and an auxiliary function. We will impose the secular term condition to the exponentially localized part to make use of linear stability property of line solitary waves. We split small solutions of (1.1) from solutions around line solitary waves so that the remainder part is exponentially localized because as in [23], resonant continuous eigenmodes of the adjoint linearized operator grow exponentially as $x\to\infty$ and the secular term condition makes sense only for exponentially localized perturbations. Since $\varphi_{c}(-\infty)$ differs as $c$ varies, we need a correction term to keep the remainder terms in the energy class. In Section 5, we compute time derivative of the secular term condition and derive a system of PDEs that describe the motion of the local amplitude $c(t,y)$ and the local phase shift $\gamma(t,y)$ whose linearized equation is

[TABLE]

The modulation equation derived from the secular term condition has a critical nonlinear term whose $L^{1}(\mathbb{R}_{y})$-norm decays like $t^{-1}$ and a term coming from the freely propagating remainder part whose $L^{1}(\mathbb{R}_{y})$-norm is expected to grow as $t\to\infty$. These terms are harmful to estimate modulation parameters $c(t,y)$ and $\gamma(t,y)$. As in [23, 24], we use a change of variables to transform these terms into harmless forms. In Section 6, we prove decay estimates of the local amplitude $c(t,y)$ and the local orientation $\gamma_{y}(t,y)$ of the modulating line solitary wave $\varphi_{c(t,y)}(x-c_{0}t-\gamma(t,y))$ by applying linear estimates obtained in Section 3, where $c_{0}$ is the amplitude of the unperturbed line solitary wave. In Section 7, we estimate a renormalized energy of perturbations which we find by removing infinite energy parts from solutions around line solitary wave and making use of the orthogonality condition imposed on the exponentially localized part in Section 4.2. In Section 8, we prove virial identities for (1.1) and prove decay estimates for localized energies of small solutions to (1.1). In Section 9, we estimate an exponentially localized norm of perturbations by using linear stability property of line solitary waves (Theorem 2.1). If we linearize (1.1) around a modulating line solitary wave and use the moving coordinate $z=x-c_{0}t-\gamma(t,y)$, we have a space-time dependent advection term. Since (1.1) is a $2$-dimensional wave equation with dispersion in the low frequency regime, the smoothing effect of (1.1) is not strong enough to treat the advection term as a remainder part. To avoid the appearance of the advection term, we prove that $\|\gamma(t,\cdot)\|_{L^{\infty}(\mathbb{R}_{y})}$ remains small for polynomially localized perturbations to line solitary waves by using an estimate similar to d’Alembert’s formula following the idea of [24]. In our paper, it remains open whether the phase shift $\gamma(t,y)$ can grow as $t$, $y\to\pm\infty$ for perturbations in the energy class. See [18] for the growth of phase shifts of gKdV solitary waves. In Section 10, we show that the large time behavior of $c(t,y)$ and $\gamma_{y}(t,y)$ can be expressed by a linear combination of self-similar solutions of the Burgers’ equation with spatial phase shift $\pm\lambda_{1,c_{0}}t$ and prove transverse stability of line solitary waves for polynomially localized perturbations.

Finally, let us introduce several notations. We denote by $\sigma(T)$ the spectrum of the operator $T$. For Banach spaces $V$ and $W$, let $B(V,W)$ be the space of all linear continuous operators from $V$ to $W$ and $\|T\|_{B(V,W)}=\sup_{\|u\|_{V}=1}\|Tu\|_{W}$ for $A\in B(V,W)$. We abbreviate $B(V,V)$ as $B(V)$. For $f\in\mathcal{S}(\mathbb{R}^{n})$ and $m\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$, let

[TABLE]

and $(m(D)f)(x)=(2\pi)^{-n/2}(\check{m}*f)(x)$. We denote $\langle f,g\rangle$ by

[TABLE]

for $\mathbb{C}^{m}$-valued functions $f=(f_{1},\cdots,f_{m})$ and $g=(g_{1},\cdots,g_{m})$.

Let $L^{2}_{\alpha}(\mathbb{R}^{2})=L^{2}(\mathbb{R}^{2};e^{2\alpha x}dxdy)$, $L^{2}_{\alpha}(\mathbb{R})=L^{2}(\mathbb{R};e^{2\alpha x}\,dx)$. For $k\geq 1$, let $H^{k}_{\alpha}(\mathbb{R}^{2})$ and $H^{k}_{\alpha}(\mathbb{R})$ be Hilbert spaces with the norms

[TABLE]

and let $\mathbf{X}_{k}=H^{k+1}_{\alpha}(\mathbb{R}^{2})\times H^{k}_{\alpha}(\mathbb{R}^{2})$ with the norm

[TABLE]

and let $\mathbf{X}=\mathbf{X}_{0}$.

The symbol $\langle x\rangle$ denotes $\sqrt{1+x^{2}}$ for $x\in\mathbb{R}$. Let $\mathbf{1}_{A}$ be the characteristic function of the set $A$.

We use $a\lesssim b$ and $a=O(b)$ to mean that there exists a positive constant such that $a\leq Cb$. Let $a\wedge b=\min\{a,b\}$ and $a\vee b=\max(a,b)$.

2. Statement of results

2.1. Linear stability of line solitary waves

To begin with, we recall the linear stability of line solitary waves for (1.1) ([26]). Let $\phi_{1}=\phi$, $\phi_{2}=\partial_{t}\phi$, $A=I-a\Delta$ and $B=I-b\Delta$. Then the Benney-Luke equation (1.1) can be rewritten as a system

[TABLE]

[TABLE]

Since (1.1) is isotropic and translation invariant, we may assume $\theta=\gamma=0$ in (1.3) without loss of generality. Let $r_{c}(x)=-cq_{c}(x)$ and $\Phi_{c}=(\varphi_{c},r_{c})^{T}$. By (1.4),

[TABLE]

and $\Phi_{c}(x-ct)$ is a planar traveling wave solution of (2.1). Linearizing (2.1) around $\Phi_{c}$, we have in the moving coordinate $z=x-ct$,

[TABLE]

where $\mathcal{L}_{c}=c\partial_{z}+L+V_{c}$ and

[TABLE]

Let $A(\eta)=1+a\eta^{2}-a\partial_{z}^{2}$, $B(\eta)=1+b\eta^{2}-b\partial_{z}^{2}$ and

[TABLE]

Then $\mathcal{L}_{c}(\eta)=e^{-iy\eta}\mathcal{L}_{c}(\,\cdot\,)=c\partial_{x}+L(\eta)+V_{c}(\eta)$. We expand $L(\eta)$ and $\mathcal{L}_{c}(\eta)$ as

[TABLE]

where $L_{0}=L(0)$, $A_{0}=A(0)$, $B_{0}=B(0)$ and $E_{ij}$ ($i$, $j=1$, $2$) is a $2\times 2$ matrix whose entry in row $i$ and column $j$ equals to $1$ and the other entries are zero. We will abbreviate $E_{ii}$ as $E_{i}$.

By Theorem 2.1 in [26], there exist an $\eta_{0}>0$, $\lambda_{c}(\eta)\in C^{\infty}([-\eta_{0},\eta_{0}])$ and

[TABLE]

such that for $\eta\in[-\eta_{0},\eta_{0}]$ and $z\in\mathbb{R}$,

[TABLE]

where $\varphi_{c}^{*}(x)=\varphi_{c}(x)+2\beta(c)$ and

[TABLE]

We remark that $\mathcal{L}_{c}(0)$ is the same with the linearized operator of $1$-dimensional Benney-Luke equation around solitary waves and that $\mathcal{L}_{c}(0)\zeta_{1,c}=0$, $\mathcal{L}_{c}(0)\zeta_{2,c}=\zeta_{1,c}$.

By [26, (6.4)–(6.6), (6.9)–(6.11)],

[TABLE]

Now we introduce spectral projections associated with continuous eigenvalues $\{\lambda(\eta)\}_{-\eta_{0}\leq\eta\leq\eta_{0}}$. As in [26, Section 8], let $\kappa_{c}(\eta)=\frac{1}{2}\Im\langle g(\cdot,\eta,c),g^{*}(\cdot,\eta,c)\rangle$ and

[TABLE]

Note that $g_{k}(z,\eta,c)$ and $g_{k}^{*}(z,\eta,c)$ are $\mathbb{R}^{2}$-valued functions that are even in $\eta$ and that

[TABLE]

Let $\alpha\in(0,\alpha_{c})$ and $\eta_{0}$ be a small positive number such that for $k=1$, $2$,

[TABLE]

Then for $\eta\in[-\eta_{0},\eta_{0}]$,

[TABLE]

Let $\mathcal{P}_{c}(\eta_{0})$ and $\mathcal{Q}_{c}(\eta_{0}):\mathbf{X}\to\mathbf{X}$ be projections defined by

[TABLE]

Then $\mathcal{P}_{c}(\eta_{0})$ is a spectral projection for $\mathcal{L}_{c}$ corresponding to a family of continuous eigenvalues $\{\lambda_{c}(\eta)\}_{-\eta_{0}\leq\eta\leq\eta_{0}}$. Let $\mathbf{Z}=\mathcal{Q}_{c}\mathbf{X}$ and $\mathcal{L}_{c}|_{\mathbf{Z}}$ be the restriction of the operator $\mathcal{L}_{c}$ to $\mathbf{Z}$. Assuming spectral stability of $\mathcal{L}_{c}|_{\mathbf{Z}}$, we have exponential stability of $e^{t\mathcal{L}_{c}}\mathcal{Q}_{c}$.

Theorem 2.1**.**

([26, Theorem 2.2])*

Let $0<a<b$, $c>1$ and $\alpha\in(0,\alpha_{c})$. Consider the operator $\mathcal{L}_{c}$ in the space $\mathbf{X}$. Assume that there exist positive constants $\beta$ and $\eta_{0}$ such that*

[TABLE]

Then for any $\beta^{\prime}<\beta$, there exists a positive constant $C$ such that

[TABLE]

Remark 2.1*.*

If $c>1$ is sufficiently close to $1$, then the assumption (S) is valid and the spectrum of $\mathcal{L}_{c}$ near [math] is similar to that of the linearized KP-II operator around a line soliton solution. See [26, Theorem 2.4].

2.2. Main Result

Now let us introduce our main result.

Theorem 2.2**.**

Let $0<a<b$ and $c_{0}>1$. Assume (S) for $c=c_{0}$. Suppose that $\Phi(t,x,y)$ is a solution of (2.1) satisfying

[TABLE]

Then there exist positive constants $C$ and $\epsilon_{0}$ such that if

[TABLE]

then there exist $C^{1}$-functions $c(t,y)$ and $\gamma(t,y)$ such that for every $t\geq 0$ and $k\geq 0$,

[TABLE]

and for any $R>0$,

[TABLE]

Moreover, there exists a $\gamma_{\infty}\in\mathbb{R}$ such that for any $\delta>0$,

[TABLE]

In the case where $\gamma_{\infty}\neq 0$ in (2.33), the distance between the solution $u$ and the set of line solitary waves in the energy space grows like $t^{1/2}$ or faster.

Corollary 2.3**.**

Let $c_{0}>1$. Suppose that (S) holds for $c=c_{0}$. Then for any $\epsilon_{0}>0$, there exists a solution of (2.1) satisfying (2.27), (2.28) and

[TABLE]

where $\mathcal{K}=\{\Phi_{c}(x\cos\theta+y\sin\theta-ct+\gamma)\mid\pm c>1\,,\quad\gamma\in\mathbb{R}\,,\quad\theta\in[0,2\pi)\}$.

3. Decay estimates for linearized modulation equations

Modulation of the local amplitude and the local phase shift of line solitary waves can be described by a system of Burgers’ equations. In this subsection, we introduce decay estimates for linearized modulation equations following [22, 24].

Let $\nu$ be a real number, \omega(\eta)=\bigl{\{}1-(\nu/\lambda_{1,c_{0}})^{2}\eta^{2}\bigr{\}}^{1/2} and

[TABLE]

Then $\mathcal{A}_{*}(\eta)\Pi(\eta)=\operatorname{diag}(\lambda_{+}(\eta),\lambda_{-}(\eta))\Pi(\eta)$ and

[TABLE]

Let $\eta_{0}$ be a positive number satisfying

[TABLE]

and let $\chi_{1}(\eta)$ and $\chi_{2}(\eta)$ be nonnegative smooth functions such that $\chi_{1}(\eta)+\chi_{2}(\eta)=1$, $\chi_{1}(\eta)=1$ if $|\eta|\leq\frac{1}{2}\eta_{0}$ and $\chi_{1}(\eta)=0$ if $|\eta|\geq\frac{3}{4}\eta_{0}$. Then

[TABLE]

Next, we will estimate the low frequency part of $e^{t\mathcal{A}_{*}(\eta)}$. Let

[TABLE]

Then

[TABLE]

We can prove the following estimates for $K_{1}$, $K_{2}$ and $K_{3}$ in the same way as [24, Lemma 2.2].

Lemma 3.1**.**

Suppose (3.2) and $k\geq 0$. Then for every $t\geq 0$,

[TABLE]

Let $Y$ and $Z$ be closed subspaces of $L^{2}(\mathbb{R})$ defined by

[TABLE]

and let $Y_{1}=\mathcal{F}^{-1}_{\eta}Z_{1}$ and $Z_{1}=\{U\in Z\mid\|U\|_{Z_{1}}:=\|U\|_{L^{\infty}}<\infty\}$. By the definition,

[TABLE]

Especially, we have $\|U\|_{L^{\infty}}\lesssim\|U\|_{L^{2}}$ for any $U\in Y$. Let $\widetilde{P}_{1}$ be a projection defined by $\widetilde{P}_{1}U=\mathcal{F}_{\eta}^{-1}\mathbf{1}_{[-\eta_{0},\eta_{0}]}\mathcal{F}_{y}U$. Then $\|\widetilde{P}_{1}U\|_{Y_{1}}\leq(2\pi)^{-1/2}\|U\|_{L^{1}}$ for any $U\in L^{1}(\mathbb{R})$. In particular, for any $U_{1}$, $U_{2}\in Y$,

[TABLE]

Let $\chi(\eta)$ be a smooth function such that $\chi(\eta)=1$ if $\eta\in[-\frac{\eta_{0}}{4},\frac{\eta_{0}}{4}]$ and $\chi(\eta)=0$ if $\eta\not\in[-\frac{\eta_{0}}{2},\frac{\eta_{0}}{2}]$. We will use the following estimates to investigate large time behavior of modulation parameters.

Lemma 3.2**.**

Let $k\geq 0$. Then for every $t\geq 0$,

[TABLE]

where $c_{1}$ is a positive constant.

Lemma 3.2 follows immediately from Lemma 3.1, (3.3) and (3.4).

Now we will show decay estimates for linearly perturbed equations of $\partial_{t}u=\mathcal{A}_{*}u$. Suppose that $\delta_{1}$, $\delta_{2}$ and $\kappa$ are positive constants and that $d_{ij}(\eta)$ and $b_{ij}(t,\eta)$ are continuous functions satisfying for $\eta\in[-\eta_{0},\eta_{0}]$ and $t\geq 0$,

[TABLE]

Let $\mathcal{A}(t,D_{y})=\mathcal{A}_{0}(D_{y})+\mathcal{A}_{1}(t,D_{y})$,

[TABLE]

and let $U(t,s)$ a solution operator of

[TABLE]

Then we have the following.

Lemma 3.3**.**

Let $k\geq 0$. If $\delta_{1}$ is sufficiently small, then for every $t\geq s\geq 0$ and $f=(f_{1},f_{2})^{T}\in Y\times Y$,

[TABLE]

where $\mu(t,s)=\exp\left(-\int_{s}^{t}b_{22}(s,0)\,ds\right)$, $C=C(\eta_{0})$ and $\limsup_{\eta_{0}\downarrow 0}C(\eta_{0})<\infty$.

Remark 3.1*.*

Since $\chi(D_{y})\widetilde{P}_{1}=\chi(D_{y})$ is bounded on $L^{1}(\mathbb{R})$, we have $\chi(D_{y})f_{2}\in L^{1}(\mathbb{R})$ if $f_{2}=\widetilde{P}_{1}\tilde{f}_{2}$ and $\tilde{f}_{2}\in L^{1}(\mathbb{R})$.

Proof of Lemma 3.3.

Let $u=(u_{1},u_{2})^{T}$ be a solution of (3.16) and $v=(\partial_{y}u_{1},u_{2})$. Then(3.16) can be read as

[TABLE]

Applying a standard energy method to (3.25) as in [22, Lemma 4.2], we have (3.17) and (3.18). We have (3.19) and (3.20) immediately from (3.17) and (3.18).

Next, we will prove (3.21). Let $w=(w_{1},w_{2})^{T}$ and $w(t,y)=\mu(t,s)u(t,y)$. Since $Y\subset L^{\infty}$ and $0<\inf_{t\geq s\geq 0}\mu(t,s)\leq\sup_{t\geq s\geq 0}\mu(t,s)<\infty$, it suffices to show $\|w_{1}(t)\|_{L^{\infty}}\lesssim\|f_{2}\|_{Y_{1}}+\|\chi(D_{y})f_{2}\|_{L^{1}}$. By (3.16),

[TABLE]

and it follows from (H) and (3.17) that

[TABLE]

Thus by Gronwall’s inequality,

[TABLE]

Since $|E_{1}B(t,\eta)|+|E_{2}\eta^{-1}B(t,\eta)|=O(\delta_{2}e^{-\kappa t})$, it follows from Lemma 3.2, (H), (3.18) and (3.26) that

[TABLE]

Next, we will prove (3.22). Using Lemma 3.2 and (3.19), we have for $w=(w_{1},w_{2})^{T}=\mu(t,s)U(t,s)\operatorname{diag}(1,\partial_{y})f$,

[TABLE]

We can prove (3.23) and (3.24) in the same way. Thus we complete the proof. ∎

Lemma 3.4**.**

Let $f=(f_{1},f_{2})^{T}$. For every $t\geq 0$,

[TABLE]

[TABLE]

[TABLE]

where $H_{t}(y)=(4\pi t)^{-1/2}\exp(-y^{2}/4t)$ and $W_{t}(y)=(2\lambda_{1,c_{0}})^{-1}\mathbf{1}_{[-\lambda_{1,c_{0}}t,\lambda_{1,c_{0}}t]}(y)$.

To investigate the large time behavior of $\gamma(t,y)$, we need the following.

Lemma 3.5**.**

Suppose that $f\in L^{1}(\mathbb{R}_{+}\times\mathbb{R})$. Then for any $\delta>0$,

[TABLE]

where $\gamma_{*}=(2\lambda_{1,c_{0}})^{-1}\int_{0}^{\infty}\int_{\mathbb{R}}f(s,y)\,dyds$.

Lemmas 3.4 and 3.5 can be shown in exactly the same way as [24, (2.20)–(2.22) and Lemma 2.4].

4. Decomposition of the perturbed line solitary waves

4.1. Ansatz for solutions around line solitary waves

Let us decompose a solution around $\varphi_{c}(x-c_{0}t)$ into a sum of a modulating line solitary wave and a dispersive part plus a small wave which is caused by amplitude changes of the line solitary wave:

[TABLE]

Here $h$ is a large positive constant (see Lemma 4.3). The modulation parameters $c(t_{0},y_{0})$ and $\gamma(t_{0},y_{0})$ denote the speed and the phase shift of the modulating line solitary wave $q_{c(t,y)}(x-c_{0}t-\gamma(t,y))$ along the line $y=y_{0}$ at the time $t=t_{0}$, $U$ is a remainder part which is expected to behave like an oscillating tail and $\Psi_{c}$ is an auxiliary function such that

[TABLE]

with $\psi\in C_{0}^{\infty}(-1,1)$ and $\int_{\mathbb{R}}\psi(x)\,dx=1$. Note that

[TABLE]

We need the adjustment by $\Psi_{c}$ in order to obtain the energy identity in Section 7.

Substituting (4.1) into (2.1), we have

[TABLE]

where $\ell=\ell_{1}+\ell_{2}$, $\ell_{2}=\ell_{21}+\ell_{22}+\ell_{23}$ and

[TABLE]

[TABLE]

Note that $N(\Psi_{c})=\mathbf{0}$. Let $\tilde{c}(t,y)=c(t,y)-c_{0}$ and $N_{0}(\Phi)=-B_{0}^{-1}(\phi_{2}\partial_{x}^{2}\phi_{1}+2\partial_{x}\phi_{1}\partial_{x}\phi_{2})\mathbf{e_{2}}$ with $\Phi=\phi_{1}\mathbf{e_{1}}+\phi_{2}\mathbf{e_{2}}$, $\mathbf{e_{1}}=(1,0)^{T}$ and $\mathbf{e_{2}}=(0,1)^{T}$. In view of (2.2) and the fact that $L=L_{0}-\partial_{y}^{2}L_{1}$, we have $\ell_{1}=\ell_{11}+\ell_{12}+\ell_{13}$,

[TABLE]

Now we introduce a symplectic orthogonality condition to fix the decomposition (4.1). Since the adjoint resonant modes are exponentially increasing as $x\to\infty$, we further decompose $U$ into a small solution of (2.1) and an exponentially localized part following the idea of [21] and [25, 27, 23] in order to use exponential linear stability in [26].

Let $\Phi(t,x,y)$ be a solution of (2.1) with $\Phi(0,x,y)=\Phi_{c_{0}}(x)+U_{0}(x,y)$ and let $U_{1}$ be a solution of

[TABLE]

and

[TABLE]

By (4.3), (4.6) and (4.7),

[TABLE]

where $N_{1,0}=N(\Phi_{c(t,y)}(z)-\Psi_{c(t,y)}(z_{1})+U)-N(\Phi_{c(t,y)}(z)-\Psi_{c(t,y)}(z_{1}))-N(U)$ and

[TABLE]

Lemma 4.1 below implies that $U_{2}(t,x,y)$, as well as $\widetilde{U}_{2}(t,z,y)$, are exponentially localized as $x\to\infty$ provided the phase shift $\gamma(t,y)$ is uniformly bounded.

Lemma 4.1**.**

Let $\alpha\in(0,\alpha_{c_{0}})$ and $U_{0}\in\mathbf{E}$. Suppose that $\Phi(t)$ is a solution of (2.1) with $\Phi(0)=\Phi_{c_{0}}+U_{0}$ and that $U_{1}(t)$ is a solution of (4.6). Then

[TABLE]

Proof of Lemma 4.1.

First, we remark that $c_{0}\partial_{x}+L$ generates $C^{0}$-semigroup on $\mathbf{E}$ and that $U_{1}(t)$, $W(t)\in C([0,\infty);\mathbf{E})$ follows from a standard argument.

Let

[TABLE]

By (2.1), (2.2), (4.6) and the fact that $N$ is quadratic nonlinearity,

[TABLE]

and

[TABLE]

where $W=(w_{1},w_{2})$ and $g=G\cdot\mathbf{e_{2}}$.

Let $p_{n}(x)=e^{2\alpha n}(1+\tanh\alpha(x-n))$ and

[TABLE]

By (4.11),

[TABLE]

We remark that $0<p_{n}^{\prime}(x)\leq 2\alpha p_{n}(x)\leq 4\alpha e^{2\alpha x}$. By Claim A.2,

[TABLE]

[TABLE]

In the last line, we use the energy identity

[TABLE]

Combining the above, we have

[TABLE]

Using Gronwall’s inequality, we see that for any $T>0$, there exists a $C>0$ such that

[TABLE]

By passing to the limit $n\to\infty$, we have $\sup_{t\in[0,T]}\|W(t)\|_{\mathbf{X}_{1}}\lesssim\|U_{0}\|_{\mathbf{E}}$ since $0<p_{n}(x)\uparrow 2e^{2\alpha x}$. Moreover, we can prove that $\|W(t)\|_{\mathbf{X}_{1}}$ is continuous. Since $W(t)\in C([0,\infty;\mathbf{E})$ and $W(t)$ is weakly continuous in $\mathbf{X}_{1}$, we have (4.9). Thus we complete the proof. ∎

4.2. The orthogonality condition

We impose a secular term condition for $\widetilde{U}_{2}(t,z,y)$ to fix the decomposition (4.1) with (4.7).

[TABLE]

where $\eta_{0}$ is a sufficiently small positive number.

Now we introduce functionals to prove the existence of the decomposition (4.1) that satisfies (4.7) and (4.13). For $U\in\mathbf{X}$ and $\gamma$, $\tilde{c}\in Y$ and $h\geq 0$, let $c(y)=c_{0}+\tilde{c}(y)$ and

[TABLE]

To begin with, we will show that $F=(F_{1},F_{2})$ maps $L^{2}_{\alpha}(\mathbb{R}^{2})\times Y\times Y\times\mathbb{R}$ into $Z\times Z$.

Lemma 4.2**.**

Let $\alpha\in(0,\alpha_{c_{0}})$, $U\in L^{2}_{\alpha}(\mathbb{R}^{2})$, $\tilde{c}$, $\gamma\in Y$ and $h\geq 0$. Then there exists a $\delta>0$ such that if $\|\tilde{c}\|_{Y}+\|\gamma\|_{Y}\leq\delta$, then $F_{k}[u,\tilde{c},\gamma,h]\in Z$ for $k=1$, $2$.

Proof.

Let $U\in C_{0}^{\infty}(\mathbb{R}^{2};\mathbb{R}^{2})$ and

[TABLE]

Then

[TABLE]

where

[TABLE]

By (2.23),

[TABLE]

and it follows from the Plancherel theorem and (4.14) that

[TABLE]

Since $\sup_{y}(|\tilde{c}(y)|+|\gamma(y)|)\lesssim\|\tilde{c}\|_{Y}+\|\gamma\|_{Y}$, we have

[TABLE]

where $C$ is a positive constant depending only on $\delta$.

Combining the above, we obtain

[TABLE]

Since $C_{0}^{\infty}(\mathbb{R}^{2})$ is dense in $L^{2}_{\alpha}(\mathbb{R}^{2})$, it follows that for any $u\in L^{2}_{\alpha}(\mathbb{R}^{2})$,

[TABLE]

Thus we complete the proof. ∎

Next, we will prove the existence of parameters $c$ and $\gamma$ that satisfy (4.1) and (4.13).

Lemma 4.3**.**

Let $\alpha\in(0,\alpha_{c_{0}})$. There exist positive constants $\delta_{0}$, $\delta_{1}$, $h_{0}$ and $C$ such that if $\|U\|_{L^{2}_{\alpha}}<\delta_{0}$ and $h\geq h_{0}$, then there exists a unique $(\tilde{c},\gamma)$ with $c=c_{0}+\tilde{c}$ satisfying

[TABLE]

Moreover, the mapping $\{U\in L^{2}_{\alpha}(\mathbb{R}^{2})\mid\|U\|_{L^{2}_{\alpha}(\mathbb{R}^{2})}<\delta_{0}\}\ni U\mapsto(\tilde{c},\gamma)=:\Omega(U)\in Y\times Y$ is $C^{1}$.

Proof.

We have $(F_{1},F_{2})\in C^{1}(L^{2}_{\alpha}(\mathbb{R}^{2})\times Y\times Y\times\mathbb{R};Z\times Z)$ and for $\tilde{c}$, $\tilde{\gamma}\in Y$,

[TABLE]

where

[TABLE]

and by (2.9), (2.10), (2.13), (2.14) and (2.23),

[TABLE]

Suppose $\eta_{0}$ and $e^{-h\alpha}$ are sufficiently small. Then $D_{(c,\gamma)}(F_{1},F_{2})(0,0,0,h)\in B(Y\times Y,Z\times Z)$ has a bounded inverse and by the implicit function theorem, there exists a unique $(\tilde{c},\gamma)\in Y\times Y$ satisfying (4.15) and (4.16) for any $U$ satisfying $\|U\|_{L^{2}_{\alpha}(\mathbb{R}^{2})}<\delta_{0}$ and the mapping $(\tilde{c},\gamma)=\Omega(U)$ is $C^{1}$. ∎

Remark 4.1*.*

If $U_{0}\in\mathbf{E}$ and $\Phi(0,x,y)=\Phi_{c_{0}}(x)+U_{0}(x,y)$, then a solution of (2.1) is in the class

[TABLE]

Combining the above with (4.9), we have

[TABLE]

and it follows from Lemma 4.3 that $(\tilde{c}(t),\gamma(t))=\Omega(W(t))\in C^{1}$ as long as $\|W(t)\|_{L^{2}_{\alpha}}$ remains small. Since $W(0)=0$, there exists a $T>0$ such that the decomposition (4.1) satisfying (4.7) and (4.13) persists for $t\in[0,T]$ and

[TABLE]

By a standard argument, we have the following continuation principle for the decomposition (4.1) satisfying (4.7) and (4.13).

Proposition 4.4**.**

Let $\alpha$, $\delta_{0}$ and $h$ be the same as in Lemma 4.3 and let $\Phi(t)$ and $U_{1}(t)$ be as in Lemma 4.1. Then there exists a constant $\delta_{2}>0$ such that if (4.1), (4.7) and (4.13) hold for $t\in[0,T)$ and

[TABLE]

then either $T=\infty$ or $T$ is not the maximal time of the decomposition (4.1) satisfying (4.7), (4.13), (4.21) and (4.22).

5. Modulation equations

In this section, we will derive a system of PDEs which describe the motion of $c(t,y)$ and $\gamma(t,y)$.

Let $\widetilde{U}_{1}(t,z,y)=U_{1}(t,x,y)$, $\widetilde{U}(t,z,y)=U(t,x,y)$ and $\tau$ be a shift operator defined by $(\tau_{\gamma}f)(x,y)=f(x+\gamma,y)$. Then (4.6) and (4.8) are transformed into

[TABLE]

where

[TABLE]

Suppose that the decomposition (4.1), (4.7) satisfying (4.13) persists for $t\in[0,T]$. Differentiating (4.13) with respect to $t$, we have in $L^{2}(-\eta_{0},\eta_{0})$,

[TABLE]

where

[TABLE]

The modulation equations of $c(t,y)$ and $\gamma(t,y)$ can be obtained by taking the inverse Fourier transform of (5.3).

5.1. The dominant part of modulation equations

In view of (4.5), we have

[TABLE]

and $\ell_{1}=\ell_{11}+\tilde{\ell}_{12}+\partial_{y}^{2}\tilde{\ell}_{1r}$,

[TABLE]

where $L_{2}(\eta)=\eta^{-2}(L_{1}(\eta)-L_{1}(0))$ for $\eta\neq 0$ and $L_{2}(0)=\frac{1}{2}\frac{d^{2}}{d\eta^{2}}L_{1}(\eta)|_{\eta=0}$. Since $g_{k}^{*}(z,0,c)=\zeta_{k,c}^{*}(z)$, the leading terms of

[TABLE]

are $G_{k}(t,y):=\int_{\mathbb{R}}(\ell_{11}+\tilde{\ell}_{12})\cdot\zeta_{k,c(t,y)}^{*}(z)\,dz$.

Lemma 5.1**.**

[TABLE]

where $B_{1}(c)=\beta_{1}(c)I+\beta_{2}(c)E_{12}$,

[TABLE]

[TABLE]

Proof of Lemma 5.1.

By (2.13) and (2.14),

[TABLE]

By a straightforward computation, we have $\tilde{\ell}_{12}=\tilde{\ell}_{12L}+\tilde{\ell}_{12N}$,

[TABLE]

[TABLE]

and

[TABLE]

We have $m_{25}(c)=c\langle(bc^{2}-2a)q_{c}^{\prime\prime\prime}+q_{c}^{\prime}+3cq_{c}q_{c}^{\prime},q_{c}\rangle=0$ by parity. By (2.15),

[TABLE]

Thus we complete the proof. ∎

The equation $(G_{1},G_{2})\simeq(0,0)$ is a dissipative wave equation. Indeed, let $a_{1j}(c)=\{\beta_{2}(c)m_{2j}(c)-\beta_{1}(c)m_{1j}(c)\}/\beta_{1}(c)^{2}$, $a_{2j}(c)=-m_{2j}(c)/\beta_{1}(c)$ and

[TABLE]

Then by (2.11) and (2.12),

[TABLE]

Solutions of the linearized equation

[TABLE]

decays like $\|\partial_{y}^{k+1}\gamma\|_{Y}+\|\partial_{y}^{k}\tilde{c}\|_{Y}\lesssim\langle t\rangle^{-(2k+1)/4}$ for $k\geq 0$ (see Lemma 3.2). Therefore we expect that the modulation equations for $c(t,y)$ and $\gamma(t,y)$ can be reduced to

[TABLE]

To estimate $\|\tilde{c}(t)\|_{Y}$ by using Lemma 3.2, we need to translate the nonlinear term $2m_{23}(c)c_{y}\gamma_{y}$ into a divergent form because $\|m_{23}(c)c_{y}\gamma_{y}\|_{L^{1}}=O(t^{-1})$. Let

[TABLE]

We remark that $\rho^{\prime}(c_{0})=1$ and $b\simeq\tilde{c}$ (see Claim C.1 in Appendix C).

By (5.15) and (5.17),

[TABLE]

where $p_{1}=-\frac{1}{2}\rho^{\prime\prime}(c_{0})$ and

[TABLE]

Let $\tilde{a}_{22}(c)=\rho^{\prime}(c)a_{22}(c)$. Since $\tilde{a}_{21}(c)=\rho^{\prime}(c)a_{21}(c)$ and $\tilde{a}_{21}^{\prime}(c)=2\rho^{\prime}(c)a_{23}(c)$,

[TABLE]

where $p_{2}(c)=\tilde{a}_{21}(c)-\tilde{a}_{21}(c_{0})=O(\tilde{c})$ and $R^{G,1}_{2}=\tilde{a}_{22}(c)c_{yy}-\tilde{a}_{22}(c_{0})b_{yy}$.

Now we will translate the principal part of $(G_{1},G_{2})$ in terms of $b$ and $\gamma$. Let $B_{2}(c)=\operatorname{diag}(1,\rho^{\prime}(c))$, $B_{3}(c)=B_{2}(c)B_{1}(c)^{-1}$, $B_{4}(c)=\widetilde{P}_{1}B_{2}(c)\widetilde{P}_{1}(\widetilde{P}_{1}B_{1}(c)\widetilde{P}_{1})^{-1}\widetilde{P}_{1}$,

[TABLE]

By (5.11) and (5.17),

[TABLE]

where

[TABLE]

with $p_{2}(c)=O(\tilde{c})$ and $R^{G}=R^{G,1}+R^{G,2}$, $R^{G,1}=(R^{G,1}_{1},R^{G,1}_{2})^{T}$,

[TABLE]

5.2. Expressions of remainder terms

To describe the remainder terms of (5.10), we introduce operators $S_{k}$ $(k=1,2)$. For $Q_{c}=\zeta_{1,c}$, $\zeta_{2,c}$ and so on, let

[TABLE]

where $\delta Q_{c}(z)=\{Q_{c}(z)-Q_{c_{0}}(z)\}/(c-c_{0})$, $g_{k1}^{*}(z,\eta,c)=\eta^{-2}\{g_{k}^{*}(z,\eta,c)-g_{k}^{*}(z,0,c)\}$ and

[TABLE]

For $j=1$ and $2$, let

[TABLE]

Then by (4.5), (5.12) and (5.13),

[TABLE]

[TABLE]

Let

[TABLE]

We set $R^{\ell_{1},j}=(R^{\ell_{1},j}_{1},R^{\ell_{1},j}_{2})^{T}$ for $j=1$, $2$, $r$ and $R^{\ell_{1}}=\partial_{y}^{2}(R^{\ell_{1},1}+R^{\ell_{1},2})+R^{\ell_{1},r}$.

We rewrite $\ell_{2}$ as $\ell_{2}=c_{t}\partial_{c}\Psi_{c_{0}}-\tilde{c}\widetilde{\Psi}_{c_{0}}+\tilde{\ell}_{2N}+\partial_{y}^{2}\tilde{\ell}_{2r}$,

[TABLE]

Let $S^{j}_{k}[p]=S^{j}_{k1}[p]-\partial_{y}^{2}S^{j}_{k2}[p]$ for $j=3$, $4$, $p(z)\in C_{0}^{\infty}(\mathbb{R};\mathbb{R}^{2})$ and

[TABLE]

[TABLE]

Then $\widetilde{S}_{3}=\widetilde{S}_{31}-\partial_{y}^{2}\widetilde{S}_{32}$, $A_{1}(t,D_{y})=A_{1}(t,0)-\partial_{y}^{2}A_{2}(t,D_{y})$ and

[TABLE]

where $R^{\ell_{2}}=R^{\ell_{2},1}-\partial_{y}^{2}R^{\ell_{2},2}+\partial_{y}R^{\ell_{2},r}$, $R^{\ell_{2},j}=(R^{\ell_{2},j}_{1},R^{\ell_{2},j}_{2})^{T}$ for $j=1$, $2$ and $r$,

[TABLE]

We set $R^{j}_{k}(t,y)=\frac{1}{2\pi}\int_{-\eta_{0}}^{\eta_{0}}II^{j}_{k}(t,\eta)e^{iy\eta}\,d\eta$ and $R^{j}=(R^{j}_{1},R^{j}_{2})^{T}$ for $0\leq j\leq 5$.

5.3. Modulation equations

Let $p_{\alpha}(x)=1+\tanh\alpha x$,

$\|U\|_{\mathbf{W}(t)}=\left\|p_{\alpha}(z_{1})^{1/2}\mathcal{E}(U)^{1/2}\right\|_{L^{2}(\mathbb{R}^{2})}$ and $\mathbb{M}_{\infty}(T)=\sup_{0\leq t\leq T,y\in\mathbb{R}}|\gamma(t,y)|$,

[TABLE]

Suppose that $\|U_{0}\|_{\mathbf{E}}$, $\mathbb{M}_{c,\gamma}(T)$, $\mathbb{M}_{1}(T)$, $\mathbb{M}_{2}(T)$ and $\mathbb{M}_{\infty}(T)$ and $\mathbb{M}_{2}(T)$ are sufficiently small. Then by Proposition 4.4, the decomposition (4.1) satisfying (4.7) and (4.13) persists for $t\in[0,T]$ and it follows from (5.3), (5.11), (5.19), (5.21), (5.22) and (5.27) that for $t\in[0,T]$,

[TABLE]

where $B_{5}=I-B_{4}(c_{0})(\partial_{y}^{2}\widetilde{S}_{1}-\widetilde{S}_{3})$,

[TABLE]

Next, we will show that $B_{5}^{-1}$ can be decomposed into a sum of a time-dependent matrix multiplied by $\widetilde{P}_{1}$ and an operator which belongs to $\partial_{y}^{2}(B(Y)\cap B(Y_{1}))$. Let

[TABLE]

By the definition of $\widetilde{S}_{1}$ and (5.23), $\left[\partial_{y},\overset{\bullet}{B}_{5}\right]=\left[\partial_{y},\overset{\circ}{B}_{5}\right]=O$ and $B_{5}^{-1}=\overset{\bullet}{B}_{5}-\partial_{y}^{2}\overset{\circ}{B}_{5}$.

Claim 5.2**.**

There exist positive constants $\eta_{1}$, $h_{0}$, $\delta$ and $C$ such that if $\eta_{0}\in(0,\eta_{1}]$, $h\geq h_{0}$ and $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $t\in[0,T]$,

[TABLE]

Using Claim B.3 in Appendix B, we can prove Claim 5.2 in the same way as [24, Claim 5.2].

Finally, we consider $B_{5}^{-1}B_{4}(c)R^{3}$. In order to prove $\|\tilde{c}(t,\cdot)\|_{Y}+\|\gamma_{y}(t,\cdot)\|_{Y}\lesssim\langle t\rangle^{-1/4}$ by using Lemma 3.3, a crude estimate (C.5) is insufficient. To distinguish a problematic part of $B_{5}^{-1}B_{4}(c)R^{3}$ in Section 6, we decompose $B_{5}^{-1}B_{4}(c)R^{3}$ as follows. Let

[TABLE]

Then $B_{5}^{-1}B_{4}(c)R^{3}=-\mathcal{N}_{U_{1}}+III_{1}+III_{2}-\partial_{y}^{2}III_{3}$. By Claims 5.2, C.2 and (C.5),

[TABLE]

and $III_{i}$ $(i=1,\,2,\,3)$ are good terms. Using a change of variable, we will eliminate the bad part from $\mathcal{N}_{U_{1}}$ in Section 6 following the lines of [23, 24].

It follows from Claims C.3–C.13 and the above that that nonlinear terms of (5.28) satisfy

[TABLE]

[TABLE]

[TABLE]

Proposition 5.3**.**

There exists a positive number $\delta_{3}$ such that if $\mathbb{M}_{c,\gamma}(T)+\mathbb{M}_{1}(T)\linebreak+\mathbb{M}_{2}(T)+\eta_{0}+e^{-\alpha h}<\delta_{3}$, then

[TABLE]

where $\mathcal{A}(t,D_{y})=E_{12}+B_{5}^{-1}\left\{A(c_{0},D_{y})-E_{12}+B_{4}(c_{0})A_{1}(t,D_{y})+\partial_{y}^{4}B_{4}(c_{0})\widetilde{S}_{1}^{\prime}\right\}$and $\mathcal{N}_{j}$ $(2\leq j\leq 4)$ satisfy (5.29)–(5.31).

6. À priori estimates for modulation parameters

In this section, we will estimate $\mathbb{M}_{c,\gamma}(T)$ and $\mathbb{M}_{\infty}(T)$.

Lemma 6.1**.**

There exist positive constants $\delta_{4}$ and $C$ such that if $\mathbb{M}_{c,\gamma}(T)+\mathbb{M}_{1}(T)+\mathbb{M}_{2}(T)+\eta_{0}+e^{-\alpha h}\leq\delta_{4}$, then

[TABLE]

Proof.

Let $\mathcal{A}_{*}(D_{y})=A(c_{0},D_{y})$, $\mathcal{A}(t,D_{y})=\mathcal{A}_{0}(D_{y})+\mathcal{A}_{1}(t,D_{y})$ and

[TABLE]

Then $\mathcal{A}(t,D_{y})$ satisfies (H) and we can apply Lemma 3.3 to the solution operator $U(t,s)$ of (3.16).

The $Y_{1}$-norm of $E_{2}\mathcal{N}_{U_{1}}$ does not necessarily decay as $t\to\infty$. We will express a bad part of $\mathcal{N}_{U_{1}}$ as a time derivative of

[TABLE]

and eliminate that part from (5.32) by the change of variable

[TABLE]

where $\quad\tilde{k}(t,y)=(\beta_{1}(c_{0})+S^{3}_{21})^{-1}\widetilde{P}_{1}k(t,y)$. Since $E_{2}\overset{\bullet}{B}_{5}=(1+\beta_{1}(c_{0})^{-1}S^{3}_{21})^{-1}E_{2}$ and $E_{2}B_{4}(c_{0})=\beta_{1}(c_{0})^{-1}E_{2}\widetilde{P}_{1}$, it follows from Claim C.9 that

[TABLE]

and

[TABLE]

where $\mathcal{A}_{2}(t,\eta)=\eta^{-2}\{\mathcal{A}_{1}(t,\eta)-\mathcal{A}_{1}(t,0)\}$ and

[TABLE]

Now we will estimate the right hand side of

[TABLE]

by using Lemma 3.3. Since $\|\tilde{k}(0)\|_{Y_{1}}+\|\chi(D_{y})\tilde{k}(0)\|_{L^{1}}\lesssim\|\langle y\rangle U_{0}\|_{L^{2}(\mathbb{R}^{2})}$, it follows that for $k\geq 0$ and $t\geq 0$,

[TABLE]

By (5.20), we have $\|\partial_{y}^{k}n_{1}(s)\|_{Y}+\|\partial_{y}^{k}n_{2}(s)\|_{Y}\lesssim\mathbb{M}_{c,\gamma}(T)^{2}\langle s\rangle^{-(2k+3)/4}$ and for $t\in[0,T]$,

[TABLE]

It follows from (5.29)–(5.31) that for $k=0$, $1$, $2$ and $t\in[0,T]$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since $|S^{3}_{21}|+|\partial_{t}S^{3}_{21}|\lesssim e^{-\alpha((c_{0}-1)t/2+h)}$ and

[TABLE]

it follows from Claims C.9, D.1 and D.2 that for $t\in[0,T]$ and $k$ satisfying $0\leq k\leq 2$,

[TABLE]

and that

[TABLE]

[TABLE]

[TABLE]

By the definition and Claim C.1, we have for $k=0$, $1$, $2$ and $t\in[0,T]$,

[TABLE]

Thus we have Lemma 6.1. ∎

7. The energy identity

In this section, we will derive an energy estimate for solutions around modulating line solitary waves.

Lemma 7.1**.**

Let $\alpha\in(0,\alpha_{c_{0}})$ and $\delta_{4}$ be as in Lemma 6.1. Suppose that $\mathbb{M}_{c,\gamma}(T)+\mathbb{M}_{1}(T)+\mathbb{M}_{2}(T)+\eta_{0}+e^{-\alpha h}\leq\delta_{4}$. Then there exists a positive constant $C$ such that

[TABLE]

To prove Lemma 7.1, we will derive an energy identity of (4.3). Let $A_{y}=1-a\partial_{y}^{2}$ and

[TABLE]

and let $\|U\|_{\mathbf{E}_{x}}=\langle U,U\rangle_{\mathbf{E}_{x}}^{1/2}$. The energy of solutions around line solitary waves is infinite since line solitary waves do not decay in the $y$-direction. Moreover, the velocity potential $\varphi_{c}(x)$ tends to a negative constant as $x\to-\infty$. Subtracting infinite energy part from $E(\Phi)$ and having the orthogonality condition (4.13) for $\eta=0$ and $k=2$ in mind, we find the quantity

[TABLE]

Lemma 7.2**.**

[TABLE]

Proof.

Let $\Phi=(\phi_{1},\phi_{2})^{T}$, $\Psi=(\psi_{1},\psi_{2})^{T}$ and $J=B^{-1}(E_{12}-E_{21})$. Since $L\Phi=JE^{\prime}(\Phi)$ and $J$ is skew-symmetric,

[TABLE]

Similarly,

[TABLE]

By integration by parts,

[TABLE]

We remark that $N^{\prime}(\Phi_{c})\Psi_{c}=-B^{-1}(r_{c}\Delta\tilde{\psi}_{c}+2\nabla r_{c}\cdot\nabla\tilde{\psi}_{c})\mathbf{e_{2}}\in\mathbf{E}$ and $E_{1}N^{\prime}(\Psi_{c})=O$.

Since $N$ is quadratic,

[TABLE]

and it follows from (4.3), (7.1), (7.3) and (7.4) that

[TABLE]

By (4.3), (7.1), (7.5) and the fact that $N\cdot\mathbf{e_{1}}=0$ and $\Psi_{c}\cdot\mathbf{e_{2}}=0$,

[TABLE]

Combining the above, we have

[TABLE]

Since $(d\partial_{x}+L_{0})\Phi_{c}+N_{0}(\Phi_{d})=0$ for $d=c_{0}$ and $d=c(t,y)$, it follows from (4.3) and (4.5) that

[TABLE]

By (7.1) and (7.2),

[TABLE]

and

[TABLE]

Since $\Psi_{c}\cdot\mathbf{e_{2}}=0$, $N\cdot\mathbf{e_{1}}=N_{0}\cdot\mathbf{e_{1}}=0$ and $E_{1}LE_{1}=O$,

[TABLE]

Moreover,

[TABLE]

Combining the above, we have

[TABLE]

Thus we prove Lemma 7.2. ∎

By Proposition 5.3, Claim C.1 and (C.5), we have the following.

Claim 7.3**.**

Let $\delta_{2}$ be as in proposition 5.3. Suppose $\mathbb{M}_{c,\gamma}(T)+\mathbb{M}_{1}(T)+\mathbb{M}_{2}(T)+\eta_{0}+e^{-\alpha L}<\delta_{2}$ for a $T\geq 0$. Then for $t\in[0,T]$,

[TABLE]

Now we are in a position to prove Lemma 7.1.

Proof of Lemma 7.1.

First, we remark $\|\Psi_{c}\|_{\mathbf{E}_{x}}\lesssim\|\tilde{c}\|_{Y}$ and that for $k\geq 1$,

[TABLE]

Since $\partial_{y}\{\varphi_{c(t,y)}(z)-\varphi_{c_{0}}(z)\}=c_{y}\partial_{c}\varphi_{c(t,y)}-\gamma_{y}(q_{c(t,y)}(z)-q_{c_{0}}(z))$,

[TABLE]

and $r_{c}(z)$, $q_{c}(z)=\varphi_{c}^{\prime}(z)$ and their derivatives are exponentially localized in $z$, we have

[TABLE]

[TABLE]

Hence it follows that

[TABLE]

Since $\|e^{\alpha|z|}\operatorname{diag}(\partial_{x},1)\ell_{11}\|_{H^{1}}\lesssim\|c_{t}\|_{Y}+\|\gamma_{t}-\tilde{c}\|_{Y}$, it follows from Claim 7.3 that

[TABLE]

Simlilarly, we have

[TABLE]

By (4.5), (5.8) and the definition of $\Phi_{c}$,

[TABLE]

Combining the above with (7.9), we have

[TABLE]

We have $|\langle c\partial_{x}\Psi_{c},\Psi_{c}\rangle_{\mathbf{E}_{y}}|\lesssim\mathbb{M}_{c,\gamma}(T)^{2}\langle t\rangle^{-3/2}$ because

[TABLE]

Since $\mathbf{e_{2}}\cdot\Phi_{c}=r_{c}=O(e^{-2\alpha_{c}|z|})$, $N\cdot\mathbf{e_{1}}=0$ and $E_{1}N^{\prime}(\Psi_{c})=O$,

[TABLE]

In view of $\partial_{y}^{2}\Phi_{c_{0}}(z)=-\gamma_{yy}\partial_{x}\Phi{c_{0}}(z)+(\gamma_{y})^{2}\partial_{x}^{2}\Phi_{c_{0}}(z)$, we have

$\|\operatorname{sech}(\alpha z)\partial_{y}^{2}L_{1}\Phi_{c_{0}}\|_{\mathbf{E}}\lesssim\|\gamma_{yy}\|_{Y}+\|(\gamma_{y})^{2}\|_{Y}$ and

[TABLE]

We can estimate the rest of the terms in a similar manner. ∎

8. Virial identities

In this section, we will prove virial identities for small solutions to (1.1). They give bounds on the transport of energy and the rate of decay of the energy density in the region $x>c_{1}t$ with $c_{1}>1$.

If the initial data is small in the energy space and polynomially localized, we can prove time decay estimates by using virial identities.

Lemma 8.1**.**

Let $\Phi(t)$ be a solution of (2.1). For any $c_{1}>1$, there exist positive constants $\alpha_{0}$ and $\delta$ such that if $\alpha\in(0,\alpha_{0})$ and $\|\Phi(0)\|_{\mathbf{E}}<\delta$, then

[TABLE]

[TABLE]

where $\rho_{1}$ and $\rho_{2}$ are constants satisfying $\rho_{2}>\rho_{1}\geq 0$.

To start with, we observe the energy conservation law of the Benney-Luke equation. Let $\mathcal{E}$ and $\mathcal{F}_{quad}$ be as in Section 4.1 and let

[TABLE]

By a straightforward computation,

[TABLE]

In view of (2.1),

[TABLE]

Combining the above, we have

[TABLE]

To prove Lemma 8.1, we need the following.

Claim 8.2**.**

Suppose that $\Phi=(\phi_{1},\phi_{2})^{T}\in\mathbf{E}$. Then

[TABLE]

Proof.

Let $U(\xi,\eta)=\left((\xi^{2}+\eta^{2})^{1/2}\mathcal{F}_{y}\phi_{1}(\xi,\eta),\mathcal{F}_{y}\phi_{2}(\xi,\eta)\right)^{T}$ and

[TABLE]

By the Plancherel theorem,

[TABLE]

Since $b>a>0$, the eigenvalues $\kappa_{1}(\xi,\eta)$ and $\kappa_{2}(\xi,\eta)$ of the Hermitian matrix $D(\xi,\eta)$ satisfies

[TABLE]

Thus $D(\xi,\eta)$ is non-negative and Claim 8.2 follows. ∎

Now we are in a position to prove Lemma 8.1.

Proof of Lemma 8.1.

Let

[TABLE]

By (8.3),

[TABLE]

Now let $\tilde{p}_{\alpha}=\operatorname{sech}\alpha(x-c_{1}t)$. Then $p_{\alpha}^{\prime}(x-c_{1}t)=\alpha\tilde{p}_{\alpha}^{2}$ and it follows from Claims 8.2 and A.3 and (A.1) that

[TABLE]

Here we use the fact that $\alpha\|u\|_{L^{2}_{\alpha}(\mathbb{R}^{2})}\leq\|\partial_{x}u\|_{L^{2}(\mathbb{R}^{2})}$. Moreover,

[TABLE]

Since $\|\Phi(t)\|_{\mathbf{E}}=\|\Phi(0)\|_{\mathbf{E}}$ by the energy conservation law, we have for $t\geq 0$,

[TABLE]

with $\mu=(c_{1}-1)/2$ provided $\alpha_{0}$ and $\delta_{5}$ are sufficiently small.

Using (8.4), we can prove (8.1) and (8.2) in the same way as [24, Lemma 7.2]. Thus we complete the proof. ∎

Lemma 8.1 and (4.12) give an upper bound of $\mathbb{M}_{1}(T)$.

Lemma 8.3**.**

There exist positive constants $\alpha_{0}$, $\delta_{5}$ and $C$ such that if $\alpha\in(0,\alpha_{0})$ and $\|U_{0}\|_{\mathbf{E}}+\mathbb{M}_{\infty}(T)\leq\delta_{5}$, then $\mathbb{M}_{1}(T)\leq C\left\|\langle x\rangle^{2}\mathcal{E}(U_{0})^{1/2}\right\|_{L^{2}(\mathbb{R}^{2})}$.

9. The decay estimate for the exponentially localized

perturbations

In this section, we will estimate $\mathbb{M}_{2}(T)$ using the exponential linear stability of $e^{t\mathcal{L}_{c_{0}}}\mathcal{Q}_{c_{0}}$ (Theorem 2.1).

Lemma 9.1**.**

Let $\alpha\in(0,\alpha_{c_{0}})$ and $\eta_{0}$ be a sufficiently small positive number. Then there exist positive constants $\delta_{6}$ and $C$ such that if $\|\langle x\rangle^{2}\mathcal{E}(U_{0})^{1/2}\|_{L^{2}}+\mathbb{M}_{c,\gamma}(T)+\mathbb{M}_{\infty}(T)+\mathbb{M}_{2}(T)+\mathbb{M}_{U}(T)\leq\delta_{6}$,

[TABLE]

Proof.

Applying Theorem 2.1 to (4.8), we have

[TABLE]

By (4.4), (4.5) and Claims 7.3 and A.1,

[TABLE]

Since $N_{1}=\{N^{\prime}(\Phi_{c(t,y)}(z)-\Phi_{c_{0}}(x))-N^{\prime}(\Psi_{c(t,y)}(z_{1}))\}U_{2}$,

[TABLE]

By (4.12),

[TABLE]

Since $|\nabla^{j}\Phi_{c(t,y)}(z)|+|\nabla^{j}\Psi_{c(t,y)}(z_{1})|\lesssim e^{-2\alpha z}\wedge 1$ for $j\geq 0$,

[TABLE]

Combining the above, we have for $t\in[0,T]$,

[TABLE]

Since $\|\mathcal{P}_{c_{0}}(\eta_{0})U_{2}\|_{\mathbf{X}_{1}}\lesssim(\mathbb{M}_{c,\gamma}(T)+\mathbb{M}_{\infty}(T))\|U_{2}\|_{\mathbf{X}_{1}}$ by (4.13), we have

[TABLE]

Combining the above with Lemma 8.3, we have (9.1) provided $\delta_{6}$ is sufficiently small. Thus we complete the proof. ∎

10. Large time behavior of the phase shift of line solitary waves

In this section, we will prove Theorem 2.2 and Corollary 2.3. To begin with, we remark that $\mathbb{M}_{c,\gamma}(T)$, $\mathbb{M}_{\infty}(T)$, $\mathbb{M}_{1}(T)$, $\mathbb{M}_{2}(T)$ and $\mathbb{M}_{U}(T)$ remain small for every $T\in[0,\infty]$ provided the initial perturbation $U_{0}$ is sufficiently small. Combining Proposition 4.4 and Lemmas 6.1, 7.1, 8.3 and 9.1, we have the following.

Proposition 10.1**.**

There exist positive constants $\epsilon_{0}$ and $C$ such that if $\epsilon:=\|(1+x^{2}+y^{2})\mathcal{E}(U_{0})^{1/2}\|_{L^{2}(\mathbb{R}^{2})}<\epsilon_{0}$, then $\mathbb{M}_{c,\gamma}(\infty)+\mathbb{M}_{1}(\infty)+\mathbb{M}_{2}(\infty)+\mathbb{M}_{U}(\infty)\leq C\epsilon$.

We see that (2.29)–(2.32) follows immediately from Proposition 10.1. To prove (2.33), we need the first order asymptotics of $\gamma_{y}$ and $\tilde{c}$ as $t\to\infty$. Let $\nu=(a_{11}(c_{0})-a_{22}(c_{0}))/2$ and let $\omega(\eta)$, $\lambda_{\pm}(\eta)$ and $\Pi(\eta)$ be as in Section 3 and $\widetilde{\Pi}(\eta)=\operatorname{diag}(i\eta,1)\Pi(\eta)$. Let

[TABLE]

where $b_{22}(t,\eta)$ is the $(2,2)$ entry of $\mathcal{A}_{1}(t,\eta)$ and $\sigma_{3}=\operatorname{diag}(1,-1)$. Then (6.1) is translated into

[TABLE]

where $\tilde{\omega}(\eta)=\omega(\eta)-1$, $\widetilde{\mathcal{N}}=e^{-\lambda_{1,c_{0}}t\sigma_{3}\partial_{y}}\widetilde{\Pi}(D_{y})^{-1}(n_{1},n_{2})^{T}$ and

[TABLE]

Note that $\operatorname{diag}(\partial_{y},1)\mathcal{N}_{2}=\mathcal{N}_{2}$ and $\operatorname{diag}(\partial_{y},1)\mathcal{N}_{3}=\partial_{y}\mathcal{N}_{3}$ since $E_{2}\mathcal{N}_{2}=\mathcal{N}_{2}$ and $E_{1}\mathcal{N}_{3}=\mathcal{N}_{3}$. We have for $\eta\in[-\eta_{0},\eta_{0}]$,

[TABLE]

If $\eta_{0}$ is sufficiently small, then $\widetilde{\Pi}(D_{y})$ and its inverse belong to $B(Y)$ and it follows from Claim D.1 and the definitions of $\mathbf{b}$ and $\mathbf{d}$ that

[TABLE]

Moreover, we have $\|\mathbf{d}(0)\|_{Y_{1}}+\|\chi(D_{y})\mathbf{d}(0)\|_{L^{1}}\lesssim\epsilon$.

We will investigate the asymptotic behavior of solutions by using the compactness argument in [16]. More precisely, we consider the rescaled solution $\mathbf{d}_{\lambda}(t,y)=\lambda\mathbf{d}(\lambda^{2}t,\lambda y)$ and prove that for any $t_{1}$ and $t_{2}$ satisfying $0<t_{1}<t_{2}<\infty$,

[TABLE]

where $\mathbf{d}_{\infty}(t,y)={}^{t}(d_{\infty,+}(t,y),d_{\infty,-}(t,y))$ and $d_{\infty,\pm}(t,y)$ are self-similar solutions of the Burgers’ equation

[TABLE]

satisfying

[TABLE]

First, we will show that rescaled solutions $\mathbf{d}_{\lambda}$ are uniformly bounded with respect to $\lambda\geq 1$.

Lemma 10.2**.**

Let $\epsilon$ be as in Proposition 10.1. Then there exists a positive constants $C$ such that for any $\lambda\geq 1$ and $t\in(0,\infty)$,

[TABLE]

Proof.

By Proposition 10.1 and (10.4), we have and (10.8). Let

[TABLE]

Then

[TABLE]

and it follows from (H), (5.29)–(5.31) and the fact that $\mathcal{A}_{0}(\eta)-\mathcal{A}_{*}(\eta)=O(\eta^{4})$ that

[TABLE]

Combining (10.10)–(10.13) with (10.8), we have (10.9). ∎

By the standard compactness argument, we have the following.

Corollary 10.3**.**

There exists a sequence $\{\lambda_{n}\}_{n\geq 1}$ satisfying $\lim_{n\to\infty}\lambda_{n}=\infty$ and $\mathbf{d}_{\infty}(t,y)$ such that

[TABLE]

[TABLE]

where $C$ is a constant given in Lemma 10.2. Moreover, for any $R>0$ and $t_{1}$, $t_{2}$ with $0<t_{1}\leq t_{2}<\infty$,

[TABLE]

To prove (10.5), we need the following.

Lemma 10.4**.**

Suppose that $\epsilon$ is sufficiently small. Then for every $t_{1}$ and $t_{2}$ satisfying $0<t_{1}\leq t_{2}<\infty$, there exist a positive constant $C$ and a function $\tilde{\delta}(R)$ satisfying $\lim_{R\to\infty}\tilde{\delta}(R)=0$ such that

[TABLE]

Since we can prove Lemma 10.4 in the same way as [24, Lemma 10.4], we omit the proof.

The initial data of $\mathbf{d}_{\infty}(t,y)$ at $t=0$ is a constant multiple of the delta function. Using the change of variable

[TABLE]

we can transform (10.1) into a conservative system

[TABLE]

where $\widetilde{\mathcal{N}}^{\prime\prime\prime}=\lambda_{2,c_{0}}\bar{\mathbf{d}}+\partial_{y}^{-1}\tilde{\omega}(D_{y})\sigma_{3}\mathbf{d}$.

Lemma 10.5**.**

[TABLE]

Using (10.11)–(10.13), we can prove Lemma 10.5 in the same way as [16]. See also the proof of [24, Lemma 10.3].

Combining Lemmas 10.4 and 10.5, Corollary 10.3 with Claim C.1 and (10.4), we have the following.

Proposition 10.6**.**

Suppose $c_{0}>1$ and that (S) holds for $c=c_{0}$. Let $\Phi(t,x,y)$ be as in Theorem 2.2. Then there exist positive constants $\epsilon_{0}$ and $C$ such that if $\epsilon:=\|(1+x^{2}+y^{2})\mathcal{E}(\Phi_{0})^{1/2}\|_{L^{2}(\mathbb{R}^{2})}<\epsilon_{0}$, then

[TABLE]

as $t\to\infty$, where

[TABLE]

and $m_{\pm}\in(-2,2)$ are constants satisfying

[TABLE]

See e.g. [24, Proof of Theorem 1.4] for the proof.

Now we are in a position to prove (2.33).

Proof of (2.33).

By Lemmas 3.3 and 3.4,

[TABLE]

and

[TABLE]

where $\gamma_{\infty,1}=-(2\lambda_{1,c_{0}})^{-1}\mu(0)\int_{\mathbb{R}}\tilde{k}(0,y)\,dy$. By Proposition 10.1, (6.3), (6.6) and (6.7),

[TABLE]

It follows from Lemmas 3.3–3.5, (5.29) and (6.5) that as $t\to\infty$,

[TABLE]

[TABLE]

for any $\delta>0$, where \gamma_{\infty,2}=(2\lambda_{1,c_{0}})^{-1}\int_{\mathbb{R}_{+}\times\mathbb{R}}\mu(s)\bigl{(}\mathcal{N}_{2}(s)+\overset{\bullet}{\mathcal{N}}_{U_{1}}(s)\bigr{)}\cdot\mathbf{e_{2}}\,dsdy and

[TABLE]

Finally, we will prove

[TABLE]

Let $\mathcal{N}_{1,0}=(n_{1,0},\partial_{y}n_{2,0})^{T}$ and

[TABLE]

By Propositions 10.1 and 10.6,

[TABLE]

where $\delta(t)$ is a function that tends to [math] as $t\to\infty$. Hence it follows from Lemma 3.3 that as $t\to\infty$,

[TABLE]

Moreover, by Lemmas 3.3 and 3.4,

[TABLE]

[TABLE]

Let $\tilde{\delta}(t)$ be a function satisfying $\lim_{t\downarrow 0}\tilde{\delta}(t)=0$. Since $|u^{B}_{\pm}(s,y)|\lesssim H_{\lambda_{2,c_{0}}s}(y)$, we have for $y$ satisfying $|y+\lambda_{1,c_{0}}t|\wedge|y-\lambda_{1,c_{0}}t|\geq 4\{\lambda_{2,c_{0}}t/\tilde{\delta}(t)\}^{1/2}$,

[TABLE]

as $t\to\infty$. Combining the above, we have (10.20). Thus we complete the proof. ∎

Proof of Corollary 2.3.

Let $\zeta\in C_{0}^{\infty}(-\eta_{0},\eta_{0})$ such that $\zeta(0)=1$ and let

[TABLE]

Then $\mathbf{b}(0)=\epsilon(1+\beta_{1}(c_{0})^{-1}S^{3}_{21}(0))^{-1}(\mathcal{F}_{\eta}^{-1}\zeta)(y)\mathbf{e_{2}}$ and

[TABLE]

Combining the above with (10.19), we have $\gamma_{\infty}\gtrsim\epsilon$, where $\gamma_{\infty}$ is a constant in (2.33). Corollary 2.3 follows immediately from (2.33) and the fact that $\gamma_{\infty}\gtrsim\epsilon$. Thus we complete the proof. ∎

Appendix A Miscellaneous estimates for operator norms

Claim A.1**.**

Suppose that $\alpha\in(0,1/\sqrt{b})$. Then

[TABLE]

Proof.

Since $\|f\|_{L^{2}_{\alpha}(\mathbb{R}^{2})}=\|\hat{f}(\xi+i\alpha,\eta)\|_{L^{2}(\mathbb{R}^{2})}$, we can prove (A.1) and (A.2) by using the Plancherel theorem (see [25]). We can obtain (A.3) and (A.4) by using (A.2). ∎

Claim A.2**.**

Let $\alpha\in(0,1/\sqrt{b})$ and $1<q<2$. Then there exists a positive constant $C$ such that for any $\gamma\in\mathbb{R}$,

[TABLE]

Proof.

The green kernel $K(x,y)$ of the operator $B$ satisfies that for $(x,y)\neq(0,0)$,

[TABLE]

Let $\widetilde{K}(x,y,x_{1},y_{1})=e^{\alpha(x-x_{1})}K(x-x_{1},y-y_{1})$. Since

[TABLE]

and $\sup_{x,y}\|\widetilde{K}(x,y,x_{1},y_{1})\|_{(L^{1}\cap L^{2})(\mathbb{R}^{2}_{x_{1},y_{1}})}+\sup_{x_{1},y_{1}}\|\widetilde{K}(x,y,x_{1},y_{1})\|_{(L^{1}\cap L^{2})(\mathbb{R}^{2}_{x,y})}<\infty$ for $\alpha\in(0,1/\sqrt{b})$, we have (A.5). We can prove the rest in the same way. ∎

Claim A.3**.**

Let $\alpha\in(0,1/\sqrt{b})$. There exists a positive constant $C$ such that for any $\gamma\in\mathbb{R}$,

[TABLE]

Proof.

Using (A.9), we can prove Claim A.3 in the same way as [25, Claim 11.2]. ∎

Appendix B Operator norms of $S^{j}_{k}$

Claim B.1**.**

Let $\alpha\in(0,\alpha_{c_{0}})$. There exist positive constants $\eta_{1}$ and $C$ such that for $\eta_{0}\in(0,\eta_{1}]$, $j\in\mathbb{Z}_{\geq 0}$, $k=1$, $2$, $t\geq 0$ and $f\in L^{2}(\mathbb{R})$,

[TABLE]

Claim B.2**.**

Let $\alpha\in(0,\alpha_{c_{0}})$. There exist positive constants $\eta_{1}$, $\delta$ and $C$ such that if $\eta_{0}\in(0,\eta_{1}]$ and $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $k=1$, $2$, $t\in[0,T]$ and $f\in L^{2}(\mathbb{R})$,

[TABLE]

[TABLE]

[TABLE]

We can prove Claims B.1 and B.2 in exactly the same way as [22, Claims B.1 and B.2].

Claim B.3**.**

Let $\alpha\in(0,\alpha_{c_{0}})$. There exist positive constants $C$ and $\eta_{1}$ such that for $\eta\in(0,\eta_{1}]$, $h\geq 0$, $k=1$, $2$ and $t\geq 0$,

[TABLE]

Claim B.4**.**

There exist positive constants $\eta_{1}$, $\delta$ and $C$ such that if $\eta_{0}\in(0,\eta_{1}]$ and $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $k=1$, $2$, $t\in[0,T]$, $h\geq 0$ and $f\in L^{2}$,

[TABLE]

Let $S^{j}_{k}=S^{j}_{k1}-\partial_{y}^{2}S^{j}_{k2}$ for $j=5$, $6$ and

[TABLE]

Claim B.5**.**

There exist positive constants $\eta_{1}$, $\delta$ and $C$ such that if $\eta_{0}\in(0,\eta_{1}]$ and $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $k=1$, $2$, $t\in[0,T]$ and $f\in L^{2}$,

[TABLE]

We can prove Claims B.3–B.5 in the same way as [24, Claims A.1–A.3].

Appendix C Estimates of $R^{j}$

To start with, we estimate the difference between $b$ and $\tilde{c}$ and the operator norms of $\delta B(c)$.

Claim C.1**.**

There exist positive constants $\delta$ and $C$ such that if $\sup_{t\in[0,T]}\|\tilde{c}(t)\|_{Y}\leq\delta$, then for $t\in[0,T]$,

[TABLE]

Proof.

Since $b-\tilde{c}=\widetilde{P}_{1}\left(\rho(c)-\rho(c_{0})-\rho^{\prime}(c_{0})\tilde{c}\right)$, we can prove Claim C.1 in exactly the same way as [22, Claim D.6]. ∎

Claim C.2**.**

There exist positive constants $C$ and $\delta$ such that if $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $t\in[0,T]$,

[TABLE]

Proof.

The first two estimates follow immediately from the definition of $B_{4}$. To prove the last two estimates, we use (5.18), the fact that $\partial_{y}^{-1}(I-\widetilde{P}_{1})\in B(Y)$ and

[TABLE]

∎

Using Claims C.1 and C.2, we have the following.

Claim C.3**.**

Let $R^{G,1}_{2,1}=\tilde{a}_{22}^{\prime}(c)(c_{y})^{2}$, $R^{G,1}_{2,2}=\left(\tilde{a}_{22}(c)-\tilde{a}_{22}(c_{0})\right)c_{y}-\tilde{a}_{22}(c_{0})(b_{y}-c_{y})$. Then $R^{G,1}_{2}=\partial_{y}R^{G,1}_{2,2}-R^{G,1}_{2,1}$ and there exist positive constants $\delta$ and $C$ such that if $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $t\in[0,T]$,

[TABLE]

To prove the last estimate, we use the fact that $B_{4}(c)$ is an upper triangular matrix.

Let

[TABLE]

Then $R^{\ell_{1},r}_{k}=R^{\ell_{1},r}_{k1}+\partial_{y}R^{\ell_{1},r}_{k2}-\partial_{y}^{2}R^{\ell_{1},r}_{k3}$.

Claim C.4**.**

There exist positive constants $\delta$ and $C$ such that if $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $t\in[0,T]$,

[TABLE]

Proof.

We see that (C.1) immediately follows from Claims B.1, B.2 and 7.3. Since

[TABLE]

we have (C.2) and (C.3). ∎

Claim C.5**.**

Let $\alpha\in(0,\alpha_{c_{0}})$. There exist positive constants $C$, $\delta$, $\eta_{1}$ and $h_{0}$ such that such that if $\eta\in(0,\eta_{1}]$, $\mathbb{M}_{c,\gamma}(T)\leq\delta$ and $h\geq h_{0}$, then for $k=1$, $2$ and $t\in[0,T]$,

[TABLE]

[TABLE]

Proof.

By Claims B.3 and B.4,

[TABLE]

Combining the above with Claims 7.3, C.1 and

[TABLE]

we have Claim C.5. ∎

Next, we will estimate $R^{j}_{k}$ ($0\leq j\leq 7$).

Claim C.6**.**

There exist $R^{0}_{kj}$ ($k=1$, $2$, $j=0$, $1$, $2$) satisfying $R^{0}_{k}=R^{0}_{k0}-2\partial_{y}R^{0}_{k1}-\partial_{y}^{2}R^{0}_{k2}$ and positive constants $C$ and $\delta$ such that if $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $t\in[0,T]$,

[TABLE]

Proof.

By (2.21), (2.22) and (4.13),

[TABLE]

Let $L_{12}(\eta)=B^{-1}B(\eta)^{-1}\{(b-a)\Delta+BA(\eta)\}E_{12}-bV_{c(t,y)}^{*}B(\eta)^{-1}$,

[TABLE]

[TABLE]

and $R^{0}_{k2}=\partial_{y}^{-2}(R^{0}_{k}-R^{0}_{k0}+2\partial_{y}R^{0}_{k1})$. Since

[TABLE]

we have $\|R^{0}_{k1}\|_{Y_{1}}\lesssim\|\widetilde{U}_{2}\|_{\mathbf{X}}(\|\gamma_{y}\|_{Y}+\|c_{y}\|_{Y})$ and

[TABLE]

Thus we complete the proof. ∎

Claim C.7**.**

There exist positive constants $C$, $\delta$, $h_{0}$ and $R^{1}_{k,j}$ $(j=1,\,2)$ satisfying

[TABLE]

such that if $\mathbb{M}_{c,\gamma}(T)+\mathbb{M}_{\infty}(T)\leq\delta$ and $h\geq h_{0}$, then for $t\in[0,T]$,

[TABLE]

where $R^{1}_{k3}=\int_{-\eta_{0}}^{\eta_{0}}\int_{\mathbb{R}^{2}}\widetilde{N}_{1}\cdot g_{k1}^{*}(z,\eta,c(t,y))e^{-iy\eta}\,dzdy$.

We remark that $R^{1}_{k}=R^{1}_{k1}+\partial_{y}R^{1}_{k2}-\partial_{y}^{2}R^{1}_{k3}$.

Proof of Claim C.7.

We have

[TABLE]

and for example,

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We can estimate the rest in the same way. ∎

Claim C.8**.**

Let

[TABLE]

Then $R^{2}_{k}=R^{2}_{k1}-\partial_{y}^{2}R^{2}_{k2}$ and there exist positive constants $C$ and $\delta$ such that if $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $t\in[0,T]$,

[TABLE]

By the definition of $R^{3}$ and Lemma 8.3, we have

[TABLE]

More precisely, we have the following.

Claim C.9**.**

Let $\alpha\in(0,\alpha_{c_{0}})$. There exist positive constants $C$, $\delta$ and $R^{3}_{2j}$ $(j=1,2)$ satisfying

[TABLE]

such that if $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $t\in[0,T]$,

[TABLE]

Proof.

Let

[TABLE]

Then $R^{3}_{k}=R^{U_{1},1}_{k}-\partial_{y}^{2}R^{U_{1},2}_{k}$ and

[TABLE]

Obviously, we cannot expect that $\|R^{U_{1},0}_{2}\|_{Y_{1}}$ decays as $t\to\infty$. The bad part of $R^{U_{1},0}_{2}$ can be extracted as spatial and time derivatives of $L^{2}$ functions. In fact, we have $R^{U_{1},0}_{2}=R^{U_{1}}_{a}+\partial_{y}^{2}R^{U_{1}}_{b}$ with $\|R^{U_{1}}_{b}\|_{Y}\lesssim\mathbb{M}_{1}(T)\langle t\rangle^{-2}$ and

[TABLE]

As in [23], we can decompose $R^{U_{1}}_{a}$ into a sum of an integrable function and a time derivative of an $L^{2}$-function. Since $\mathcal{L}_{c_{0}}(0)^{*}\zeta_{2,c_{0}}^{*}=0$, it follows from (5.1) that

[TABLE]

where $R^{U_{1}}_{c}=\langle L_{1}\widetilde{U}_{1},\zeta_{2,c_{0}}^{*}\rangle$, $R^{U_{1}}_{d}=\langle N(U_{1}),\zeta_{2,c_{0}}^{*}\rangle+\left\langle\widetilde{U}_{1},(\gamma_{t}\partial_{z}-[L,\tau_{\gamma}]\tau_{-\gamma})\zeta_{2,c_{0}}^{*}\right\rangle$ and

[TABLE]

Thus we complete the proof. ∎

Let $R^{4}_{kj}=S_{kj}^{5}(c_{t})-S_{kj}^{6}(\gamma_{t}-\tilde{c})$ for $j=1$, $2$. Then $R^{4}_{k}=R^{4}_{k1}-\partial_{y}^{2}R^{4}_{k2}$. By Claims B.4, B.5 and 7.3, we have the following.

Claim C.10**.**

Let $\alpha\in(0,\alpha_{c_{0}})$. There exist positive constants $C$ and $\delta$ such that if $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $k=1$, $2$ and $t\in[0,T]$,

[TABLE]

Claim C.11**.**

Let $\alpha\in(0,\alpha_{c_{0}})$. There exist positive constants $\eta_{1}$, $C$, $\delta$ and $R^{5}_{j}$ $(j=1,\,2)$ such that if $\eta_{0}\in(0,\eta_{1}]$ and $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $k=1$, $2$ and $t\in[0,T]$, $R^{5}_{k}=R^{5}_{k1}+\partial_{y}R^{5}_{k2}$ and

[TABLE]

Proof.

Since $L^{*}=B^{-1}A\Delta E_{12}+E_{21}$ and $[\partial_{x},\tau_{-\gamma(t,y)}]=0$,

[TABLE]

Combining the above with $\tau_{-\gamma(t,y)}[\partial_{y}^{2},\tau_{\gamma(t,y)}]=(\gamma_{y})^{2}\partial_{z}^{2}+\gamma_{yy}\partial_{z}+2\gamma_{y}\partial_{zy}^{2}$, we can find $R^{5}_{k1}$ and $R^{5}_{k2}$ satisfying $R^{5}_{k}=R^{5}_{k1}+\partial_{y}R^{5}_{k2}$ and (C.9). ∎

Claim C.12**.**

Let $R^{6,1}=[\partial_{y}^{2},B_{4}(c)]r_{6}$ and $R^{6,2}=(B_{4}(c)-B_{4}(c_{0}))r_{6}$. Then $R^{6}=R^{6,1}+\partial_{y}^{2}R^{6,2}$ and there exist positive constants $C$ and $\delta$ such that if $\mathbb{M}_{c,\gamma}(T)\leq\delta$, then for $t\in[0,T]$,

[TABLE]

Proof.

Since $\|r_{6}\|_{Y}\lesssim\mathbb{M}_{c,\gamma}(T)\left(\mathbb{M}_{c,\gamma}(T)+\mathbb{M}_{1}(T)+\mathbb{M}_{2}(T)^{2}\right)\langle t\rangle^{-3/4}$ by ClaimsB.1, 7.3 and C.1, we have Claim C.12. ∎

Claim C.13**.**

There exist positive constants $C$, $\delta$ and $h_{0}$ such that if $\mathbb{M}_{c,\gamma}(T)\leq\delta$ and $h\geq h_{0}$, then for $t\in[0,T]$,

[TABLE]

Claim C.13 follows immediately from Claims B.3, 7.3, C.1 and C.2.

Appendix D Estimates for $k(t,y)$

By Lemma 8.3, the $L^{2}$-norm of $k(t,y)$ decays like $t^{-2}$ as $t\to\infty$.

Claim D.1**.**

Suppose that $\sup_{y}|\gamma(t,y)|\leq 1$ for $t\in[0,T]$. There exist positive constants $\delta$ and $C$ such that if $\|\langle x\rangle^{2}\mathcal{E}(U_{0})^{1/2}\|_{L^{2}}<\delta$, then

[TABLE]

Next, we will give an upper bound of the growth rate of $\|k(t,y)\|_{L^{1}}$ when $U_{0}(x,y)$ is polynomially localized in $\mathbb{R}^{2}$.

Claim D.2**.**

Suppose $\sup_{y}|\gamma(t,y)|\leq 1$ for $t\in[0,T]$ and that $U_{1}$ is a solution of (4.6). Then there exists a positive constant $C$ such that

[TABLE]

Proof.

Since $U_{1}$ is a solution of (4.6), it follows from (8.3) that

[TABLE]

Combining the above with (4.12), we have

[TABLE]

Since $\|\langle y\rangle k(t,y)\|_{L^{2}(\mathbb{R}_{y})}\lesssim\|\langle y\rangle\mathcal{E}(U_{1})^{1/2}\|_{L^{2}(\mathbb{R}^{2})}$, we have Claim D.2. ∎

Acknowledgment

T. M. is supported by JSPS KAKENHI Grant Number 17K05332. Y. S. would like to express his gratitude to Institute of Mathematics, Academia Sinica in Taiwan where he worked on this research.