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The integrable nonlocal nonlinear Schrödinger equation with oscillatory boundary conditions: long-time asymptotics

Yan Rybalko, Dmitry Shepelsky, Shou-Fu Tian

Abstract

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation \[ \I q_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \] subject to the step-like initial data: $q(x,0)\to0$ as $x\to-\infty$ and $q(x,0)\simeq Ae^{2\I Bx}$ as $x\to\infty$, where $A>0$ and $B\in\mathbb{R}$. The goal is to study the long-time asymptotic behavior of the solution of this problem assuming that $q(x,0)$ is close, in a certain spectral sense, to the ``step-like'' function $q_{0,R}(x)= \begin{cases} 0, &x\leq R,\\ Ae^{2\I Bx}, &x>R, \end{cases}$ with $R>0$. A special attention is paid to how $B\ne0$ affects the asymptotics.

The integrable nonlocal nonlinear Schrödinger equation with oscillatory boundary conditions: long-time asymptotics

Abstract

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation subject to the step-like initial data: as and as , where and . The goal is to study the long-time asymptotic behavior of the solution of this problem assuming that is close, in a certain spectral sense, to the ``step-like'' function with . A special attention is paid to how affects the asymptotics.

Paper Structure

This paper contains 15 sections, 9 theorems, 130 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Inverse scattering transform and the Riemann-Hilbert problem
  3. Eigenfunctions
  4. Scattering data
  5. Pure step initial data
  6. Basic Riemann-Hilbert problem
  7. Long-time asymptotics
  8. Phase function and jump factorizations
  9. Asymptotics in Case II with $n=0$
  10. Region $|\xi|>-B$, $B<0$
  11. Region $0\leq|\xi|<-B$, $B<0$
  12. Region $|\xi|>B$, $B>0$
  13. Region $0\leq|\xi|<B$, $B>0$
  14. Asymptotics in Case I with $n=0$
  15. Concluding remarks

Key Result

Proposition 1.1

Consider the Cauchy problem IVP with boundary conditions bc2. Assume that the initial datum $q_0(x)$ has the form shst with $0<R<\frac{\pi}{2\left(4B^2+A^2\right)^{1/2}}$, $0< 4|B|R\leq\pi$ and that either (i) $4B^2-A^2<0$ or (ii) $4B^2-A^2>0$ and $\pi^{-2}R^2(4B^2-A^2)$ is not a squared integer. Th Here $\delta(-B,\xi)$ and $\hat{\delta}(-B,\xi;B)$ are given in ddef and hat-del, respectively, in

Figures (5)

  • Figure 1: Asymptotics of the pure step initial data, described in Proposition \ref{['RA1']}. The left figure illustrates the case $B>0$, and the right corresponds to $B<0$.
  • Figure 2: Signature table of $\theta(k,\xi)$.
  • Figure 3: Contour $\hat{\Gamma} =\hat{\gamma}_1\cup\dots \cup\hat{\gamma}_4$ and domains $\hat{\Omega}_j$, $j=0,\dots,4$ in Section \ref{['R1']}.
  • Figure 4: Contour $\hat{\Gamma} =\hat{\gamma}_1\cup\dots \cup\hat{\gamma}_4$ and domains $\hat{\Omega}_j$, $j=0,\dots,4$ in Section \ref{['R2']}.
  • Figure 5: Contour $\hat{\Gamma} =\hat{\gamma}_1\cup\dots \cup\hat{\gamma}_4$ and domains $\hat{\Omega}_j$, $j=0,\dots,4$ in Section \ref{['S2R2']}.

Theorems & Definitions (24)

  • Proposition 1.1: Rough asymptotics for pure step initial values
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4: Zeros of $a_1(k)$
  • proof
  • ...and 14 more