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Long-time asymptotics of the coupled nonlinear Schödinger equation in a weighted Sobolev space

Yubin Huang, Liming Ling, Xiaoen Zhang

Abstract

We study the Cauchy problem for the focusing coupled nonlinear Schrödinger (CNLS) equation with initial data $\mathbf{q}_0$ lying in the weighted Sobolev space and the scattering data having $n$ simple zeros. Based on the corresponding $3\times3$ matrix spectral problem, we deduce the Riemann-Hilbert problem (RHP) for CNLS equation through inverse scattering transform. We remove discrete spectrum of initial RHP using Darboux transformations. By applying the nonlinear steepest-descent method for RHP introduced by Deift and Zhou, we compute the long-time asymptotic expansion of the solution $\mathbf{q}(x,t)$ to an (optimal) residual error of order $\mathcal{O}\left(t^{-3 / 4+1/(2p)}\right)$ where $2\le p<\infty$. The leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton-soliton and soliton-radiation interactions. Our work strengthens and extends the earlier work regarding long-time asymptotics for solutions of the nonlinear Schrödinger equation with a delta potential and even initial data by Deift and Park.

Long-time asymptotics of the coupled nonlinear Schödinger equation in a weighted Sobolev space

Abstract

We study the Cauchy problem for the focusing coupled nonlinear Schrödinger (CNLS) equation with initial data lying in the weighted Sobolev space and the scattering data having simple zeros. Based on the corresponding matrix spectral problem, we deduce the Riemann-Hilbert problem (RHP) for CNLS equation through inverse scattering transform. We remove discrete spectrum of initial RHP using Darboux transformations. By applying the nonlinear steepest-descent method for RHP introduced by Deift and Zhou, we compute the long-time asymptotic expansion of the solution to an (optimal) residual error of order where . The leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton-soliton and soliton-radiation interactions. Our work strengthens and extends the earlier work regarding long-time asymptotics for solutions of the nonlinear Schrödinger equation with a delta potential and even initial data by Deift and Park.
Paper Structure (7 sections, 18 theorems, 268 equations, 4 figures)

This paper contains 7 sections, 18 theorems, 268 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Results of scattering theory for focusing CNLS
  3. The Long-Time Asymptotics
  4. Decomposition of continuous spectrum and discrete spectrum
  5. Factorization of the jump matrix
  6. Contour deformation
  7. Solving model problem

Key Result

Theorem 1

Let $\mathbf{q}(x,t)$, $x\in\mathbb{R}$, $t\ge0$, solve CNLS equation CNLS with $\mathbf{q}_0=\mathbf{q}(x,0)$ satisfying Assumption assumption-q0. Then for $2\le p<\infty$, as $t\rightarrow\infty$, where $\xi=-\frac{x}{2t}$, $\Gamma(\cdot)$ is the Gamma function, the vector-valued function $\mathbf{R}(\lambda)$ is defined by equation def-R-no-soliton, the constant $\pmb{A}(\xi)$ is given by equa

Figures (4)

  • Figure 1: The regions of decay of the exponential factor $\mathrm{e}^{\pm2\mathrm{i} t\theta}$ for large $t>0$.
  • Figure 2: The contours $\Sigma_k$ and regions $\Omega_k$.
  • Figure 3: The contours $\mathbb{R}_\xi$.
  • Figure 4: The contours $\Gamma_\xi$.

Theorems & Definitions (34)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • ...and 24 more