Large time asymptotics for a cubic nonlinear Schr\"odinger system in one space dimension

TL;DR

This paper studies the long-term behavior of a two-component cubic nonlinear Schrödinger system in one dimension, revealing that solutions resemble free solutions over time with specific profile restrictions due to long-range interactions.

Contribution

It demonstrates the asymptotic free behavior of solutions and uncovers the influence of long-range nonlinear interactions on profile restrictions.

Findings

Solutions behave like free solutions at large times

Profiles of components are strongly restricted by nonlinear interactions

Long-range effects significantly influence asymptotic behavior

Abstract

We consider a two-component system of cubic nonlinear Schr\"odinger equations in one space dimension. We show that each component of the solutions to this system behaves like a free solution in the large time, but there is a strong restriction between the profiles of them. This turns out to be a consequence of non-trivial long-range nonlinear interactions.

Equations

$$\displaystyle\left\{\begin{array}[]{ll}\begin{array}[]{l}\mathcal{L}u_{1}=-i|u_{2}|^{2}u_{1},\\ \mathcal{L}u_{2}=-i|u_{1}|^{2}u_{2},\end{array}&(t,x)\in(0,\infty)\times\mathbb{R},\end{array}\right.$$

$$u_j(0,x)={\phi }_j^0(x),x\in \mathbb{R},j=1,2,$$

$$\mathcal{L}u=\lambda |u{|}^{2}u$$

$$u(t,x)=\frac{1}{\sqrt{it}}{\alpha }^{\pm }(x/t)e^{i\{\frac{x^2}{2t}-\lambda |{\alpha }^{\pm }(x/t){|}^2\log t\}}+o(t^{-1/2})\text{mbox}in{L}^{\infty }({\mathbb{R}}_x)$$

$$\frac{d}{dt}\Bigl{(}\|u_{1}(t)\|_{L^{2}}^{2}+\|u_{2}(t)\|_{L^{2}}^{2}\Bigr{)}=-2\int_{\mathbb{R}}|u_{1}(t,x)|^{2}|u_{2}(t,x)|^{2}\,dx$$

$$\displaystyle\frac{d}{dt}\Bigl{(}\|u_{1}(t)\|_{L^{2}}^{2}-\|u_{2}(t)\|_{L^{2}}^{2}\Bigr{)}=0.$$

$$\displaystyle\left\{\begin{array}[]{ll}\begin{array}[]{l}(\partial_{t}^{2}-\Delta)v_{1}=-|\partial_{t}v_{2}|^{2}\partial_{t}v_{1},\\ (\partial_{t}^{2}-\Delta)v_{2}=-|\partial_{t}v_{1}|^{2}\partial_{t}v_{2},\end{array}&(t,x)\in(0,\infty)\times\mathbb{R}^{2},\end{array}\right.$$

$$\mathcal{F}\varphi (\xi )=\hat{\varphi }(\xi )=\frac{1}{\sqrt{2\pi }}{\int }_{\mathbb{R}}{e}^{-iy\xi }\varphi (y)dy,\xi \in \mathbb{R}.$$

$$\underset{t\to +\infty }{\lim }\Vert u_j(t)-\mathcal{U}(t){\phi }_j^{+}{\Vert }_{L^2}=0,j=1,2.$$

$${\hat{\phi }}_1^{+}(\xi )\cdot {\hat{\phi }}_2^{+}(\xi )=0,\xi \in \mathbb{R}.$$

$$\mathcal{J}=\mathcal{U}(t)x\mathcal{U}(-t),$$

$$\Vert \varphi {\Vert }_{L^{\infty }}\le C{t}^{-1/2}\Vert \varphi {\Vert }_{L^2}^{1/2}\Vert \mathcal{J}\varphi {\Vert }_{L^2}^{1/2},t>0,$$

$$\mathcal{J}({\varphi }_1{\varphi }_2\overline{{\varphi }_3})=(\mathcal{J}{\varphi }_1){\varphi }_2\overline{{\varphi }_3}+{\varphi }_1(\mathcal{J}{\varphi }_2)\overline{{\varphi }_3}-{\varphi }_1{\varphi }_2\overline{\mathcal{J}{\varphi }_3}.$$

$$\displaystyle\alpha_{j}(t,\xi)=\mathcal{F}\Bigl{[}\mathcal{U}(-t)u_{j}(t,\cdot)\Bigr{]}(\xi)$$

$${\partial }_t{\alpha }_1=-i\mathcal{F}\mathcal{U}(-t)\mathcal{L}{u}_1=-\mathcal{F}\mathcal{U}(-t)(|u_2{|}^2{u}_1)=-\frac{1}{t}|{\alpha }_2{|}^2{\alpha }_1+R_1,$$

$$\displaystyle R_{1}=\frac{1}{t}|\alpha_{2}|^{2}\alpha_{1}-\mathcal{F}\mathcal{U}(-t)\bigl{[}|u_{2}|^{2}u_{1}\bigr{]}.$$

$${\partial }_t{\alpha }_2=-\frac{1}{t}|{\alpha }_1{|}^2{\alpha }_2+R_2,$$

$$\displaystyle R_{2}=\frac{1}{t}|\alpha_{1}|^{2}\alpha_{2}-\mathcal{F}\mathcal{U}(-t)\bigl{[}|u_{1}|^{2}u_{2}\bigr{]}.$$

$$\displaystyle|R(t,\xi)|\leq\frac{C}{t^{5/4}\langle\xi\rangle}\bigl{(}\|u(t)\|_{H^{1}}+\|\mathcal{J}u(t)\|_{H^{1}}\bigr{)}^{3}.$$

$$\Vert u(t){\Vert }_{H^2}+\Vert \mathcal{J}u(t){\Vert }_{H^1}\le C\epsilon ⟨t{⟩}^{\gamma }$$

$$|\alpha (t,\xi )|\le C\epsilon ⟨\xi {⟩}^{-1}$$

$$|R(t,\xi )|$$

$${\partial }_t{\alpha }_1=-\frac{1}{t}|{\alpha }_2{|}^2{\alpha }_1,{\partial }_t{\alpha }_2=-\frac{1}{t}|{\alpha }_1{|}^2{\alpha }_2$$

$$\underset{t\to +\infty }{\lim }\Vert {\alpha }_j(t)-{\alpha }_j^{+}{\Vert }_{L^2}=0$$

$$\Vert u_j(t)-\mathcal{U}(t){\phi }_j^{+}{\Vert }_{L^2}=\Vert \mathcal{F}(\mathcal{U}(-t)u_j(t)-{\phi }_j^{+}){\Vert }_{L^2}=\Vert {\alpha }_j(t)-{\alpha }_j^{+}{\Vert }_{L^2}\to 0$$

$$\frac{d\Phi }{dt}(t)\le -\frac{C_0}{t}{\left|\Phi (t)\right|}^p+\frac{C_1}{t^q}$$

$$\Phi (t)\le \frac{C_2}{(\log t{)}^{p^{\ast }-1}}$$

$$C_2=\frac{1}{\log 2}\left((\log t_0{)}^{p^{\ast }}\Phi (t_0)+C_1{\int }_2^{\infty }\frac{(\log \tau {)}^{p^{\ast }}}{{\tau }^q}d\tau \right)+{\left(\frac{p^{\ast }}{C_{0}p}\right)}^{p^{\ast }-1}.$$

$$\frac{dy}{dt}(t)=\lambda (t)y(t)+Q(t)$$

$$|y(t)-y^{+}|\le C_3{\int }_t^{\infty }(|y^{+}||\lambda (\tau )|+|Q(\tau )|)d\tau$$

$$C_3=\exp \left({\int }_{t_0}^{\infty }|\lambda (\tau )|d\tau \right)$$

Full text

Large time asymptotics for a cubic nonlinear Schrödinger system

in one space dimension

Chunhua Li

Department of Mathematics, College of Science, Yanbian University. 977 Gongyuan Road, Yanji, Jilin 133002, China. (E-mail: sxlch@ybu.edu.cn)

  

Yoshinori Nishii

Department of Mathematics, Graduate School of Science, Osaka University. 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan. (E-mail: y-nishii@cr.math.sci.osaka-u.ac.jp)   

Yuji Sagawa

Department of Mathematics, Graduate School of Science, Osaka University. 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan.

  

Hideaki Sunagawa

Department of Mathematics, Graduate School of Science, Osaka University. 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan. (E-mail: sunagawa@math.sci.osaka-u.ac.jp)

Abstract: We consider a two-component system of cubic nonlinear Schrödinger equations in one space dimension. We show that each component of the solutions to this system behaves like a free solution in the large time, but there is a strong restriction between the profiles of them. This turns out to be a consequence of non-trivial long-range nonlinear interactions.

Key Words: Nonlinear Schrödinger system, asymptotically free, long-range interaction.

2010 Mathematics Subject Classification: 35Q55, 35B40.

1 Introduction

This paper deals with the global in time behavior of solutions $u=(u_{1}(t,x),u_{2}(t,x))$ to

[TABLE]

with the initial condition

[TABLE]

where $i=\sqrt{-1}$, $\mathcal{L}=i\partial_{t}+(1/2)\partial_{x}^{2}$, and $\varphi^{0}=(\varphi^{0}_{1}(x),\varphi_{2}^{0}(x))$ is a given $\mathbb{C}^{2}$-valued function of $x\in\mathbb{R}$ which belongs to an appropriate weighted Sobolev space and satisfies a suitable smallness condition.

First of all, let us summarize the backgrounds briefly. As is well-known, cubic nonlinearity gives a critical situation when we consider large time behavior of solutions to the nonlinear Schrödinger equation in $\mathbb{R}$. In general, cubic nonlinearity should be regarded as a long-range perturbation. For example, according to Hayashi–Naumkin [3], the small data solution $u(t,x)$ to

[TABLE]

with $\lambda\in\mathbb{R}\backslash\{0\}$ behaves like

[TABLE]

as $t\to\pm\infty$, where $\alpha^{\pm}$ is a suitable $\mathbb{C}$-valued function on $\mathbb{R}$. An important consequence of this asymptotic expression is that the solution to (1.6) decays like $O(|t|^{-1/2})$ uniformly in $x\in\mathbb{R}$, while it does not behave like the free solution (unless $\lambda=0$). In other words, the additional logarithmic correction in the phase reflects a typical long-range character of the cubic nonlinear Schrödinger equations in one space dimension. If $\lambda\in\mathbb{C}$ in (1.6), another kind of long-range effect can be observed. For instance, according to [15] (see also [8], [5], [1], etc.), the small data solution $u(t,x)$ to (1.6) decays like $O(t^{-1/2}(\log t)^{-1/2})$ in $L^{\infty}(\mathbb{R}_{x})$ as $t\to+\infty$ if $\operatorname{\rm Im}\lambda<0$. This gain of additional logarithmic time decay should be interpreted as another kind of long-range effect. There are various extensions of these results. In the previous works [10] and [11], several structural conditions on the nonlinearity have been introduced under which the small data global existence holds for a class of cubic nonlinear Schrödinger systems in $\mathbb{R}$, and large time asymptotic behavior of the global solutions have also been investigated (see also [7], [14] and the references cited therein for related works). We do not state these conditions here, but we only point out that the small data global existence for (1.4) follows from the results of [10] and [7] but the large time asymptotic behavior of solutions is not covered by these results. We note that the system (1.4) possesses two conservation laws

[TABLE]

and

[TABLE]

However, these are not enough to say something about the large time asymptotics for $u(t)$, and this is not trivial at all. To the authors’ knowledge, there are no previous results which cover the asymptotic behavior of solutions to (1.4)–(1.5).

Our motivation of considering (1.4) comes from the recent work [12], in which the system of semilinear wave equations

[TABLE]

has been treated in connection with the Agemi-type structural condition (that is a kind of weak null conditions). From the viewpoint of conservation laws, there are a lot of similarities between (1.4) and (1.11). It has been shown in [12] that global solutions to (1.11) with small data behaves like solutions to the free wave equations, but there is a strong restriction in the profiles. Although the approach of [12] does not use the conservation laws directly, it may be natural to expect that an analogous phenomenon can be observed for solutions to (1.4). The aim of the present paper is to reveal it.

Before stating the main result, let us introduce some notations. For $s$, $s^{\prime}\geq 0$, we denote by $H^{s}$ the $L^{2}$-based Sobolev space of order $s$, and the weighted Sobolev space $H^{s,s^{\prime}}$ is defined by $\{\phi\in L^{2}\,|\,\langle\,\cdot\,\rangle^{s^{\prime}}\phi\in H^{s}\}$ equipped with the norm $\|\phi\|_{H^{s,s^{\prime}}}=\|\langle\,\cdot\,\rangle^{s^{\prime}}\phi\|_{H^{s}}$, where $\langle x\rangle=\sqrt{1+x^{2}}$. The Fourier transform $\mathcal{F}$ is defined by

[TABLE]

We also set $\mathcal{U}(t)=\exp(i\frac{t}{2}\partial_{x}^{2})$, so that $\mathcal{U}(t)\phi=:w(t)$ solves the free Schrödinger equation $\mathcal{L}w=0$ with $w(0)=\phi$.

The main result is as follows.

Theorem 1.1**.**

Suppose that $\varphi^{0}=(\varphi_{1}^{0},\varphi_{2}^{0})\in H^{2}\cap H^{1,1}$ and $\varepsilon=\|\varphi^{0}\|_{H^{2}\cap H^{1,1}}$ is suitably small. Let $u=(u_{1},u_{2})\in C([0,\infty);H^{2}\cap H^{1,1})$ be the solution to (1.4)–(1.5). Then there exists $\varphi^{+}=(\varphi_{1}^{+},\varphi_{2}^{+})\in L^{2}$ with $\hat{\varphi}^{+}=(\hat{\varphi}_{1}^{+},\hat{\varphi}_{2}^{+})\in L^{\infty}$ such that

[TABLE]

Moreover we have

[TABLE]

Remark 1.1*.*

We emphasize that (1.12) should be regarded as a consequence of non-trivial long-range nonlinear interactions because such a phenomenon does not occur in the usual short-range situation. To complement this point, we will give auxiliary results on the final state problem for (1.4) in Appendix B.

Remark 1.2*.*

In the case where $u_{1}=u_{2}$, the system (1.4) is reduced to the single equation (1.6) with $\lambda=-i,$ so we can adapt the result of [1] to see that $\|u(t)\|_{L^{2}}$ converges to [math] as $t\to+\infty$ without restrictions on the size of the initial data. However, this is an exceptional case. We are interested in general situations of (1.4)–(1.5). When $\|\varphi_{1}^{0}\|_{L^{2}}\neq\|\varphi_{2}^{0}\|_{L^{2}}$ in (1.5), it follows from the conservation law (1.7) and the $L^{2}$-unitarity of $\mathcal{U}(t)$ and $\mathcal{F}$ that at least one of $\hat{\varphi}_{1}^{+}$ or $\hat{\varphi}_{2}^{+}$ does not identically vanish. This implies that solutions $u=(u_{1},u_{2})$ to (1.4)–(1.5) do not converge to [math] as $t\to+\infty$ in $L^{2}$ for generic initial data being suitably small. In this sense, our problem is much more delicate than the single case (1.6) with $\lambda=-i$.

Remark 1.3*.*

It is worthwhile to note that the presence of $-i$ in the right-hand sides of (1.4) is essential for our result. Indeed, if we drop $-i$ from the right-hand sides of (1.4) (that is, $\mathcal{L}u_{1}=|u_{2}|^{2}u_{1}$ and $\mathcal{L}u_{2}=|u_{1}|^{2}u_{2}$), we can show that the solutions have logarithmic phase corrections as in the single case (1.6) with $\lambda\in\mathbb{R}\backslash\{0\}$ (see e.g. [16] for detail).

Remark 1.4*.*

The above theorem concerns only the forward Cauchy problem (i.e., for $t>0$). In the backward case (or, equivalently, in the case where $-i$ is replaced by $+i$ in the right-hand sides of (1.4)), even the small data global existence may fail in general. See [13] and the references cited therein for more information and the related works on this issue.

We close this section with the contents of this paper. We introduce some preliminary lemmas in Section 2. Theorem 1.1 is proved in Section 3. In Appendix A, we give a proof of a technical lemma. Appendix B is devoted to auxiliary observations on the final state problem for (1.4).

2 Preliminaries

In this section, we collect several identities and inequalities which are useful in the proof of Theorem 1.1. In what follows, we will denote various positive constants by the same letter $C$, which may vary from one line to another. We set $\mathcal{J}=x+it\partial_{x}$. It is well-known that $[\mathcal{L},\mathcal{J}]=0$ and $[\partial_{x},\mathcal{J}]=1$, where $[\cdot,\cdot]$ stands for the commutator, that is, $[\mathcal{A},\mathcal{B}]=\mathcal{A}\mathcal{B}-\mathcal{B}\mathcal{A}$ for two linear operators $\mathcal{A}$ and $\mathcal{B}$. Also we have

[TABLE]

and

[TABLE]

Let $u=(u_{1},u_{2})$ be a smooth solution to (1.4)–(1.5) on $[0,\infty)\times\mathbb{R}$. We define $\alpha=(\alpha_{1},\alpha_{2})$ by

[TABLE]

for $j=1,2$. Then from (1.4) it follows that

[TABLE]

where

[TABLE]

Similarly we have

[TABLE]

where

[TABLE]

Concerning estimates for $R=(R_{1},R_{2})$, we have the following estimate.

Lemma 2.1**.**

Let $R$ be as above. For $t\geq 1$, we have

[TABLE]

This estimate is not a new one (see e.g. Lemma 5.2 in [10]). For the convenience of the readers, we will give a proof in Appendix A.

Next we recall the basic decay estimates for global solutions $u$ to (1.4)–(1.5). From the argument of [10], we already know the following result.

Lemma 2.2**.**

Let $0<\gamma<1/12$. Suppose that $\varepsilon=\|\varphi^{0}\|_{H^{2}\cap H^{1,1}}$ is suitably small. Then the solution $u$ to (1.4)–(1.5) satisfies

[TABLE]

for $t\geq 0$ and

[TABLE]

for $t\geq 0$, $\xi\in\mathbb{R}$, where $\alpha$ is given by (2.3).

As a by-product of Lemmas 2.1 and 2.2, we have

[TABLE]

for $t\geq 1$. Roughly speaking, this means that the evolution of $\alpha=(\alpha_{1},\alpha_{2})$ may be governed by

[TABLE]

up to the harmless remainders. We also note that $u(t)=\mathcal{U}(t)\mathcal{F}^{-1}\alpha(t)$. This point of view, whose original idea goes back to Hayashi–Naumkin [3], is the key of our approach.

3 Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. The main step is to show the following.

Proposition 3.1**.**

Let $\alpha=(\alpha_{1}(t,\xi),\alpha_{2}(t,\xi))$ be given by (2.3) for the solution $u=(u_{1},u_{2})$ to (1.4) satisfying the assumptions of Theorem 1.1. There exists $\alpha^{+}=(\alpha_{1}^{+}(\xi),\alpha_{2}^{+}(\xi))\in L^{2}\cap L^{\infty}$ such that

[TABLE]

for $j=1,2$. Moreover we have $\alpha_{1}^{+}(\xi)\cdot\alpha_{2}^{+}(\xi)=0$ for $\xi\in\mathbb{R}$.

Once this proposition is obtained, we can derive Theorem 1.1 immediately by setting $\varphi_{j}^{+}=\mathcal{F}^{-1}\alpha_{j}^{+}$. Indeed we have

[TABLE]

as $t\to+\infty$.

In the rest of this section, we will prove Proposition 3.1. Note that many parts of the arguments below are simliar to those in [12], though we need several modifications to fit for the present situation. Before going into the proof, let us recall the following lemmas.

Lemma 3.1**.**

Let $C_{0}>0$, $C_{1}\geq 0$, $p>1$, $q>1$ and $t_{0}\geq 2$. Suppose that $\Phi(t)$ satisfies

[TABLE]

for $t\geq t_{0}$. Then we have

[TABLE]

for $t\geq t_{0}$, where $p^{*}$ is the Hölder conjugate of $p$ (i.e., $1/p+1/p^{*}=1$), and

[TABLE]

For the proof, see Lemma 4.1 in [6].

Lemma 3.2**.**

Let $t_{0}>0$ be given. For $\lambda$, $Q\in C\cap L^{1}([t_{0},\infty))$, assume that $y(t)$ satisfies

[TABLE]

for $t\geq t_{0}$. Then we have

[TABLE]

for $t\geq t_{0}$, where

[TABLE]

and

[TABLE]

For the proof, see Lemma 4.2 in [12].

Proof of Proposition 3.1. We first show the pointwise convergence of $\alpha(t,\xi)$ as $t\to+\infty$. We fix $\xi\in\mathbb{R}$ and introduce

[TABLE]

Then it follows from (2.4) and (2.5) that

[TABLE]

Also (2.6) and (2.7) lead to

[TABLE]

for $0<\gamma<1/12$. Therefore we obtain

[TABLE]

where

[TABLE]

and

[TABLE]

for $t\geq 2$. Note that

[TABLE]

and

[TABLE]

for $0<\gamma<1/12$. Now we divide the argument into three cases according to the sign of $m(\xi)$ as follows.

• •

Case 1: $m(\xi)>0$.**  First we focus on the asymptotics for $\alpha_{2}$. By (3.2), we can rewrite (2.5) as**

[TABLE]

for $t\geq 2$. So we have

[TABLE]

for $t\geq 2$, whence

[TABLE]

Integration in $t$ leads to

[TABLE]

for $t\geq 2$. Therefore we see that

[TABLE]

In particular, $\alpha_{2}(t,\xi)\to 0$ as $t\to+\infty$. Next we turn our attentions to the asymptotics for $\alpha_{1}$. Since (2.4) can be viewed as

[TABLE]

with $\lambda(t)=-|\alpha_{2}(t,\xi)|^{2}/{t}$ and $Q(t)=R_{1}(t,\xi)$, we can apply Lemma 3.2 to obtain

[TABLE]

for $t\geq 2$, where

[TABLE]

By (2.6), (2.7) and (3.4), we have

[TABLE]

and

[TABLE]

for $t\geq 2$. Therefore we conclude that $\alpha_{1}(t,\xi)\to\beta_{1}^{+}(\xi)$ as $t\to+\infty$.

• •

Case 2: $m(\xi)<0$.** Similarly to the previous case, we have**

[TABLE]

for each fixed $\xi\in\mathbb{R}$, where

[TABLE]

Remark that $|\beta_{2}^{+}(\xi)|\leq C\varepsilon\langle\xi\rangle^{-1}$.

• •

Case 3: $m(\xi)=0$.** By (2.4), (2.6), (2.7), (3.2) and (3.3), we have**

[TABLE]

for $t\geq 2$, and $0<\gamma<1/12$. Thus we can apply Lemma 3.1 with $\Phi(t)=|\alpha_{1}(t,\xi)|^{2}$ to obtain

[TABLE]

Also (3.2) gives us $|\alpha_{2}(t,\xi)|=\sqrt{|\alpha_{1}(t,\xi)|^{2}+r(t,\xi)}\to 0$ as $t\to+\infty$.

Summing up the three cases above, we deduce that $\alpha(t,\xi)$ converges as $t\to+\infty$ for each fixed $\xi\in\mathbb{R}$. To obtain (3.1), we set

[TABLE]

and $\alpha^{+}(\xi)=(\alpha^{+}_{1}(\xi),\alpha^{+}_{2}(\xi))$ for $\xi\in\mathbb{R}$, where $\beta_{1}^{+}(\xi)$ and $\beta_{2}^{+}(\xi)$ are shown in Cases 1 and 2, respectively. Then it is obvious that $\alpha_{1}^{+}(\xi)\cdot\alpha_{2}^{+}(\xi)=0$ for $\xi\in\mathbb{R}$. Also, by virtue of (3.5), we have $\alpha^{+}\in L^{2}\cap L^{\infty}(\mathbb{R})$ and

[TABLE]

for $t\geq 2$. Moreover, it holds that

[TABLE]

for each fixed $\xi\in\mathbb{R}$. Therefore Lebesgue’s dominated convergence theorem yields (3.1). ∎

Appendix A Proof of Lemma 2.1

We give a proof of Lemma 2.1. For this purpose, we introduce some notations. We define the operators $\mathcal{M}(t)$, $\mathcal{D}(t)$ and $\mathcal{W}(t)$ by

[TABLE]

so that $\mathcal{U}(t)=\exp(i\frac{t}{2}\partial_{x}^{2})$ is decomposed into

[TABLE]

An important estimate is

[TABLE]

which comes from the Gagliardo-Nirenberg inequality $\|\phi\|_{L^{\infty}}\leq C\|\phi\|_{L^{2}}^{1/2}\|\partial_{x}\phi\|_{L^{2}}^{1/2}$ and the inequality

[TABLE]

with $\theta=x^{2}/(2t),\sigma=1/2$. Note also that

[TABLE]

and

[TABLE]

where the constant $C$ is independent of $t$ (see e.g., [10] for the proof). In what follows, we will occasionally omit “$(t)$” from $\mathcal{M}(t)$, $\mathcal{D}(t)$, $\mathcal{W}(t)$ if it causes no confusion.

Let $\alpha$ be given by (2.3). By (A.1), we have

[TABLE]

whence

[TABLE]

Therefore (A.2), (A.4), (A.5) and the Sobolev imbedding $H^{1}(\mathbb{R})\hookrightarrow L^{\infty}(\mathbb{R})$ lead to

[TABLE]

Next we observe that

[TABLE]

where $\alpha_{j}^{(1)}=i\xi\alpha_{j}$. Then we see as before that

[TABLE]

The desired estimate for $R_{1}$ follows immediately from (A.6) and (A.7). The estimate for $R_{2}$ can be shown in the same way. ∎

Appendix B Final state problem for (1.4)

To complement Remark 1.1, we give two auxiliary results on the final state problem for (1.4), that is, finding a solution $u=(u_{1},u_{2})$ to (1.4) which satisfies

[TABLE]

for a prescribed final state $\psi^{+}=(\psi_{1}^{+},\psi_{2}^{+})$. Roughly speaking, the propositions below imply that (B.1) holds if and only if

[TABLE]

This indicates that our problem must be distinguished from the usual short-range situation, because (B.1) should hold in the short-range case regardless of whether (B.2) is true or not (see e.g. [2]).

The precise statements are as follows.

Proposition B.1**.**

Let $T_{0}\geq 1$ be given, and let $u$ be a solution to (1.4) for $t\geq T_{0}$ satisfying

[TABLE]

with some $\gamma\in(0,1/12)$. If there exists $\psi^{+}\in L^{2}$ with $\hat{\psi}^{+}\in L^{\infty}$ such that (B.1) holds, then we must have (B.2).

Proposition B.2**.**

Suppose that $\psi^{+}$ satisfies $\hat{\psi}^{+}\in H^{0,s}\cap L^{\infty}$ with some $s>1$, and that $\delta=\|\hat{\psi}^{+}\|_{L^{\infty}}$ is suitably small. If (B.2) holds, then there exist $T\geq 1$ and a unique solution $u$ to (1.4) for $t\geq T$ satisfying $\mathcal{U}(-t)u\in C([T,\infty);H^{0,1})$ and (B.1).

We are going to give a proof of them. Note that the arguments below are essentially the same as those given in Section 5 of [9]. We also remark that the regularity assumptions in these propositions are certainly not optimal. It may be possible to relax them (see e.g. [4]), but that is out of the present purpose.

Proof of Proposition B.1. In what follows, we write $N_{1}(v)=|v_{2}|^{2}v_{1}$, $N_{2}(v)=|v_{1}|^{2}v_{2}$ and $N(v)=(N_{1}(v),N_{2}(v))$ for $v=(v_{1},v_{2})$. Let $\alpha$ be given by (2.3) for the solution $u$ to (1.4). Then, similarly to (2.4), we have

[TABLE]

where

[TABLE]

and

[TABLE]

Now we shall argue by contradiction. If (B.2) is not true, then we can take $\eta>0$ such that $\|N_{j}(\hat{\psi}^{+})\|_{L^{2}}\geq\eta$ for $j=1,2$. By (B.3) and Lemma 2.1, we have $\|R_{j}(t)\|_{L^{2}}\leq Ct^{-5/4+3\gamma}$ for $t\geq T_{0}$. We also note that

[TABLE]

whence, by (B.1), we can take $T^{*}\geq T_{0}$ such that $\|S_{j}(t)\|_{L^{2}}\leq\eta/(2t)$ for $t\geq T^{*}$. Summing up, we obtain

[TABLE]

for $t\geq T^{*}$. Letting $t\to\infty$, we have

[TABLE]

which is the desired contradiction. ∎

Proof of Proposition B.2.  With $T\geq 1$ to be fixed, let us introduce the function space

[TABLE]

and the norm

[TABLE]

where $0<\mu<(s_{0}-1)/2$ and $s_{0}=\min\{2,s\}$. For $v=(v_{1},v_{2})\in\mathfrak{X}_{T}$, we set

[TABLE]

and $\Phi[v]=(\Phi_{1}[v],\Phi_{2}[v])$. We also put $w^{\sharp}(t)=\mathcal{M}(t)\mathcal{D}(t)\mathcal{F}\psi^{+}$, $w^{\flat}(t)=\mathcal{U}(t)\psi^{+}-w^{\sharp}(t)$, $\kappa=\|\psi^{+}\|_{H^{0,s_{0}}}$ and

[TABLE]

Since (B.2) yields $N(\hat{\psi}^{+})=0$, it follows from (A.1) that

[TABLE]

We observe the basic estimates for $w^{\sharp}(t)$ and $w^{\flat}(t)$:

[TABLE]

where we have used (2.1), (A.1) and (A.3) with $\sigma=s_{0}/2$ or $(s_{0}-1)/2$.

Now we are going to show that $\Phi$ is a contraction mapping on $\mathfrak{Y}_{T}$ by choosing $\delta$ and $T$ appropriately. Let $v\in\mathfrak{Y}_{T}$. By using (B.5), we rewrite (B.4) as

[TABLE]

It follows from (2.2) that

[TABLE]

So we have

[TABLE]

Therefore

[TABLE]

Also, because of the estimate

[TABLE]

we obtain

[TABLE]

Combining (B.6) and (B.7), we arrive at

[TABLE]

Hence we have $\Phi[v]\in\mathfrak{Y}_{T}$ if we choose $\delta$ so small and $T$ so large that the term $(*)$ does not exceed $1$. Next we take $v$, $\tilde{v}\in\mathfrak{Y}_{T}$. Then we have

[TABLE]

and we can see as before that

[TABLE]

by choosing $\delta$ and $T$ suitably. Therefore $\Phi:\mathfrak{Y}_{T}\to\mathfrak{Y}_{T}$ is a contraction mapping, and thus, admits a unique fixed point. In other words, there exists $u\in\mathfrak{Y}_{T}$ such that

[TABLE]

which gives the desired solution to (1.4) for $t\geq T$. Moreover we have

[TABLE]

as $t\to+\infty$. This completes the proof of Proposition B.2. ∎

Acknowledgments

The work of H. S. is supported by Grant-in-Aid for Scientific Research (C) (No. 17K05322), JSPS.