# On the NLS approximation for the nonlinear Klein-Gordon equation

Seokchang Hong  Department of Mathematics, Chung-Ang University, Seoul 06974,
Korea<EMAIL_ADDRESS>and  Younghun Hong  Department of Mathematics,
Chung-Ang University, Seoul 06974, Korea<EMAIL_ADDRESS>
###### Abstract.

In this paper, developing a new approach based on Fourier analysis methods for
dispersive PDEs, we establish a low regularity NLS approximation for the one-
dimensional cubic Klein-Gordon equation. Our main result includes energy class
solutions which are formally asymptotically in $L^{2}(\mathbb{R})$. A precise
rate of convergence is also obtained assuming more regularity.

##  1\. Introduction

The nonlinear Schrödinger equation (NLS) is a universal model describing the
envelope dynamics of slowly modulating small amplitude wave packets. By multi-
scaling analysis, the NLS is derived from various Hamiltonian systems,
including the the Korteweg-de Vries equation [19, 21], the Boussinesq equation
[7], water wave problems [32, 10, 4, 24, 25, 26, 3, 31, 17, 30, 8, 12, 6],
nonlinear optics [5] and discrete models [22, 20, 14, 11]. For extensive
references on the universality of the NLS, we refer to the books by Schneider
and Uecker [23] and by Sulem and Sulem [27].

The purpose of this article is to develop a new approach to justify the NLS
approximation based on Fourier analysis methods for dispersive PDEs. In this
way, we aim to include a larger class of solutions compared to the previously
known dynamical system approach as well as exploring a possibility of
extending the interval of approximation using the conservation laws.

To make our discussion concrete, we restrict ourselves to the one-dimensional
nonlinear Klein-Gordon equation (NLKG), that is, the simplest expository
textbook example [23, Chapter 11], given by

$\partial_{t}^{2}u-\partial_{x}^{2}u+u+u^{3}=0,$ (1.1)

where $u=u(t,x):\mathbb{R}\times\mathbb{R}\to\mathbb{R}$. For a small
parameter $\epsilon>0$, we seek a solution of the form

$u_{\epsilon}(t,x)=\epsilon
A_{\epsilon}(\epsilon^{2}t,\epsilon(x-c_{g}t))e^{i(kx-\omega t)}+c.c.,$ (1.2)

where $A_{\epsilon}=A_{\epsilon}(t,x):\mathbb{R}\times\mathbb{R}\to\mathbb{C}$
is the amplitude function and c.c. stands for the complex conjugate of the
former term. Here, the dispersion relation and the group velocity are chosen
respectively as

$\omega=\langle k\rangle\quad\textup{and}\quad
c_{g}=\omega^{\prime}(k)=\frac{k}{\langle k\rangle},$

where $\langle\cdot\rangle=\sqrt{1+\cdot^{2}}$ denotes the Japanese bracket,
to cancel out $O(\epsilon)$\- and $O(\epsilon^{2})$-terms. Then, from the next
$O(\epsilon^{3})$-order terms, the cubic NLS

$2i\omega\partial_{t}\psi^{\textup{(NLS)}}+(1-c_{g}^{2})\partial_{x}^{2}\psi^{\textup{(NLS)}}-3|\psi^{\textup{(NLS)}}|^{2}\psi^{\textup{(NLS)}}=0,$
(1.3)

where
$\psi^{\textup{(NLS)}}=\psi^{\textup{(NLS)}}(t,x):\mathbb{R}\times\mathbb{R}\to\mathbb{C}$,
is derived so that the sum of two wave packets

$\epsilon\psi^{\textup{(NLS)}}(\epsilon^{2}t,\epsilon(x-c_{g}t))e^{i(kx-\omega
t)}+c.c.$

approximates the NLKG flow. For a more detailed formal derivation and a
rigorous proof for sufficiently regular flows, we refer to [23, Chapter 11].

In this paper, we include a larger class of solutions, namely the energy
class, for the NLS approximation. Note that solutions to the NLKG (1.1)
preserve the energy

$\mathcal{E}(u)=\frac{1}{2}\left(\|\partial_{x}u\|_{L^{2}(\mathbb{R})}^{2}+\|\partial_{t}u\|_{L^{2}(\mathbb{R})}^{2}+\|u\|_{L^{2}(\mathbb{R})}^{2}\right)+\frac{1}{4}\|u\|_{L^{4}(\mathbb{R})}^{4}.$

By the $\epsilon$-scaling in the ansatz (1.2), it is natural to introduce the
rescaled Sobolev space $H_{\epsilon}^{s}(\mathbb{R})$, for $s\in\mathbb{R}$
and $\epsilon\in(0,1]$, equipped with the norm

$\|u\|_{H_{\epsilon}^{s}(\mathbb{R})}=\|\langle\epsilon
D\rangle^{s}u\|_{L^{2}(\mathbb{R})},$

where $D=-i\partial_{x}$, and define $m(D)$ as the Fourier multiplier operator
with symbol $m(\xi)$, i.e., $\widehat{m(D)f}(\xi)=m(\xi)\hat{f}(\xi)$. In
particular, we call $H_{\epsilon}^{1}(\mathbb{R})$ the energy class.

Throughout the article, we assume that initial data satisfies

$\sup_{\epsilon\in(0,1]}\|\psi_{\epsilon,0}\|_{H_{\epsilon}^{1}(\mathbb{R})}\leq
R<\infty$ (H1)

and

$\lim_{\epsilon\to
0}\|\mathbf{1}_{|D|>\delta\epsilon^{-1/3}}\psi_{\epsilon,0}\|_{H_{\epsilon}^{1}(\mathbb{R})}=0\quad\textup{for
any }\delta>0,$ (H2)

where $\mathbf{1}_{|D|>N}$ is a high frequency cut-off with symbol
$\mathbf{1}_{|\xi|>N}$.

######  Remark 1.1.

$(i)$ Some regularity must be imposed on initial data, because the NLKG (1.1)
is well-posed in $H^{s}(\mathbb{R})$ only if $s\geq\frac{1}{2}$ (see [15,
Appendix D]). In (H1), the regularity $s$ is chosen to be one, but we may say
it is asymptotically zero due to the formal norm convergence
$\|\cdot\|_{H_{\epsilon}^{1}(\mathbb{R})}\to\|\cdot\|_{L^{2}(\mathbb{R})}$ as
$\epsilon\to 0$ even though $H_{\epsilon}^{1}(\mathbb{R})=H^{1}(\mathbb{R})$
as sets.
$(ii)$ The energy class solutions are considered with a hope to find potential
applications of the energy conservation law, for instance, extending the
interval of approximation. Certainly, there is a room to reduce the regularity
further in the assumption (H1), but such low regularities will not be pursued
here to avoid additional technical complications.
$(iii)$ (H2) ensures that initial data $\epsilon\psi_{0}(\epsilon
x)e^{ikx}+c.c.$ is tightly localized at two frequencies $\pm k$, with $|\xi\mp
k|\ll\epsilon^{-\frac{2}{3}}$, and thus the two-wave packet structure (1.2)
remains for long time.

Our first main result asserts that the NLS approximation is valid under the
above two assumptions.

######  Theorem 1.2 (NLS approximation to the NLKG).

We assume (H1)-(H2) for $\\{\psi_{\epsilon,0}\\}_{\epsilon\in(0,1]}$. For
$\epsilon\in(0,1]$, let $u_{\epsilon}(t)\in
C_{t}(\mathbb{R};H^{1}(\mathbb{R}))$ be the solution to the NLKG (1.1) with
initial data

$\big{(}u_{\epsilon}(0),\partial_{t}u_{\epsilon}(0)\big{)}=\big{(}\epsilon\psi_{\epsilon,0}(\epsilon\cdot)e^{ikx}+c.c.,\langle
D\rangle(-i\epsilon\psi_{\epsilon,0}(\epsilon\cdot)e^{ikx}+c.c.)\big{)}\in
H^{1}(\mathbb{R})\times L^{2}(\mathbb{R}),$ (1.4)

and let $\psi_{\epsilon}^{(\textup{NLS})}(t)\in
C_{t}(\mathbb{R};H_{\epsilon}^{1}(\mathbb{R}))$ be the solution to the NLS
(1.3) with initial data $\psi_{\epsilon,0}\in H_{\epsilon}^{1}(\mathbb{R})$.
Then, there exists $T>0$, independent of $\epsilon\in(0,1]$, such that

$\lim_{\epsilon\to
0}\frac{1}{\sqrt{\epsilon}}\left\|u_{\epsilon}(t,x)-\left(\epsilon\psi_{\epsilon}^{(\textup{NLS})}(\epsilon^{2}t,\epsilon(x-c_{g}t))e^{i(kx-\omega
t)}+c.c.\right)\right\|_{C_{t}([-\frac{T}{\epsilon^{2}},\frac{T}{\epsilon^{2}}];H^{1}(\mathbb{R}))}=0.$
(1.5)

######  Remark 1.3.

$(i)$ The convergence (1.5) is well-known for higher regularity solutions. By
a dynamical system approach, it is shown provided that the NLS flow
$\psi^{(\textup{NLS})}(t)$ is in $C_{t}([-T,T];H^{s}(\mathbb{R}))$ for $s\geq
5$ [23, Theorem 11.2.6]. In [18], the regularity is reduced to $s>1$. In our
main theorem, the required regularity is reduced to one, but which is formally
asymptotically zero (see Remark 1.1 $(i)$).
$(ii)$ Higher-order corrections will not be discussed in this paper, because
our main focus is on including rough solutions while higher-order corrections
are typically valid for more regular solutions (see [23, Theorem 11.2.6]).

######  Remark 1.4 (Optimality).

The assumptions (H1)-(H2) are optimal in the sense that the linear part of the
reformulated equation can be approximated by the linear Schrödinger flow only
under (H1)-(H2) (see Lemma 3.1 and Remark 3.2).

######  Remark 1.5 (Rate of convergence in (1.5)).

$(i)$ By the $\epsilon$-scaling, both the NLKG flow $u_{\epsilon}(t)$ and the
NLS ansatz
$\epsilon\psi_{\epsilon}^{(\textup{NLS})}(\epsilon^{2}t,\epsilon(x-c_{g}t))e^{i(kx-\omega
t)}+c.c.$ are of $O(\sqrt{\omega})$ in $H^{1}(\mathbb{R})$. The main result
(1.5) justifies the NLS approximation with $o(\sqrt{\omega})$-difference.
$(ii)$ It does not seem possible to improve the $o(\sqrt{\omega})$-difference
in (1.5) for general solutions in Theorem 1.2, because our proof relies on a
density argument (see Remark 3.2).

The next theorem provides a precise rate of convergence assuming more
regularity.

######  Theorem 1.6 (NLS approximation to the NLKG; rate of convergence).

In Theorem 1.2, we further assume that for some $s>0$,

$\sup_{\epsilon\in(0,1]}\|\psi_{\epsilon,0}\|_{H^{s}(\mathbb{R})}<\infty.$
(H2’)

Then, for any small $\eta>0$, we have

$\left\|u_{\epsilon}(t,x)-\left(\epsilon\psi_{\epsilon}^{(\textup{NLS})}(\epsilon^{2}t,\epsilon(x-c_{g}t))e^{i(kx-\omega
t)}+c.c.\right)\right\|_{C_{t}([-\frac{T}{\epsilon^{2}},\frac{T}{\epsilon^{2}}];L^{2}(\mathbb{R}))}\lesssim\epsilon^{\min\\{\frac{s}{3}+\frac{1}{2},\frac{3}{2}-\eta\\}}.$

######  Remark 1.7.

$(i)$ (H2’) is a stronger assumption, because (H2’) implies (H2).
$(ii)$ From the linear flow approximation, one can see that the
$O(\epsilon^{\frac{s}{3}+\frac{1}{2}})$-rate of convergence is optimal (see
Remark 3.2).

The main contribution of this paper is to introduce a new approach for the NLS
approximation which we think is robust. Our approach is based on the
reformulation of the problem as a system of integral equations (2.4), or the
Duhamel representation (see Section 2). It turns out that this integral
representation has several crucial advantages in reducing regularity. First of
all, we note that in a dynamical system approach [23, Section 11.2], it
involves estimating the residual

$\textup{Res}(u)=-\partial_{t}^{2}u+\partial_{x}^{2}u-u-u^{3},\quad
u(t,x)=\epsilon\psi^{(\textup{NLS})}(\epsilon^{2}t,\epsilon(x-c_{g}t))e^{i(kx-\omega
t)}+c.c.,$

where $\psi^{(\textup{NLS})}$ is a solution to the NLS (1.3). Thus, proving
smallness of the residual in $H^{1}(\mathbb{R})$ requires high Sobolev norm
bounds, $\|\psi^{(\textup{NLS})}\|_{H^{5}(\mathbb{R})}<\infty$. However, such
derivative terms do not appear in the integral equation (2.4).

Secondly, the reformulation provides more detailed information about the limit
procedure. Indeed, derivation of the integral equations (2.4) leads us to
notice that the amplitude $A_{\epsilon}(t)$ in (1.2) includes very high
frequency waves, and thus the amplitude must be separated into the core
profile $\psi_{\epsilon}$ and the high frequency remainder $r_{\epsilon}$ (see
Remark 2.1). Consequently, they must be measured separately in Sobolev norms
with different scales.

Lastly, we mention that from the integral representation (2.4), dispersive
effects can be captured properly. By the reformulation, we see that the linear
evolution in (2.4) is given by the propagator
$S_{\epsilon}(t)=e^{-\frac{it}{\epsilon^{2}}(\langle 1+\epsilon
D\rangle-\sqrt{2}-\frac{\epsilon}{\sqrt{2}}D)}$. Then, employing well-known
Fourier analysis methods as Strichartz estimates and multilinear estimates
involving Fourier restriction norms, one can deduce uniform bounds for
nonlinear solutions which are useful to prove the NLS approximation.

######  Remark 1.8.

The NLS is derived from the NLKG in a different context, namely as a non-
relativistic limit [16], but it does not have the technical issue caused by
the high frequency remainder.

######  Remark 1.9.

An interesting question is whether the $O(\frac{1}{\epsilon^{2}})$-interval of
approximation can be extended. In [9], a positive answer is given for more
complicated quadratic NLKG in a periodic setting but with more regular
solutions. As an alternative approach, one may attempt to use the energy
conservation law to extend the interval of approximation. Nevertheless, we are
currently unable to do that. Indeed, our proof heavily relies on the two-wave
packet structure $\epsilon\psi_{\epsilon,0}(\epsilon\cdot)e^{ikx}+c.c.$ at
initial time, but the structure will be broken up as time goes, because the
nonlinearity immediately generates different frequency modes. Unfortunately,
the energy conservation law does not seem to control this procedure
straightforwardly. Therefore, it will be left to our future work.

###  1.1. Organization of the paper

In Section 2, we derive the system of integral equations (2.4). In Section 3
and 4, we investigate the properties of the linear part of (2.4) and use them
to prove basic well-posedness of the system. Then, in Section 5 and 6, we
prove more refined estimates for the core profile (smallness of high
frequencies) and the remainder (smallness). Finally, in Section 7, we complete
the proof of the main results.

###  1.2. Acknowledgement

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (NRF-2020R1A2C4002615).

##  2\. Reformulation of the problem

To begin with, we present a reformulation of the cubic nonlinear Klein-Gordon
equation (1.1) in a way that Fourier analysis methods can be properly
employed.

###  2.1. Derivation of the equation for the amplitude function

For numerical simplicity, we fix $k=1$ by scaling, and look for a solution of
the form

$u_{\epsilon}(t,x)=\epsilon
A_{\epsilon}(\epsilon^{2}t,\epsilon(x-\tfrac{t}{\sqrt{2}}))e^{i(x-\sqrt{2}t)}+c.c..$

In order to find the equation for the amplitude $A_{\epsilon}(t,x)$, rescaling
by
$v_{\epsilon}(t,x)=\frac{1}{\epsilon}u_{\epsilon}(\frac{t}{\epsilon^{2}},\frac{x}{\epsilon})$,
we deduce

$\partial_{t}^{2}v_{\epsilon}+\frac{1}{\epsilon^{4}}(1-\epsilon^{2}\partial_{x}^{2})v_{\epsilon}-\frac{1}{\epsilon^{2}}v_{\epsilon}^{3}=0$
(2.1)

with initial data
$(v_{\epsilon}(0),\partial_{t}v_{\epsilon}(0)\big{)}=\big{(}2\textup{Re}(e^{\frac{ix}{\epsilon}}\psi_{\epsilon,0}),\tfrac{2}{\epsilon^{2}}\langle\epsilon\partial_{x}\rangle\textup{Im}(e^{\frac{ix}{\epsilon}}\psi_{\epsilon,0}))$.
We note that the equation (2.1) in a strong form is given by

$\displaystyle v_{\epsilon}(t)$
$\displaystyle=\left\\{e^{-\frac{it}{\epsilon^{2}}\langle\epsilon
D\rangle}e^{\frac{ix}{\epsilon}}\psi_{\epsilon,0}-\frac{i}{2}\int_{0}^{t}e^{-\frac{i(t-t_{1})}{\epsilon^{2}}\langle\epsilon
D\rangle}\frac{1}{\langle\epsilon
D\rangle}v_{\epsilon}(t_{1})^{3}dt_{1}\right\\}+c.c.$

with

$v_{\epsilon}(t,x)=e^{-\frac{itD}{\epsilon\sqrt{2}}}e^{\frac{i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}A_{\epsilon}(t,x)+c.c..$

Next, complexifying the equation, we impose that the amplitude function
$A_{\epsilon}(t)$ obeys the equation

$\displaystyle A_{\epsilon}(t)$
$\displaystyle=e^{-\frac{i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}e^{-\frac{it}{\epsilon^{2}}(\langle\epsilon
D\rangle-\frac{\epsilon
D}{\sqrt{2}})}e^{\frac{ix}{\epsilon}}\psi_{\epsilon,0}$
$\displaystyle\quad-\frac{i}{2}\int_{0}^{t}e^{-\frac{i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}e^{-\frac{i(t-t_{1})}{\epsilon^{2}}(\langle\epsilon
D\rangle-\frac{\epsilon D}{\sqrt{2}})}\frac{1}{\langle\epsilon
D\rangle}\left(e^{\frac{i}{\epsilon}(x-\frac{t_{1}}{\epsilon\sqrt{2}})}A_{\epsilon}(t_{1})+c.c.\right)^{3}dt_{1}.$

We observe that
$e^{-\frac{i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}e^{-\frac{it}{\epsilon^{2}}(\langle\epsilon
D\rangle-\frac{\epsilon D}{\sqrt{2}})}e^{\frac{ix}{\epsilon}}$ is the Fourier
multiplier of symbol $e^{-itp_{\epsilon}(\xi)}$, where

$p_{\epsilon}(\xi):=\frac{1}{\epsilon^{2}}\left(\langle
1+\epsilon\xi\rangle-\sqrt{2}-\frac{\epsilon}{\sqrt{2}}\xi\right),$ (2.2)

and similarly,
$e^{-\frac{i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}e^{-\frac{i(t-t_{1})}{\epsilon^{2}}(\langle\epsilon
D\rangle-\frac{\epsilon D}{\sqrt{2}})}\frac{1}{\langle\epsilon
D\rangle}e^{\frac{i}{\epsilon}(x-\frac{t_{1}}{\epsilon\sqrt{2}})}$ has symbol
$\frac{e^{-i(t-t_{1})p_{\epsilon}(\xi)}}{\langle 1+\epsilon\xi\rangle}$. Thus,
introducing the linear propagator

$S_{\epsilon}(t)=e^{-itp_{\epsilon}(D)},$

the equation can be written as

$\displaystyle A_{\epsilon}(t)$
$\displaystyle=S_{\epsilon}(t)\psi_{\epsilon,0}-\frac{i}{2}\int_{0}^{t}S_{\epsilon}(t-t_{1})\frac{1}{\langle
1+\epsilon
D\rangle}e^{-\frac{i}{\epsilon}(x-\frac{t_{1}}{\epsilon\sqrt{2}})}\left(A_{\epsilon}(t_{1})e^{\frac{i}{\epsilon}(x-\frac{t_{1}}{\epsilon\sqrt{2}})}+c.c.\right)^{3}dt_{1}.$

Finally, expanding the nonlinear term as

$e^{-\frac{i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}\left(A_{\epsilon}e^{\frac{i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}+c.c.\right)^{3}=3\big{(}|A_{\epsilon}|^{2}A_{\epsilon}+\mathcal{R}_{\epsilon}(A_{\epsilon})\big{)},$

where

$\mathcal{R}_{\epsilon}(A_{\epsilon}):=\frac{1}{3}e^{\frac{2i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}A_{\epsilon}^{3}+e^{-\frac{2i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}|A_{\epsilon}|^{2}\bar{A}_{\epsilon}+\frac{1}{3}e^{-\frac{4i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}\bar{A}_{\epsilon}^{3},$

we obtain the amplitude equation in a compact form

$A_{\epsilon}(t)=S_{\epsilon}(t)\psi_{\epsilon,0}-\frac{3i}{2}\int_{0}^{t}S_{\epsilon}(t-t_{1})\frac{1}{\langle
1+\epsilon
D\rangle}\left(|A_{\epsilon}|^{2}A_{\epsilon}+\mathcal{R}_{\epsilon}(A_{\epsilon})\right)(t_{1})dt_{1}.$
(2.3)

###  2.2. Decomposition of the amplitude equation

It turns out, however, that the equation (2.3) by itself is rather difficult
to analyze, because on the frequency side, the solution consists of a large
localized part and small very high frequency pieces.

######  Remark 2.1.

The solution $A_{\epsilon}$ to (2.3) is never localized in frequencies of
order one. Indeed, even if $\hat{A}_{\epsilon}(t,\xi)$ is mostly localized
near the origin at time $t=0$, the nonlinear term

$-\frac{i}{2}\int_{0}^{t}S_{\epsilon}(t-t_{1})\frac{1}{\langle 1+\epsilon
D\rangle}\left(e^{\frac{2i}{\epsilon}(x-\frac{t_{1}}{\epsilon\sqrt{2}})}A_{\epsilon}^{3}+3e^{-\frac{2i}{\epsilon}(x-\frac{t_{1}}{\epsilon\sqrt{2}})}|A_{\epsilon}|^{2}\bar{A}_{\epsilon}+e^{-\frac{4i}{\epsilon}(x-\frac{t_{1}}{\epsilon\sqrt{2}})}\bar{A}_{\epsilon}^{3}\right)(t_{1})dt_{1}$

immediately generates very high frequencies
$\xi=\mp\frac{2}{\epsilon},-\frac{4}{\epsilon}$. One may expect that these
high frequency pieces vanish as $\epsilon\to 0$ due to fast dispersion.
However, they might be measured largely in a standard Sobolev norm
$\|\cdot\|_{H^{s}(\mathbb{R})}$. On the other hand, if one employs the
rescaled Sobolev norm $\|\cdot\|_{H_{\epsilon}^{s}(\mathbb{R})}$ for (2.3),
one becomes unable to capture how much portion of $A_{\epsilon}$ is localized
in frequencies.

A key idea to resolve the problem is to separate the amplitude function into
the core profile and the high frequency remainder,

$A_{\epsilon}(t,x)=\psi_{\epsilon}(t,x)+r_{\epsilon}(t,x),$

and impose that $(\psi_{\epsilon}(t),r_{\epsilon}(t))$ solves the system of
equations

$\left\\{\begin{aligned}
\psi_{\epsilon}(t)&=S_{\epsilon}(t)\psi_{\epsilon,0}-\frac{3i}{2}\int_{0}^{t}S_{\epsilon}(t-t_{1})\frac{1}{\langle
1+\epsilon
D\rangle}\left(|\psi_{\epsilon}|^{2}\psi_{\epsilon}\right)(t_{1})dt_{1},\\\
r_{\epsilon}(t)&=-\frac{3i}{2}\int_{0}^{t}S_{\epsilon}(t-t_{1})\frac{1}{\langle
1+\epsilon
D\rangle}\big{(}|A_{\epsilon}|^{2}(A_{\epsilon})-|\psi_{\epsilon}|^{2}\psi_{\epsilon}+\mathcal{R}_{\epsilon}(A_{\epsilon})\big{)}(t_{1})dt_{1}.\end{aligned}\right.$
(2.4)

In a sequel, they will be estimated separately.

##  3\. Linear rescaled Klein-Gordon flow

In this section, we investigate properties of the linear rescaled Klein-Gordon
flow

$S_{\epsilon}(t)=e^{-itp_{\epsilon}(D)}=e^{-\frac{it}{\epsilon^{2}}(\langle
1+\epsilon D\rangle-\sqrt{2}-\frac{\epsilon}{\sqrt{2}}D)}$

focusing on its connection to the linear Schrödinger flow
$e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}}$.

###  3.1. Convergence of the linear flow

By the formal Taylor series expansion

$\langle 1+\epsilon
D\rangle=\sqrt{2}+\frac{\epsilon}{\sqrt{2}}D+\frac{\epsilon^{2}}{4\sqrt{2}}D^{2}-\frac{\epsilon^{3}}{8\sqrt{2}}D^{3}+\cdots,$

it is expected that $S_{\epsilon}(t)$ converges to
$e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}}$ as $\epsilon\to 0$. It can be
stated rigorously as follows.

######  Lemma 3.1 (Convergence of the linear flow $S_{\epsilon}(t)$).

Let $\epsilon\in(0,1]$. $(i)$ For any sufficiently small $\delta>0$, we have

$\|S_{\epsilon}(t)u_{0}-e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}}u_{0}\|_{L^{2}(\mathbb{R})}\lesssim\delta^{3}\|u_{0}\|_{L^{2}(\mathbb{R})}+\|P_{>\delta\epsilon^{-1/3}}u_{0}\|_{L^{2}(\mathbb{R})}.$

$(ii)$ If $u_{0}\in H^{s}(\mathbb{R})$, then

$\|S_{\epsilon}(t)u_{0}-e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}}u_{0}\|_{L^{2}(\mathbb{R})}\lesssim\epsilon^{\frac{s}{3}}\|u_{0}\|_{H^{s}(\mathbb{R})}.$

###### Proof.

By the Plancherel theorem, we have

$\|S_{\epsilon}(t)u_{0}-e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}}u_{0}\|_{L^{2}}\sim\|(e^{-it(p_{\epsilon}(\xi)-\frac{1}{4\sqrt{2}}\xi^{2})}-1)\hat{u}_{0}\|_{L^{2}}.$

Note that

$|e^{-it(p_{\epsilon}(\xi)-\frac{1}{4\sqrt{2}}\xi^{2})}-1|\lesssim\min\left\\{\epsilon|\xi|^{3},1\right\\},$
(3.1)

because
$|p_{\epsilon}(\xi)-\frac{1}{4\sqrt{2}}\xi^{2}|\lesssim\epsilon|\xi|^{3}$ by
Taylor’s theorem. Thus, the lemma follows. ∎

######  Remark 3.2.

From the proof of Lemma 3.1 (see (3.1) in particular), one can see that (H2)
is the minimal requirement for the convergence
$\|S_{\epsilon}(t)u_{0}-e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}}u_{0}\|_{H_{\epsilon}^{1}(\mathbb{R})}\to
0$ as well as (H2’) is optimal for the $O(\epsilon^{\frac{s}{3}})$-rate of
convergence.

###  3.2. Uniform linear Strichartz estimates

For $\epsilon\in(0,1]$, we define the rescaled Sobolev norms by

$\|u\|_{W_{\epsilon}^{s,r}(\mathbb{R})}:=\|\langle\epsilon
D\rangle^{s}u\|_{L^{r}(\mathbb{R})}\quad\textup{and}\quad\|u\|_{H_{\epsilon}^{s}(\mathbb{R})}:=\|u\|_{W_{\epsilon}^{s,2}(\mathbb{R})}.$

Using a smooth cut-off $\chi\in C_{c}^{\infty}(\mathbb{R})$ such that
$\textup{supp}\chi\subset\\{\xi:\frac{1}{2}<|\xi|<4\\}$, $\chi\equiv 1$ on
$1\leq|\xi|\leq 2$ and $\sum_{N\in 2^{\mathbb{Z}}}\chi(\frac{\cdot}{N})\equiv
1$, we define the Littlewood-Paley projection $P_{N}$ by

$\widehat{P_{N}f}(\xi)=\chi(\tfrac{\xi}{N})\hat{f}(\xi).$

Our analysis heavily relies on dispersive properties of the linear Klein-
Gordon flow $S_{\epsilon}(t)$ in the form of Strichartz estimates. Indeed,
such estimates are well-known for the standard Klein-Gordon equation, and in
essence, the same holds for the rescaled one. However, in consideration of the
$\epsilon\to 0$ limit, possible $\epsilon$-dependences in the inequalities
must be clarified.

######  Lemma 3.3 (Uniform linear Strichartz estimates).

Let $\epsilon\in(0,1]$. Suppose that $q,r,\tilde{q},\tilde{r}\geq 2$ and

$\frac{2}{q}+\frac{1}{r}=\frac{2}{\tilde{q}}+\frac{1}{\tilde{r}}=\frac{1}{2}.$

Then, we have

$\|S_{\epsilon}(t)u_{0}\|_{L_{t}^{q}(\mathbb{R};L^{r}(\mathbb{R}))}\lesssim\|u_{0}\|_{H_{\epsilon}^{\frac{3}{q}}(\mathbb{R})}$

and

$\left\|\int_{0}^{t}S_{\epsilon}(t-t_{1})F(t_{1})dt_{1}\right\|_{L_{t}^{q}(\mathbb{R};L^{r}(\mathbb{R}))}\lesssim\|F\|_{L_{t}^{\tilde{q}^{\prime}}(\mathbb{R};W_{\epsilon}^{3(\frac{1}{q}+\frac{1}{\tilde{q}}),\tilde{r}^{\prime}}(\mathbb{R}))}.$

###### Proof.

Since $p_{\epsilon}^{\prime\prime}(\epsilon\xi)=\frac{1}{\langle
1+\epsilon\xi\rangle^{3}}$, the van der Corput lemma deduces that

$\|S_{\epsilon}(t)P_{N}u_{0}\|_{L_{x}^{\infty}}=\left\|\int_{\mathbb{R}}\left\\{\int_{\mathbb{R}}e^{-itp_{\epsilon}(\xi)}e^{i(x-y)\xi}\chi(\tfrac{\xi}{N})dy\right\\}u_{0}(y)dy\right\|_{L_{x}^{\infty}}\lesssim\frac{1}{\langle\epsilon
N\rangle^{3/2}|t|^{1/2}}\|u_{0}\|_{L^{1}}.$

Thus, the standard interpolation argument (see Keel-Tao [13] for instance),
together with the Littlewood-Paley inequality, yields the desired
inequalities. ∎

######  Remark 3.4.

Strichartz estimates for the rescaled flow $S_{\epsilon}(t)$ require some
regularities like the standard linear Klein-Gordon flow $e^{it\langle
D\rangle}$. On the other hand, in the $\epsilon\to 0$ limit, they are closely
related to the Strichartz estimates for the linear Schrödinger flow
$e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}}$, i.e.,

$\|e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}}u_{0}\|_{L_{t}^{q}(\mathbb{R};L^{r}(\mathbb{R}))}\lesssim\|u_{0}\|_{L^{2}(\mathbb{R})},$
$\left\|\int_{0}^{t}e^{\frac{i(t-t_{1})}{4\sqrt{2}}\partial_{x}^{2}}F(t_{1})dt_{1}\right\|_{L_{t}^{q}(\mathbb{R};L^{r}(\mathbb{R}))}\lesssim\|F\|_{L_{t}^{\tilde{q}^{\prime}}(\mathbb{R};L^{\tilde{r}^{\prime}}(\mathbb{R}))}$

as the rescaled Sobolev norm $\|\cdot\|_{W_{\epsilon}^{s,r}(\mathbb{R})}$
formally converges to the $L^{r}(\mathbb{R})$-norm.

Next, we introduce two types of Fourier restriction norm spaces, namely the
Bourgain spaces [1], associated with the linear propagator $S_{\epsilon}(t)$.
We define $X^{s,b}$ (resp., $X_{\epsilon}^{s,b}$) as the completion of
Schwartz functions with respect to

$\|u\|_{X^{s,b}}:=\|\langle\xi\rangle^{s}\langle\tau+p_{\epsilon}(\xi)\rangle^{b}\tilde{u}(\tau,\xi)\|_{L^{2}_{\tau,\xi}(\mathbb{R}\times\mathbb{R})}$
$\left(\textup{resp.,
}\|u\|_{X^{s,b}}:=\|\langle\epsilon\xi\rangle^{s}\langle\tau+p_{\epsilon}(\xi)\rangle^{b}\tilde{u}(\tau,\xi)\|_{L^{2}_{\tau,\xi}(\mathbb{R}\times\mathbb{R})}\right),$

where $\tilde{u}(\tau,\xi)$ denotes the space-time Fourier transform of
$u(t,x)$. Then, given $T>0$, we define $X^{s,b}([-T,T])$ (resp.,
$X_{\epsilon}^{s,b}([-T,T])$) with the norm

$\|u\|_{X^{s,b}([-T,T])}:=\inf\Big{\\{}\|v\|_{X^{s,b}}:v=u\textup{ on
}[-T,T]\Big{\\}}$ $\left(\textup{resp.,
}\|u\|_{X_{\epsilon}^{s,b}([-T,T])}:=\inf\Big{\\{}\|v\|_{X_{\epsilon}^{s,b}}:v=u\textup{
on }[-T,T]\Big{\\}}\right).$

Later, we will frequently use the following basic mapping properties (refer to
[1, 29]) and the trilinear estimate involving the Fourier restriction spaces.

######  Lemma 3.5.

$(i)$ (Inhomogeneous estimate) For any $b>\frac{1}{2}$,

$\left\|\int_{0}^{t}S_{\epsilon}(t-t_{1})F(t_{1})dt_{1}\right\|_{X^{0,b}([-T,T])}\lesssim\|F\|_{X^{0,-(1-b)}([-T,T])}.$

$(ii)$ (Transference principle) If $b>\frac{1}{2}$,
$\frac{2}{q}+\frac{1}{r}=\frac{1}{2}$ and $q,r\geq 2$, then

$\|u\|_{L^{q}_{t}([-T,T];L^{r}_{x}(\mathbb{R}))}\lesssim\|u\|_{X^{\frac{3}{q},b}_{\epsilon}([-T,T])}.$

######  Lemma 3.6 (Trilinear estimate).

For $b\in(\frac{1}{2},1]$, we have

$\displaystyle\|v_{1}v_{2}v_{3}\|_{X^{0,-(1-b)}([-T,T])}$
$\displaystyle\lesssim\|v_{1}v_{2}v_{3}\|_{L_{t}^{2b}(\mathbb{R};L_{x}^{2}(\mathbb{R})}$
$\displaystyle\lesssim
T^{\frac{1-b}{2b}}\|v_{1}\|_{X_{\epsilon}^{1,b}([-T,T])}\|v_{2}\|_{X_{\epsilon}^{1,b}([-T,T])}\|v_{3}\|_{X^{0,b}([-T,T])}.$

###### Proof.

Interpolating the two basic inequalities
$\|u\|_{L_{t}^{\infty}L_{x}^{2}}\lesssim\|u\|_{X_{\epsilon}^{0,b}}$ and
$\|u\|_{L_{t}^{2}L_{x}^{2}}\lesssim\|u\|_{X_{\epsilon}^{0,0}}$, we obtain
$\|u\|_{L_{t}^{\frac{2b}{2b-1}}L_{x}^{2}}\lesssim\|u\|_{X_{\epsilon}^{0,1-b}}$
whose dual inequality is given by
$\|u\|_{X_{\epsilon}^{0,-(1-b)}}\lesssim\|u\|_{L_{t}^{2b}L_{x}^{2}}$. Thus, by
Hölder’s inequality and Strichartz estimates, we prove that

$\displaystyle\|v_{1}v_{2}v_{3}\|_{X^{0,-(1-b)}}$
$\displaystyle\lesssim\|v_{1}v_{2}v_{3}\|_{L_{t}^{2b}L_{x}^{2}}\lesssim
T^{\frac{1-b}{2b}}\|v_{1}\|_{L_{t}^{4}L_{x}^{\infty}}\|v_{2}\|_{L_{t}^{4}L_{x}^{\infty}}\|v_{3}\|_{C_{t}L_{x}^{2}}$
$\displaystyle\lesssim
T^{\frac{1-b}{2b}}\|v_{1}\|_{X_{\epsilon}^{1,b}}\|v_{2}\|_{X_{\epsilon}^{1,b}}\|v_{3}\|_{X^{0,b}}.$

∎

##  4\. Well-posedness of the nonlinear Klein-Gordon equation

We present basic well-posedness results for the rescaled NLKG (2.1) and the
system (2.4). Consequently, we confirm that
$\psi_{\epsilon}(t,x)+r_{\epsilon}(t,x)+c.c.$, from (2.4), is a unique
solution to the NLKG (2.1) with initial data
$\big{(}2\textup{Re}(e^{\frac{ix}{\epsilon}}\psi_{\epsilon,0}),\tfrac{2}{\epsilon^{2}}\langle\epsilon\partial_{x}\rangle\textup{Im}(e^{\frac{ix}{\epsilon}}\psi_{\epsilon,0})\big{)}$
(see Remark 4.4 below).

The well-posedness of the standard NLKG (1.1) is well-known. For the rescaled
one, it is rephrased as follows.

######  Proposition 4.1 (Global well-posedness for the rescaled NLKG).

Let $\epsilon\in(0,1]$. Given initial data
$(v_{\epsilon}(0),\partial_{t}v_{\epsilon}(0))\in
H_{\epsilon}^{1}(\mathbb{R})\times L^{2}(\mathbb{R})$, there exists a unique
global strong solution $v_{\epsilon}(t)\in C_{t}(\mathbb{R};H_{\epsilon}^{1})$
to the rescaled NLKG (2.1). Moreover, $v_{\epsilon}(t)$ preserves the energy

$\mathcal{E}_{\epsilon}[v]=\frac{\epsilon^{2}}{2}\|\partial_{x}v\|_{L^{2}(\mathbb{R})}^{2}+\frac{1}{2}\|v\|_{L^{2}(\mathbb{R})}^{2}+\frac{\epsilon^{4}}{2}\|\partial_{t}v\|_{L^{2}(\mathbb{R})}^{2}+\frac{\epsilon^{2}}{4}\|v\|_{L^{4}(\mathbb{R})}^{4}.$

###### Sketch of Proof.

The proof follows from a standard contraction mapping argument. Indeed, by the
Sobolev embedding $H^{1}(\mathbb{R})\hookrightarrow L^{\infty}(\mathbb{R})$,
one can show that the corresponding integral equation is locally well-posed in
$C_{t}([-T_{\epsilon},T_{\epsilon}];H_{\epsilon}^{1})$ for sufficiently small
$T_{\epsilon}>0$ possibly depending on $\epsilon\in(0,1]$. Then, it can be
upgraded to global well-posedness, because the conservation law prevents the
solution from blowing up in finite time. ∎

######  Remark 4.2.

As for well-posedness of the rescaled NLKG (2.1), it is not necessary to
employ Strichartz estimates in the previous section. Nevertheless, if one uses
$\epsilon$-depending inequalities, e.g., the Sobolev inequality
$\|v\|_{L^{\infty}(\mathbb{R})}\lesssim\frac{1}{\epsilon}\|v\|_{H_{\epsilon}^{1}(\mathbb{R})}$,
the size of the interval $[-T_{\epsilon},T_{\epsilon}]$ in the contraction
mapping argument would shrink to $\\{0\\}$ as $\epsilon\to 0$.

Next, using Strichartz estimates, we establish well-posedness for the system
(2.4).

######  Proposition 4.3 (Local well-posedness for the system (2.4)).

Let $\epsilon\in(0,1]$ and $b\in(\frac{1}{2},1)$. We assume that
$\\{\psi_{\epsilon,0}\\}_{\epsilon\in(0,1]}$ satisfies (H1) for some $R>0$.
Then, there exists $0<T\lesssim R^{-\frac{4b}{1-b}}$, independent of
$\epsilon\in(0,1]$, such that the system (2.4) has a unique solution
$(\psi_{\epsilon}(t),r_{\epsilon}(t))\in
C_{t}([-T,T];H_{\epsilon}^{1}(\mathbb{R})\times H_{\epsilon}^{1}(\mathbb{R}))$
with

$\|\psi_{\epsilon}\|_{X_{\epsilon}^{1,b}([-T,T])}\lesssim
R\quad\textup{and}\quad\|r_{\epsilon}\|_{X_{\epsilon}^{1,b}([-T,T])}\lesssim
T^{\frac{1-b}{2b}}R^{3}.$ (4.1)

Moreover, we have

$\|\psi_{\epsilon}\|_{X^{s,b}([-T,T])}\lesssim\|\psi_{\epsilon,0}\|_{H^{s}(\mathbb{R})}.$
(4.2)

###### Proof.

We define

$\Phi_{1}(\psi)(t):=S_{\epsilon}(t)\psi_{\epsilon,0}-\frac{3i}{2}\int_{0}^{t}S_{\epsilon}(t-t_{1})\frac{1}{\langle
1+\epsilon D\rangle}|\psi|^{2}\psi(t_{1})dt_{1}.$

Then, by Strichartz estimates and the trilinear estimate (Lemma 3.6), one can
show that

$\displaystyle\|\Phi_{1}(\psi)\|_{X_{\epsilon}^{1,b}}$ $\displaystyle\leq
cR+cT^{\frac{1-b}{2b}}\|\psi\|_{X_{\epsilon}^{1,b}}^{3}\leq
cR+cT^{\frac{1-b}{2b}}(2cR)^{3},$
$\displaystyle\|\Phi_{1}(\psi)-\Phi_{1}(\psi^{\prime})\|_{X_{\epsilon}^{1,b}}$
$\displaystyle\leq
cT^{\frac{1-b}{2b}}\big{(}\|\psi\|_{X_{\epsilon}^{1,b}}^{2}+\|\psi^{\prime}\|_{X_{\epsilon}^{1,b}}^{2}\big{)}\|\psi-\psi^{\prime}\|_{X_{\epsilon}^{1,b}}$
$\displaystyle\leq
2cT^{\frac{1-b}{2b}}(2cR)^{2}\|\psi-\psi^{\prime}\|_{X_{\epsilon}^{1,b}}\leq\frac{1}{2}\|\psi-\psi^{\prime}\|_{X_{\epsilon}^{1,b}},$

provided that
$\|\psi\|_{X_{\epsilon}^{1,b}},\|\psi^{\prime}\|_{X_{\epsilon}^{1,b}}\leq
2cR$. We take small $T>0$ such that
$cT^{\frac{1-b}{2b}}(2cR)^{2}\leq\frac{1}{4}$. Then, it follows that
$\Phi_{1}$ is contractive on $\\{\psi\in
X_{\epsilon}^{1,b}:\|\psi\|_{X_{\epsilon}^{1,b}}\leq 2cR\\}$, and it has a
unique fixed point $\psi_{\epsilon}$ with
$\|\psi_{\epsilon}\|_{X_{\epsilon}^{1,b}}\leq 2cR$.

Next, we consider

$\displaystyle\Phi_{2}(r)(t):$
$\displaystyle=-\frac{3i}{2}\int_{0}^{t}S_{\epsilon}(t)\frac{1}{\langle
1+\epsilon
D\rangle}\left(|\psi_{\epsilon}+r|^{2}(\psi_{\epsilon}+r)-|\psi_{\epsilon}|^{2}\psi_{\epsilon}+\mathcal{R}_{\epsilon}(\psi_{\epsilon}+r)\right)(t_{1})dt_{1}.$

We again use Strichartz estimates and the trilinear estimate (Lemma 3.6) to
obtain

$\displaystyle\|\Phi_{2}(r)\|_{X_{\epsilon}^{1,b}}$ $\displaystyle\leq
cT^{\frac{1-b}{2b}}\big{(}\|\psi_{\epsilon}\|_{X_{\epsilon}^{1,b}}^{3}+\|r\|_{X_{\epsilon}^{1,b}}^{3}\big{)}\leq
2cT^{\frac{1-b}{2b}}(2cR)^{3},$
$\displaystyle\|\Phi_{2}(r)-\Phi_{2}(r^{\prime})\|_{X_{\epsilon}^{1,b}}$
$\displaystyle\leq
cT^{\frac{1-b}{2b}}\big{(}\|\psi_{\epsilon}\|_{X_{\epsilon}^{1,b}}^{2}+\|r\|_{X_{\epsilon}^{1,b}}^{2}+\|r^{\prime}\|_{X_{\epsilon}^{1,b}}^{2}\big{)}\|r-r^{\prime}\|_{X_{\epsilon}^{1,b}}$
$\displaystyle\leq
3cT^{\frac{1-b}{2b}}(2cR)^{2}\|r-r^{\prime}\|_{X_{\epsilon}^{1,b}}\leq\frac{3}{4}\|r-r^{\prime}\|_{X_{\epsilon}^{1,b}}$

if $\|r\|_{X_{\epsilon}^{1,b}},\|r^{\prime}\|_{X_{\epsilon}^{1,b}}\leq
2cT^{\frac{1-b}{2b}}(2cR)^{3}\leq cR$. Therefore, $r=\Phi_{2}(r)$ has a unique
solution $r_{\epsilon}$ with $\|r_{\epsilon}\|_{X_{\epsilon}^{1,b}}\leq
2cT^{\frac{1-b}{2b}}(2cR)^{3}$.

It remains to show (4.2). Indeed, by Strichartz estimates, we have

$\|\psi_{\epsilon}\|_{X^{s,b}}\lesssim\|\psi_{\epsilon,0}\|_{H^{s}}+\||\psi_{\epsilon}|^{2}\psi_{\epsilon}\|_{X^{s,-(1-b)}}.$

Slightly modifying the proof of Lemma 3.6, one can show that

$\displaystyle\||\psi_{\epsilon}|^{2}\psi_{\epsilon}\|_{X^{s,-(1-b)}}$
$\displaystyle\lesssim\||\psi_{\epsilon}|^{2}\psi_{\epsilon}\|_{L_{t}^{2b}H_{x}^{s}}\lesssim
T^{\frac{1-b}{2b}}\|\psi_{\epsilon}\|_{L_{t}^{4}L_{x}^{\infty}}^{2}\|\psi_{\epsilon}\|_{C_{t}H_{x}^{s}}$
(4.3) $\displaystyle\lesssim
T^{\frac{1-b}{2b}}\|\psi_{\epsilon}\|_{X_{\epsilon}^{1,b}}^{2}\|\psi_{\epsilon}\|_{X^{s,b}}\lesssim
T^{\frac{1-b}{2b}}R^{2}\|\psi_{\epsilon}\|_{X^{s,b}}.$

Thus, taking smaller $T\ll R^{-\frac{4b}{1-b}}$ if necessary, we obtain
$\|\psi_{\epsilon}\|_{X^{s,b}}\lesssim\|\psi_{\epsilon,0}\|_{H^{s}}$. ∎

######  Remark 4.4.

From the construction, summing two equations in the system (2.4) and their
c.c.’s, one can see that $\psi_{\epsilon}(t,x)+r_{\epsilon}(t,x)+c.c.$ is a
strong solution to the rescaled NLKG (2.1) and that it is included in
$C_{t}([-T,T];H_{\epsilon}^{1}(\mathbb{R}))$ by the transference principle.
Therefore, by uniqueness, the solution constructed from the system (2.4) is a
unique strong solution to the rescaled NLKG with initial data
$(2\textup{Re}(e^{\frac{ix}{\epsilon}}\psi_{\epsilon,0}),\frac{2}{\epsilon^{2}}\langle\epsilon\partial_{x}\rangle\textup{Im}(e^{\frac{ix}{\epsilon}}\psi_{\epsilon,0}))$.

######  Remark 4.5.

We have an analogous well-posedness result for the NLS

$i\partial_{t}\psi^{\textup{(NLS)}}+\frac{1}{4\sqrt{2}}\partial_{x}^{2}\psi^{\textup{(NLS)}}-\frac{3}{2\sqrt{2}}|\psi^{\textup{(NLS)}}|^{2}\psi^{\textup{(NLS)}}=0.$
(4.4)

Indeed, by the standard argument (see Cazenave [2]) but with the Strichartz
estimate

$\|e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}}u_{0}\|_{C_{t}(\mathbb{R};L^{2}(\mathbb{R}))\cap
L_{t}^{4}(\mathbb{R};L_{x}^{\infty}(\mathbb{R}))}\lesssim\|u_{0}\|_{L^{2}(\mathbb{R})},$

one can show that there exists $T>0$, independent of $\epsilon\in(0,1]$, such
that the following hold.
$(i)$ If (H1) holds, the solution $\psi_{\epsilon}^{\textup{(NLS)}}\in
C_{t}([-T,T];H_{\epsilon}^{1}(\mathbb{R}))$ to the NLS (4.4) with initial data
$\psi_{\epsilon,0}$ exists, and

$\|\psi_{\epsilon}^{\textup{(NLS)}}\|_{C_{t}([-T,T];H_{\epsilon}^{1}(\mathbb{R}))\cap
L_{t}^{4}([-T,T];L^{\infty}(\mathbb{R}))}\lesssim\|\psi_{\epsilon,0}\|_{H_{\epsilon}^{1}(\mathbb{R})}.$
(4.5)

$(ii)$ If we further assume (H2’) for some $s>0$, then

$\|\psi_{\epsilon}^{\textup{(NLS)}}\|_{C_{t}([-T,T];H^{s}(\mathbb{R}))}+\big{\|}|\psi_{\epsilon}^{\textup{(NLS)}}|^{2}\psi_{\epsilon}^{\textup{(NLS)}}\big{\|}_{L_{t}^{1}([-T,T];H^{s}(\mathbb{R}))}\lesssim\|\psi_{\epsilon,0}\|_{H^{s}(\mathbb{R})}.$
(4.6)

##  5\. High frequency estimate for the core profile

We recall that the convergence of the linear flow $S_{\epsilon}(t)$ requires
smallness of high frequencies (see Lemma 3.1 and Remark 3.2). Thus, one may
expect that a similar estimate would be needed for the nonlinear problem. In
this section, we prove the following proposition for the core profile.

######  Proposition 5.1 (High frequency estimates for the core profile).

Let $\epsilon\in(0,1]$ and $b\in(\frac{1}{2},1)$. We assume (H1) and (H2) for
$\\{\psi_{\epsilon,0}\\}_{\epsilon\in(0,1]}$, and let
$(\psi_{\epsilon},r_{\epsilon})\in
C_{t}([-T,T];H_{\epsilon}^{1}(\mathbb{R})\times H_{\epsilon}^{1}(\mathbb{R}))$
be the solution to the system (2.4). Then, for any $\delta>0$, we have

$\lim_{\epsilon\to
0}\left(\|P_{>\delta\epsilon^{-1/3}}\psi_{\epsilon}\|_{X_{\epsilon}^{1,b}([-T,T])}+\|P_{>\delta\epsilon^{-1/3}}(|\psi_{\epsilon}|^{2}\psi_{\epsilon})\|_{L_{t}^{2b}([-T,T];L_{x}^{2}(\mathbb{R}))}\right)=0,$

where $P_{>N}=\sum_{M>\frac{N}{2}}P_{M}$.

For the proof, one may try to estimate the $\psi_{\epsilon}$-equation in the
system (2.4) putting the high frequency projection
$P_{>\delta\epsilon^{-1/3}}$. However, one would immediately realize that the
frequency cut-off does not work properly, because frequency cut-offs are
blurred in products. Indeed, the difference between
$P_{>\delta\epsilon^{-1/3}}(|\psi_{\epsilon}|^{2}\psi_{\epsilon})$ and
$|P_{>\delta\epsilon^{-1/3}}\psi_{\epsilon}|^{2}P_{>\delta\epsilon^{-1/3}}\psi_{\epsilon}$
is nonzero in general.

To resolve the technical issue, we employ the operator $m_{N}(D)$ with
multiplier

$m_{N}(\xi):=\left\\{\begin{aligned} &\frac{|\xi|}{N}&&\textup{if }|\xi|\leq
N,\\\ &1&&\textup{if }|\xi|\geq N.\end{aligned}\right.$

######  Remark 5.2.

$(i)$ $m_{N}(D)$ acts like a high frequency cut-off in the sense that
$m_{N}(\xi)\equiv 1$ for high frequencies $|\xi|\geq N$ but it is arbitrarily
small in very low frequencies, i.e., $m_{N}(\xi)\leq\delta$ if
$|\xi|\leq\delta N$ for any small $\delta>0$.
$(ii)$ $m_{N}(\xi_{1})$ and $m_{N}(\xi_{2})$ are comparable if
$|\xi_{1}|\sim|\xi_{2}|$, i.e., $m_{N}(\xi)\sim m_{N}(2\xi)$ for all $\xi$.
For this reason, $m_{N}(D)$ has an advantage in handling the blurring effect
in a product.
$(iii)$ $m_{N}(D)$ acts like the differential operator
$\frac{1}{N}|\partial_{x}|$ in low frequencies $|\xi|\leq N$.

###### Proof of Proposition 5.1.

We claim that

$\displaystyle\|m_{N}(D)\psi_{\epsilon}\|_{X_{\epsilon}^{1,b}}$
$\displaystyle\lesssim\|m_{N}(D)\psi_{\epsilon,0}\|_{H_{\epsilon}^{1}},$ (5.1)
$\displaystyle\|m_{N}(D)(|\psi_{\epsilon}|^{2}\psi_{\epsilon})\|_{L_{t}^{2b}L_{x}^{2}}$
$\displaystyle\lesssim\|m_{N}(D)\psi_{\epsilon,0}\|_{H_{\epsilon}^{1}}.$

Indeed, applying the Fourier multiplier $m_{N}(D)$ to the
$\psi_{\epsilon}$-equation in the system (2.4) and then estimating by
Strichartz estimates, we obtain

$\displaystyle\|m_{N}(D)\psi_{\epsilon}\|_{X_{\epsilon}^{1,b}}$
$\displaystyle\lesssim\|m_{N}(D)\psi_{\epsilon,0}\|_{H_{\epsilon}^{1}}+\|m_{N}(D)(|\psi_{\epsilon}|^{2}\psi_{\epsilon})\|_{X^{0,-(1-b)}}$
(5.2)
$\displaystyle\lesssim\|m_{N}(D)\psi_{\epsilon,0}\|_{H_{\epsilon}^{1}}+\|m_{N}(D)(|\psi_{\epsilon}|^{2}\psi_{\epsilon})\|_{L_{t}^{2b}L_{x}^{2}}.$

For the nonlinear term, we decompose

$\displaystyle m_{N}(D)\big{(}|\psi_{\epsilon}|^{2}\psi_{\epsilon}\big{)}$
$\displaystyle=m_{N}(D)\big{(}|P_{\leq N}\psi_{\epsilon}|^{2}P_{\leq
N}\psi_{\epsilon}\big{)}+m_{N}(D)\big{(}|\psi_{\epsilon}|^{2}\psi_{\epsilon}-|P_{\leq
N}\psi_{\epsilon}|^{2}P_{\leq N}\psi_{\epsilon}\big{)}$
$\displaystyle=:I+I\\!I,$

where $P_{\leq N}=\sum_{M\leq\frac{N}{2}}P_{M}$. For the former term $I$, we
note that $|P_{\leq N}\psi_{\epsilon}|^{2}P_{\leq N}\psi_{\epsilon}$ is
localized in $|\xi|\leq 6N$ on the Fourier side, where $m_{N}(D)$ behaves like
$\sim\frac{|D|}{N}$ (see Remark 5.2 $(ii)$ and $(iii)$). Thus, the Leibniz
rule and the trilinear estimate (Lemma 3.6) yield

$\displaystyle\|I\|_{L_{t}^{2b}L_{x}^{2}}$
$\displaystyle\sim\frac{1}{N}\|\partial_{x}(|P_{\leq
N}\psi_{\epsilon}|^{2}P_{\leq
N}\psi_{\epsilon})\|_{L_{t}^{2b}L_{x}^{2}}\lesssim
T^{\frac{1-b}{2b}}\|\psi_{\epsilon}\|_{X_{\epsilon}^{1,b}}^{2}\frac{1}{N}\|\partial_{x}P_{\leq
N}\psi_{\epsilon}\|_{X^{0,b}}$ $\displaystyle\lesssim
T^{\frac{1-b}{2b}}\|m_{N}(D)\psi_{\epsilon}\|_{X^{0,b}}.$

For the latter term $I\\!I$, using the trivial bound $m_{N}(\xi)\leq 1$ and
the trilinear estimate (Lemma 3.6), we show that

$\displaystyle\|I\\!I\|_{L_{t}^{2b}L_{x}^{2}}$
$\displaystyle\leq\big{\|}|\psi_{\epsilon}|^{2}\psi_{\epsilon}-|P_{\leq
N}\psi_{\epsilon}|^{2}P_{\leq
N}\psi_{\epsilon}\big{\|}_{L_{t}^{2b}L_{x}^{2}}\lesssim
T^{\frac{1-b}{2b}}\|\psi_{\epsilon}\|_{X_{\epsilon}^{1,b}}^{2}\|P_{>N}\psi_{\epsilon}\|_{X^{0,b}}$
$\displaystyle\sim T^{\frac{1-b}{2b}}\|m_{N}(D)\psi_{\epsilon}\|_{X^{0,b}},$

where $m_{N}(\xi)\sim 1$ for $|\xi|\gtrsim N$ is used in the last step. Hence,
taking smaller $T>0$ if necessary, we prove that

$\|m_{N}(D)(|\psi_{\epsilon}|^{2}\psi_{\epsilon})\|_{L_{t}^{2b}L_{x}^{2}}\leq\frac{1}{2}\|m_{N}(D)\psi_{\epsilon}\|_{X^{0,b}},$

from which (5.1) follows (see (5.2)).

Now, we fix $\delta>0$ and set $N=\delta\epsilon^{-1/3}$. Then, by the claim
(5.1), it suffices to show that
$\|m_{\delta\epsilon^{-1/3}}(D)\psi_{\epsilon,0}\|_{L^{2}}\to 0$ (see Remark
5.2 $(i)$). Indeed, for any $\delta_{0}\in(0,\delta]$, we have

$\displaystyle\|m_{\delta\epsilon^{-1/3}}(D)\psi_{\epsilon,0}\|_{L^{2}}$
$\displaystyle\leq\|m_{\delta\epsilon^{-1/3}}(D)P_{\leq\delta_{0}\epsilon^{-1/3}}\psi_{\epsilon,0}\|_{L^{2}}+\|m_{\delta\epsilon^{-1/3}}(D)P_{>\delta_{0}\epsilon^{-1/3}}\psi_{\epsilon,0}\|_{L^{2}}$
$\displaystyle\lesssim\frac{\delta_{0}}{\delta}\|\psi_{\epsilon,0}\|_{L^{2}}+\|P_{>\delta_{0}\epsilon^{-1/3}}\psi_{\epsilon,0}\|_{L^{2}}.$

where we used the properties of $m_{N}(D)$ in the last step (see Remark 5.2).
Consequently, by (H1) and (H2), we obtain
$\|m_{\delta\epsilon^{-1/3}}(D)\psi_{\epsilon,0}\|_{L^{2}}\lesssim\frac{\delta_{0}}{\delta}+o_{\epsilon}(1)$.
However, since $\delta_{0}>0$ is arbitrarily, this completes the proof of the
proposition. ∎

######  Remark 5.3.

Repeating the proof of Proposition 5.1 to the NLS (4.4), one can show that for
any $\delta>0$,

$\lim_{\epsilon\to
0}\|P_{>\delta\epsilon^{-1/3}}(|\psi_{\epsilon}^{\textup{(NLS)}}|^{2}\psi_{\epsilon}^{\textup{(NLS)}})\|_{L_{t}^{1}([-T,T];H_{\epsilon}^{1}(\mathbb{R}))}=0.$
(5.3)

Indeed, the proof of (5.3) is easier, since one may employ the simpler
Strichartz norm $\|\cdot\|_{C_{t}([-T,T];H_{\epsilon}^{1}(\mathbb{R}))\cap
L_{t}^{4}([-T,T];L_{x}^{\infty}(\mathbb{R}))}$ rather than the Fourier
restriction norm.

##  6\. Remainder estimate

In Proposition 4.3, a preliminary bound
$\|r_{\epsilon}\|_{X_{\epsilon}^{1,b}([-T,T])}\lesssim
T^{\frac{1-b}{2b}}R^{3}$ is obtained for the remainder term. In this section,
we show that the remainder vanishes as $\epsilon\to 0$.

######  Proposition 6.1 (Remainder estimate).

Let $\epsilon\in(0,1]$. We assume (H1) and (H2) for
$\\{\psi_{\epsilon,0}\\}_{\epsilon\in(0,1]}$, and let
$(\psi_{\epsilon},r_{\epsilon})\in
C_{t}([-T,T];H_{\epsilon}^{1}(\mathbb{R})\times H_{\epsilon}^{1}(\mathbb{R}))$
be the solution to the system (2.4). Then, we have

$\lim_{\epsilon\to
0}\|r_{\epsilon}(t)\|_{C_{t}([-T,T];H_{\epsilon}^{1}(\mathbb{R}))}=0.$ (6.1)

If we further assume (H2’) for some $s>0$, then

$\|r_{\epsilon}(t)\|_{C_{t}([-T,T];H_{\epsilon}^{1}(\mathbb{R}))}\lesssim\epsilon^{\min\\{s,1-\eta\\}}\quad\textup{for
any small }\eta>0.$ (6.2)

The proof of the proposition is reduced to that of the following lemma.

######  Lemma 6.2 (Reduction to the core profile estimate).

Let $\epsilon\in(0,1]$, and let $\eta>0$ be a sufficiently small number which
does not depend on $\epsilon\in(0,1]$. We assume that
$\\{\psi_{\epsilon,0}\\}_{\epsilon\in(0,1]}$ satisfies (H1). Then, the
solution $(\psi_{\epsilon},r_{\epsilon})\in
X_{\epsilon}^{1,1-\eta}([-T,T])\times X_{\epsilon}^{1,1-\eta}([-T,T])$ to the
system (2.4) satisfies

$\|r_{\epsilon}\|_{X_{\epsilon}^{1,\frac{1}{2}+\eta}([-T,T])}\lesssim\epsilon^{1-2\eta}+\|P_{>\frac{1}{100\epsilon}}\psi_{\epsilon}\|_{X^{0,\frac{1}{2}+\eta}([-T,T])}.$

###### Proof of Proposition 6.1, assuming Lemma 6.2.

By Proposition 5.1, Lemma 6.2 yields (6.1). For (6.2), we recall from
Proposition 4.3 that $\|\psi_{\epsilon}\|_{X^{s,\frac{1}{2}+\eta}}\lesssim 1$
if (H2’) holds. Thus, it follows that
$\|P_{>\frac{1}{100\epsilon}}\psi_{\epsilon}\|_{X^{0,\frac{1}{2}+\eta}}\lesssim\epsilon^{s}$.
∎

For the proof of the lemma, we employ the following integral estimate.

######  Lemma 6.3.

For $\tau,\xi\in\mathbb{R}$, we define

$\mathcal{I}^{\pm}(\tau,\xi):=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\mathbf{1}_{\\{|\xi_{1}|,|\xi_{2}|,|\xi-\xi_{1}-\xi_{2}|\leq\frac{1}{50\epsilon}\\}}d\xi_{1}d\xi_{2}}{\langle\tau+\frac{1}{\epsilon^{2}}(\pm\langle
1+\epsilon\xi_{1}\rangle+\langle 1+\epsilon\xi_{2}\rangle+\langle
1+\epsilon(\xi-\xi_{1}-\xi_{2})\rangle)\rangle^{2(1-\eta)}}.$

Then, for $\epsilon\in(0,1]$ and sufficiently small $\eta>0$, we have

$\mathcal{I}^{\pm}(\tau,\xi)\lesssim\max\left\\{\frac{1}{\langle\tau+\frac{2\sqrt{2}\pm\sqrt{2}+\frac{1}{10}}{\epsilon^{2}}\rangle^{1-2\eta}},\frac{1}{\langle\tau+\frac{2\sqrt{2}\pm\sqrt{2}-\frac{1}{10}}{\epsilon^{2}}\rangle^{1-2\eta}}\right\\}.$

###### Proof.

In a sequel, we always assume that
$|\xi_{1}|,|\xi_{2}|,|\xi-\xi_{1}-\xi_{2}|\leq\frac{1}{50\epsilon}$. First, in
the integral $\mathcal{I}^{\pm}(\tau,\xi)$, we will substitute the function in
the bracket by

$y_{2}=y_{2}(\xi_{2}):=\tau+\frac{1}{\epsilon^{2}}\Big{(}\pm\langle
1+\epsilon\xi_{1}\rangle+\langle 1+\epsilon\xi_{2}\rangle+\langle
1+\epsilon(\xi-\xi_{1}-\xi_{2})\rangle\Big{)}.$

Indeed, $y_{2}(\xi_{2})$ is a convex function having its unique minimum at
$\xi_{2}=\frac{\xi-\xi_{1}}{2}$, because
$y_{2}^{\prime}(\frac{\xi-\xi_{1}}{2})=0$ and
$y_{2}^{\prime\prime}(\xi_{2})=\frac{1}{\langle
1+\epsilon\xi_{2}\rangle^{3}}+\frac{1}{\langle
1+\epsilon(\xi-\xi_{1}-\xi_{2})\rangle^{3}}\sim 1$. As a consequence, by the
mean value theorem, we have
$|\frac{dy_{2}}{d\xi_{2}}|=|y_{2}^{\prime}(\xi_{2})-y_{2}^{\prime}(\frac{\xi-\xi_{1}}{2})|\sim|\xi_{2}-\frac{\xi-\xi_{1}}{2}|$,
while by Taylor’s theorem,
$y_{2}(\xi_{2})-y_{2}(\frac{\xi-\xi_{1}}{2})=\frac{y_{2}^{\prime\prime}(c)}{2}(\xi_{2}-\frac{\xi-\xi_{1}}{2})^{2}\sim(\xi_{2}-\tfrac{\xi-\xi_{1}}{2})^{2}$.
Hence, it follows that

$\left|\frac{dy_{2}}{d\xi_{2}}\right|\sim\sqrt{y_{2}(\xi_{2})-y_{2}(\tfrac{\xi-\xi_{1}}{2})}.$

Moreover, we have

$\Big{|}y_{2}-\tau-\frac{1}{\epsilon^{2}}(2\sqrt{2}\pm\sqrt{2})\Big{|}\leq\frac{1}{10\epsilon^{2}}.$

Therefore, substituting $\xi_{2}$ by $y_{2}$ separately on
$(-\infty,\frac{\xi-\xi_{1}}{2}]$ and $[\frac{\xi-\xi_{1}}{2},\infty)$, and
switching the order of integration, we obtain

$\mathcal{I}^{\pm}(\tau,\xi)\lesssim\int_{\tau+\frac{2\sqrt{2}\pm\sqrt{2}-\frac{1}{10}}{\epsilon^{2}}}^{\tau+\frac{2\sqrt{2}\pm\sqrt{2}+\frac{1}{10}}{\epsilon^{2}}}\left\\{\int_{|\xi_{1}|\leq\frac{1}{50\epsilon}}\frac{\mathbf{1}_{\\{y_{2}\geq
y_{2}(\frac{\xi-\xi_{1}}{2})\\}}}{\langle
y_{2}\rangle^{2(1-\eta)}\sqrt{y_{2}-y_{2}(\frac{\xi-\xi_{1}}{2})}}d\xi_{1}\right\\}dy_{2}.$

Next, in the inner integral $\\{\cdots\\}$, we will substitute by

$y_{1}=y_{1}(\xi_{1}):=y_{2}(\tfrac{\xi-\xi_{1}}{2})=\tau+\frac{1}{\epsilon^{2}}\Big{(}\pm\langle
1+\epsilon\xi_{1}\rangle+2\langle
1+\epsilon(\tfrac{\xi-\xi_{1}}{2})\rangle\Big{)},$ (6.3)

whose first and second derivatives are given by

$\displaystyle y_{1}^{\prime}(\xi_{1})$
$\displaystyle=\frac{1}{\epsilon}\left(\pm\frac{1+\epsilon\xi_{1}}{\langle
1+\epsilon\xi_{1}\rangle}-\frac{1+\epsilon(\tfrac{\xi-\xi_{1}}{2})}{\langle
1+\epsilon(\tfrac{\xi-\xi_{1}}{2})\rangle}\right),$ (6.4) $\displaystyle
y_{1}^{\prime\prime}(\xi_{1})$ $\displaystyle=\pm\frac{1}{\langle
1+\epsilon\xi_{1}\rangle^{3}}+\frac{1}{2\langle
1+\epsilon(\tfrac{\xi-\xi_{1}}{2})\rangle^{3}}.$

For $\mathcal{I}^{+}(\tau,\xi)$, we note as before that $y_{1}(\xi_{1})$ is
convex, $y_{1}^{\prime}(\frac{\xi}{3})=0$,
$|\frac{dy_{1}}{d\xi_{1}}|\sim|\xi_{1}-\frac{\xi}{3}|$ and
$y_{1}(\xi_{1})-y_{1}(\tfrac{\xi}{3})\sim(\xi_{1}-\tfrac{\xi}{3})^{2}$. Thus,
the substitution $y_{1}=y_{2}(\frac{\xi-\xi_{1}}{2})$ yields

$\mathcal{I}^{+}(\tau,\xi)\lesssim\int_{\tau+\frac{3\sqrt{2}-\frac{1}{10}}{\epsilon^{2}}}^{\tau+\frac{3\sqrt{2}+\frac{1}{10}}{\epsilon^{2}}}\left\\{\int_{y_{1}(\frac{\xi}{3})}^{y_{2}}\frac{dy_{1}}{\langle
y_{2}\rangle^{2(1-\eta)}\sqrt{y_{2}-y_{1}}\sqrt{y_{1}-y_{1}(\tfrac{\xi}{3})}}\right\\}dy_{2}.$

Then, applying the elementary inequality

$\int_{\alpha}^{\beta}\frac{dy_{1}}{\sqrt{\beta-
y_{1}}\sqrt{y_{1}-\alpha}}=\int_{0}^{\beta-\alpha}\frac{dy_{1}}{\sqrt{(\beta-\alpha)-y_{1}}\sqrt{y_{1}}}=\int_{0}^{1}\frac{dy_{1}}{\sqrt{1-y_{1}}\sqrt{y_{1}}}\sim
1\quad(\alpha<\beta),$

we conclude that

$\mathcal{I}^{+}(\tau,\xi)\lesssim\int_{\tau+\frac{3\sqrt{2}-\frac{1}{10}}{\epsilon^{2}}}^{\tau+\frac{3\sqrt{2}+\frac{1}{10}}{\epsilon^{2}}}\frac{dy_{2}}{\langle
y_{2}\rangle^{2(1-\eta)}}\sim\max_{\pm}\left\\{\frac{1}{\langle\tau+\frac{3\sqrt{2}\pm\frac{1}{10}}{\epsilon^{2}}\rangle^{1-2\eta}}\right\\}.$

For $\mathcal{I}^{-}(\tau,\xi)$, we again substitute by
$y_{1}(\xi_{1})=y_{2}(\frac{\xi-\xi_{1}}{2})$, but we instead use that
$|y_{1}(\xi_{1})-\tau-\frac{\sqrt{2}}{\epsilon^{2}}|\leq\frac{1}{10\epsilon^{2}}$
(see (6.3)) and $-\frac{dy_{1}}{d\xi_{1}}\sim\frac{1}{\epsilon}$ (see (6.4)).
Then, it follows that

$\displaystyle\mathcal{I}^{-}(\tau,\xi)$
$\displaystyle\lesssim\int_{\tau+\frac{\sqrt{2}-\frac{1}{10}}{\epsilon^{2}}}^{\tau+\frac{\sqrt{2}+\frac{1}{10}}{\epsilon^{2}}}\left\\{\int_{\tau+\frac{\sqrt{2}-\frac{1}{10}}{\epsilon^{2}}}^{\tau+\frac{\sqrt{2}+\frac{1}{10}}{\epsilon^{2}}}\frac{\epsilon\mathbf{1}_{\\{y_{2}\geq
y_{1}\\}}}{\langle
y_{2}\rangle^{2(1-\eta)}\sqrt{y_{2}-y_{1}}}dy_{1}\right\\}dy_{2}$
$\displaystyle\lesssim\int_{\tau+\frac{\sqrt{2}-\frac{1}{10}}{\epsilon^{2}}}^{\tau+\frac{\sqrt{2}+\frac{1}{10}}{\epsilon^{2}}}\frac{dy_{2}}{\langle
y_{2}\rangle^{2(1-\eta)}}\sim\max_{\pm}\left\\{\frac{1}{\langle\tau+\frac{\sqrt{2}\pm\frac{1}{10}}{\epsilon^{2}}\rangle^{1-2\eta}}\right\\}.$

∎

###### Proof of Lemma 6.2.

For convenience, we denote $P_{low}=P_{\leq\frac{1}{100\epsilon}}$ and
$P_{high}=P_{>\frac{1}{100\epsilon}}$. For the $r_{\epsilon}$-equation in the
system (2.4), after taking out the low frequency piece
$\mathcal{R}_{\epsilon}(P_{low}\psi_{\epsilon})$ from
$\mathcal{R}_{\epsilon}(\psi_{\epsilon})$, we apply Strichartz estimates.
Then, it follows that

$\displaystyle\|r_{\epsilon}\|_{X_{\epsilon}^{1,\frac{1}{2}+\eta}}$
$\displaystyle\lesssim\big{\|}|A_{\epsilon}|^{2}A_{\epsilon}-|\psi_{\epsilon}|^{2}\psi_{\epsilon}+\mathcal{R}_{\epsilon}(A_{\epsilon})-\mathcal{R}_{\epsilon}(P_{low}\psi_{\epsilon})\big{\|}_{X_{\epsilon}^{0,-(\frac{1}{2}-\eta)}}+\|\mathcal{R}_{\epsilon}(P_{low}\psi_{\epsilon})\|_{X_{\epsilon}^{0,-(\frac{1}{2}-\eta)}}$
$\displaystyle=:I+I\\!I.$

For $I$, we apply the trilinear estimate (Lemma 3.6) with the bounds
$\|\psi_{\epsilon}\|_{X_{\epsilon}^{1,1-\eta}},\|r_{\epsilon}\|_{X_{\epsilon}^{1,1-\eta}}\lesssim
1$ (see Proposition 4.3). Then, it follows that

$I\lesssim
T^{\frac{1-2\eta}{2(1+2\eta)}}\big{(}\|r_{\epsilon}\|_{X^{0,\frac{1}{2}+\eta}}+\|P_{high}\psi_{\epsilon}\|_{X^{0,\frac{1}{2}+\eta}}\big{)}.$

Hence, replacing $T>0$ by smaller one if necessary, we obtain

$\|r_{\epsilon}\|_{X_{\epsilon}^{1,\frac{1}{2}+\eta}}\lesssim
I\\!I+\|P_{high}\psi_{\epsilon}\|_{X^{0,\frac{1}{2}+\eta}}.$

It remains to estimate $I\\!I$. Indeed, it is obvious that

$\displaystyle I\\!I$
$\displaystyle\lesssim\big{\|}e^{\frac{2i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}(P_{low}\psi_{\epsilon})^{3}\big{\|}_{X^{0,-(\frac{1}{2}-\eta)}}+\Big{\|}e^{\frac{2i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}\overline{(e^{\frac{2i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}P_{low}\psi_{\epsilon})^{2}}(P_{low}\psi_{\epsilon})\Big{\|}_{X^{0,-(\frac{1}{2}-\eta)}}$
$\displaystyle\quad+\Big{\|}e^{\frac{2i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}\overline{(e^{\frac{2i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}P_{low}\psi_{\epsilon})^{3}}\Big{\|}_{X^{0,-(\frac{1}{2}-\eta)}}$
$\displaystyle=:I\\!I_{1}+I\\!I_{2}+I\\!I_{3}.$

Thus, it suffices to show that for $j=1,2,3$,

$I\\!I_{j}\lesssim\epsilon^{1-2\eta}\|\psi_{\epsilon}\|_{X^{0,1-\eta}}^{3}\lesssim_{R}\epsilon^{1-2\eta}.$
(6.5)

We consider $I\\!I_{1}$ first. Note that by the Plancherel theorem and the
Cauchy-Schwarz inequality, the proof of (6.5) with $j=1$ is reduced to show a
uniform bound for the integral111It is a typical reduction. We refer to [28,
Lemma 3.1] for instance.

$\displaystyle\frac{1}{\langle\tau-\frac{\sqrt{2}}{\epsilon^{2}}+p_{\epsilon}(\xi+\frac{2}{\epsilon})\rangle^{1-2\eta}}\iint_{\mathbb{R}\times\mathbb{R}}$
$\displaystyle\iint_{\mathbb{R}\times\mathbb{R}}\frac{\mathbf{1}_{\\{|\xi_{1}|,|\xi_{2}|,|\xi-\xi_{1}-\xi_{2}|\leq\frac{1}{50\epsilon}\\}}}{\langle\tau-\tau_{1}-\tau_{2}+p_{\epsilon}(\xi-\xi_{1}-\xi_{2})\rangle^{2(1-\eta)}}$
$\displaystyle\qquad\cdot\frac{1}{\langle\tau_{1}+p_{\epsilon}(\xi_{1})\rangle^{2(1-\eta)}\langle\tau_{2}+p_{\epsilon}(\xi_{2})\rangle^{2(1-\eta)}}d\tau_{1}d\tau_{2}d\xi_{1}d\xi_{2}$

Indeed, eliminating $\tau_{1},\tau_{2}$-integrations by the elementary
inequality

$\int_{\mathbb{R}}\frac{d\tau}{\langle\tau-\alpha\rangle^{2(1-\eta)}\langle\tau-\beta\rangle^{2(1-\eta)}}\lesssim\frac{1}{\langle\alpha-\beta\rangle^{2(1-\eta)}},$

it is further reduced to show that

$\mathcal{I}_{1}:=\frac{1}{\langle\tau-\frac{\sqrt{2}}{\epsilon^{2}}+p_{\epsilon}(\xi+\frac{2}{\epsilon})\rangle^{1-2\eta}}\iint_{\mathbb{R}\times\mathbb{R}}\frac{\mathbf{1}_{\\{|\xi_{1}|,|\xi_{2}|,|\xi-\xi_{1}-\xi_{2}|\leq\frac{1}{50\epsilon}\\}}}{\langle\tau+p_{\epsilon}(\xi_{1})+p_{\epsilon}(\xi_{2})+p_{\epsilon}(\xi-\xi_{1}-\xi_{2}))\rangle^{2(1-\eta)}}d\xi_{1}d\xi_{2}$

is bounded uniformly in $\tau,\xi\in\mathbb{R}$. For its proof, we apply Lemma
6.3 with

$\displaystyle\tau+p_{\epsilon}(\xi_{1})+p_{\epsilon}(\xi_{2})+p_{\epsilon}(\xi-\xi_{1}-\xi_{2}))$
$\displaystyle=\tau-\frac{3\sqrt{2}}{\epsilon^{2}}-\frac{\xi}{\epsilon\sqrt{2}}+\frac{1}{\epsilon^{2}}\Big{(}\langle
1+\epsilon\xi_{1}\rangle+\langle 1+\epsilon\xi_{2}\rangle+\langle
1+\epsilon(\xi-\xi_{1}-\xi_{2})\rangle\Big{)}.$

Then, it follows that

$\displaystyle\mathcal{I}_{1}$
$\displaystyle=\frac{\mathbf{1}_{\\{|\xi|\leq\frac{3}{50\epsilon}\\}}}{\langle\tau-\frac{\sqrt{2}}{\epsilon^{2}}+p_{\epsilon}(\xi+\frac{2}{\epsilon})\rangle^{1-2\eta}}\mathcal{I}^{+}(\tau-\tfrac{3\sqrt{2}}{\epsilon^{2}}-\tfrac{\xi}{\epsilon\sqrt{2}},\xi)$
$\displaystyle\lesssim\frac{\mathbf{1}_{\\{|\xi|\leq\frac{3}{50\epsilon}\\}}}{\langle\tau-\frac{\xi}{\epsilon\sqrt{2}}-\frac{3\sqrt{2}}{\epsilon^{2}}+\frac{1}{\epsilon^{2}}\langle
3+\epsilon\xi\rangle\rangle^{1-2\eta}}\max_{\pm}\left\\{\frac{1}{\langle\tau-\frac{\xi}{\epsilon\sqrt{2}}\pm\frac{1}{10\epsilon^{2}}\rangle^{1-2\eta}}\right\\}\lesssim\epsilon^{2(1-2\eta)}.$

Similarly but using that the Fourier transform of
$\overline{e^{\frac{2i}{\epsilon}(x-\frac{t}{\epsilon\sqrt{2}})}A_{\epsilon}}$
is given by
$\overline{\widetilde{A_{\epsilon}}(-\tau+\frac{\sqrt{2}}{\epsilon^{2}},-\xi-\frac{2}{\epsilon})}$
and
$-\tau+\frac{\sqrt{2}}{\epsilon^{2}}+p_{\epsilon}(-\xi-\frac{2}{\epsilon})=-\tau+\frac{\sqrt{2}}{\epsilon^{2}}+\frac{\xi}{\epsilon\sqrt{2}}+\frac{1}{\epsilon^{2}}\langle
1+\epsilon\xi\rangle$, the proof of (6.5) for $j=2,3$ can be respectively
reduced to show that

$\displaystyle\sup_{\tau\in\mathbb{R},|\xi|\leq\frac{3}{50\epsilon}}\frac{1}{\langle\tau-\frac{\xi}{\epsilon\sqrt{2}}+\frac{\sqrt{2}}{\epsilon^{2}}+\frac{1}{\epsilon^{2}}\langle
1-\epsilon\xi\rangle\rangle^{1-2\eta}}\mathcal{I}^{-}\big{(}-\tau+\tfrac{\xi}{\epsilon\sqrt{2}}+\tfrac{3\sqrt{2}}{\epsilon^{2}},\xi\big{)}\lesssim\epsilon^{2(1-2\eta)},$
$\displaystyle\sup_{\tau\in\mathbb{R},|\xi|\leq\frac{3}{50\epsilon}}\frac{1}{\langle\tau-\frac{\xi}{\epsilon\sqrt{2}}+\frac{\sqrt{2}}{\epsilon^{2}}+\frac{1}{\epsilon^{2}}\langle
1-\epsilon\xi\rangle\rangle^{1-2\eta}}\mathcal{I}^{+}\big{(}-\tau+\tfrac{\xi}{\epsilon\sqrt{2}}+\tfrac{3\sqrt{2}}{\epsilon^{2}},\xi\big{)}\lesssim\epsilon^{2(1-2\eta)},$

but these integral inequalities immediately follow from Lemma 6.3. ∎

##  7\. Proof of the main results

We are ready to prove Theorem 1.2 and Theorem 1.6. Indeed, by the remainder
estimate (Proposition 6.1), it is enough to consider the core profile
$\psi_{\epsilon}$ in the system (2.4).

Subtracting the NLS

$\psi_{\epsilon}^{\textup{(NLS)}}(t)=e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}}\psi_{\epsilon,0}-\frac{3i}{2\sqrt{2}}\int_{0}^{t}e^{\frac{i(t-t_{1})}{4\sqrt{2}}\partial_{x}^{2}}\big{(}|\psi_{\epsilon}^{\textup{(NLS)}}|^{2}\psi_{\epsilon}^{\textup{(NLS)}}\big{)}(t_{1})dt_{1}$
(7.1)

from the core profile equation, the difference of the two flows is written as

$\displaystyle\psi_{\epsilon}(t)-\psi_{\epsilon}^{\textup{(NLS)}}(t)$
$\displaystyle=(S_{\epsilon}(t)-e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}})\psi_{\epsilon,0}$
$\displaystyle\quad-\frac{3i}{2}\int_{0}^{t}(S_{\epsilon}(t-t_{1})-e^{\frac{i(t-t_{1})}{4\sqrt{2}}\partial_{x}^{2}})\frac{1}{\langle
1+\epsilon D\rangle}(|\psi_{\epsilon}|^{2}\psi_{\epsilon})(t_{1})dt_{1}$
$\displaystyle\quad-\frac{3i}{2}\int_{0}^{t}e^{\frac{i(t-t_{1})}{4\sqrt{2}}\partial_{x}^{2}}\frac{1}{\langle
1+\epsilon
D\rangle}\big{(}|\psi_{\epsilon}|^{2}\psi_{\epsilon}-|\psi_{\epsilon}^{\textup{(NLS)}}|^{2}\psi_{\epsilon}^{\textup{(NLS)}}\big{)}(t_{1})dt_{1}$
$\displaystyle\quad-\frac{3i}{2}\int_{0}^{t}e^{\frac{i(t-t_{1})}{4\sqrt{2}}\partial_{x}^{2}}\left(\frac{1}{\langle
1+\epsilon
D\rangle}-\frac{1}{\sqrt{2}}\right)\big{(}|\psi_{\epsilon}^{\textup{(NLS)}}|^{2}\psi_{\epsilon}^{\textup{(NLS)}}\big{)}(t_{1})dt_{1}.$

For the third term on the right hand side, we note that by the uniform bounds
for nonlinear solutions ((4.5) for $\psi_{\epsilon}^{\textup{(NLS)}}$ and
(4.1) with the transference principle for $\psi_{\epsilon}$),

$\displaystyle\left\||\psi_{\epsilon}|^{2}\psi_{\epsilon}-|\psi_{\epsilon}^{\textup{(NLS)}}|^{2}\psi_{\epsilon}^{\textup{(NLS)}}\right\|_{L_{t}^{1}L_{x}^{2}}$
$\displaystyle\leq
T^{1/2}\left(\|\psi_{\epsilon}\|_{L_{t}^{4}L_{x}^{\infty}}^{2}+\|\psi_{\epsilon}^{\textup{(NLS)}}\|_{L_{t}^{4}L_{x}^{\infty}}^{2}\right)\|\psi_{\epsilon}(t)-\psi_{\epsilon}^{\textup{(NLS)}}\|_{C_{t}L_{x}^{2}}$
$\displaystyle\lesssim
T^{1/2}\|\psi_{\epsilon}(t)-\psi_{\epsilon}^{\textup{(NLS)}}\|_{C_{t}L_{x}^{2}},$

where $T>0$ is sufficiently small but independent of $\epsilon>0$ and the time
interval $[-T,T]$ in norms is omitted for convenience. Hence, we have that for
$s=0,1$,

$\displaystyle\|\psi_{\epsilon}(t)-\psi_{\epsilon}^{\textup{(NLS)}}(t)\|_{C_{t}H_{\epsilon}^{s}}$
$\displaystyle\lesssim\|(S_{\epsilon}(t)-e^{\frac{it}{4\sqrt{2}}\partial_{x}^{2}})\psi_{\epsilon,0}\|_{C_{t}H_{\epsilon}^{s}}$
(7.2)
$\displaystyle\quad+\int_{0}^{T}\big{\|}(S_{\epsilon}(t-t_{1})-e^{\frac{i(t-t_{1})}{4\sqrt{2}}\partial_{x}^{2}})(|\psi_{\epsilon}|^{2}\psi_{\epsilon})(t_{1})\big{\|}_{C_{t}L_{x}^{2}}dt_{1}$
$\displaystyle\quad+\left\|\left(\frac{1}{\langle 1+\epsilon
D\rangle}-\frac{1}{\sqrt{2}}\right)\big{(}|\psi_{\epsilon}^{\textup{(NLS)}}|^{2}\psi_{\epsilon}^{\textup{(NLS)}}\big{)}\right\|_{L_{t}^{1}H_{\epsilon}^{s}}.$

For Theorem 1.2, we take $s=1$ in (7.2), and recall that by the high frequency
estimates ((H2), Proposition 5.1 and (5.3)),

$\|P_{>\delta\epsilon^{-1/3}}\psi_{\epsilon,0}\|_{H_{\epsilon}^{1}},\|P_{>\delta\epsilon^{-1/3}}(|\psi_{\epsilon}|^{2}\psi_{\epsilon})\|_{L_{t}^{1}L_{x}^{2}},\|P_{>\delta\epsilon^{-1/3}}(|\psi_{\epsilon}^{\textup{(NLS)}}|^{2}\psi_{\epsilon}^{\textup{(NLS)}})\|_{L_{t}^{1}L_{x}^{2}}\to
0$

for any $\delta>0$. Thus, by Lemma 3.1, we conclude that
$\|\psi_{\epsilon}(t)-\psi_{\epsilon}^{\textup{(NLS)}}(t)\|_{C_{t}H_{\epsilon}^{1}}\to
0$.

For Theorem 1.6, we assume that (H2’) holds, and take $s=0$ in (7.2). Then, by
(4.3) and (4.6), we have that $\|\psi_{\epsilon,0}\|_{H^{s}}$,
$\||\psi_{\epsilon}|^{2}\psi_{\epsilon}\|_{L_{t}^{1}H_{x}^{s}}$ and
$\||\psi_{\epsilon}^{\textup{(NLS)}}|^{2}\psi_{\epsilon}^{\textup{(NLS)}}\|_{L_{t}^{1}H_{x}^{s}}$
are uniformly bounded. Thus, following the proof of Lemma 3.1, one can show
that
$\|\psi_{\epsilon}(t)-\psi_{\epsilon}^{\textup{(NLS)}}(t)\|_{C_{t}L_{x}^{2}}\lesssim\epsilon^{\frac{\epsilon}{3}}$.
Finally, combining with the remainder estimate (Proposition 6.1), we complete
the proof.

## References

  * [1] J. Bourgain, _Global solutions of nonlinear Schrödinger equations_ , American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI, 1999. viii+182 pp.
  * [2] T. Cazenave, _Semilinear Schrödinger equations_ , Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003.
  * [3] W. Craig, P. Guyenne and C. Sulem, _A Hamiltonian approach to nonlinear modulation of surface water waves_ , Wave Motion 47 (2010), no. 8, 552–563.
  * [4] W. Craig, C. Sulem and P.-L. Sulem, _Nonlinear modulation of gravity waves: a rigorous approach_ , Nonlinearity 5 (1992), no. 2, 497–522.
  * [5] W. Dörfler, A. Lechleiter, M. Plum, G. Schneider and C. Wieners, _The role of the nonlinear Schrödinger equation in nonlinear optics_ , Photonic Crystals: Mathematical Analysis and Numerical Approximation, Oberwolfach Seminars, Vol. 42, (2011), Springer, Basel.
  * [6] W.-P. Düll, _Validity of the nonlinear Schrödinger approximation for the two-dimensional water wave problem with and without surface tension in the arc length formulation_ , Arch. Ration. Mech. Anal. 239 (2021), no. 2, 831–914.
  * [7] W.-P. Düll and G. Schneider, _Justification of the nonlinear Schrödinger equation for a resonant Boussinesq model_ , Indiana Univ. Math. J. 55 (2006), no. 6, 1813–1834.
  * [8] W.-P. Düll, G. Schneider and C.E. Wayne, _Justification of the nonlinear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth_ , Arch. Ration. Mech. Anal. 220 (2016), no. 2, 543–602.
  * [9] R. Feola and F. Giuliani, _Long time NLS approximation for the quasilinear Klein-Gordon equation on large domains under periodic boundary conditions_ , arXiv:2206.11836.
  * [10] H. Hasimoto and H. Ono, _Nonlinear modulation of gravity waves_ , J. Phys. Soc. Jpn. 33, (1972), 805–811.
  * [11] Y. Hong and C. Yang, _Strong convergence for discrete nonlinear Schrödinger equations in the continuum limit_ , SIAM J. Math. Anal. 51 (2019), no. 2, 1297–1320.
  * [12] M. Ifrim and D. Tataru, _The NLS approximation for two dimensional deep gravity waves_ , Sci. China Math. 62, (2019), 1101–1120.
  * [13] M. Keel and T. Tao, _Endpoint Strichartz estimates_ , Amer. J. Math. 120 (1998), no. 5, 955–980.
  * [14] K. Kirkpatrick, E. Lenzmann and G. Staffilani, _On the continuum limit for discrete NLS with long-range lattice interactions_ , Comm. Math. Phys. 317 (2013), no. 3, 563–591.
  * [15] J. Krieger, E. Lenzmann, and P. Raphael, _Nondispersive solutions to the L2-critical half-wave equation_ , Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129.
  * [16] S. Machihara, K. Nakanishi and T. Ozawa, _Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations_ , Math. Ann. 322 (2002), no. 3, 603–621.
  * [17] N. Masmoudi and K. Nakanishi, _Multifrequency NLS scaling for a model equation of gravity-capillary waves_ , Comm. Pure Appl. Math. 66 (2013), no. 8, 1202–1240.
  * [18] K. Matthies and H. Uecker, _Low regularity justification results for envelope approximations of nonlinear wave packets in periodic media_ , Asymptot. Anal. 99 (2016), no. 1-2, 53–65.
  * [19] G. Schneider, _Approximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation_ , J. Differ. Equ. 147, no. 2, (1998), 333–354.
  * [20] G. Schneider, _Bounds for the nonlinear Schrödinger approximation of the Fermi-Pasta-Ulam system_ , Appl. Anal. 89 (2010), no. 9, 1523–1539.
  * [21] G. Schneider, _Justification of the NLS approximation for the KdV equation using the Miura transformation_ , Adv. Math. Phys., Art. ID 854719, (2011), 4pp.
  * [22] Y. Shi and J. E. Hearst, _The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling_ , J. Chem. Phys. 101, no. 6, (1994), 5186–5200.
  * [23] G. Schneider and H. Uecker, _Nonlinear PDEs. A dynamical systems approach_ , Graduate Studies in Mathematics, 182. American Mathematical Society, Providence, RI, 2017. xiii+575 pp.
  * [24] G. Schneider and C. E. Wayne, _The long-wave limit for the water wave problem. I. The case of zero surface tension_ , Comm. Pure Appl. Math. 53 (2000), no. 12, 1475–1535.
  * [25] G. Schneider and C. E. Wayne, _The rigorous approximation of long-wavelength capillary - gravity waves_ , Arch. Ration. Mech. Anal. 162, (2002), 247–285.
  * [26] G. Schneider and C. E. Wayne, _Justification of the NLS approximtion for a quasilinear water wave model_ , J. Differ. Equ. 251, (2011), 238–269.
  * [27] C. Sulem and P.-L. Sulem, _The nonlinear Schrödinger equation. Self-focusing and wave collapse_ , Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. xvi+350 pp.
  * [28] T. Tao, _Multilinear weighted convolution of $L^{2}$ functions, and applications to non-linear dispersive equations_, Amer. J. Math. 123, (2001), 839–908.
  * [29] T. Tao, _Nonlinear dispersive equations. Local and global analysis_ , CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373 pp.
  * [30] N. Totz, _A justification of the modulation approximation to the 3D full water wave problem_ , Comm. Math. Phys. 335 (2015), no. 1, 369–443.
  * [31] N. Totz and S. Wu, _A rigorous justification of the modulation approximation to the 2D full water wave problem_ , Comm. in Math. Phys., 310, no. 3, (2012), 817–883.
  * [32] V. E. Zakharov, _Stability of periodic waves of finite amplitude on the surface of a deep fluid_ , J. Appl. Mech. Tech. Phys. 9, 1968, 190–194.