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Direct Scattering of the Focusing Nonlinear Schrödinger Equation with Step-like Oscillatory Initial Data

Tamara Grava, Robert Jenkins, Xiaofan Zhang, Zechuan Zhang

Abstract

In this manuscript we set up the direct and inverse scattering problems for step-like traveling-wave solutions of the nonlinear Schrödinger equation. Specifically, we consider initial data $u(x,0)$ satisfying $u(x,0)\to u_0^\ell(x)$ as $x\to-\infty$ and $u(x,0)\to u_0^r(x)$ as $x\to+\infty$, where $u_0^\ell(x)$ and $u_0^r(x)$ are elliptic traveling waves. Under suitable assumptions on the initial data we formulate the direct scattering problem and establish analytic properties of the scattering data. We then formulate the inverse problem as a Riemann--Hilbert problem and prove its solvability. Finally, we observe that this Riemann--Hilbert formulation is a special case of the one arising for full soliton-gas initial data.

Direct Scattering of the Focusing Nonlinear Schrödinger Equation with Step-like Oscillatory Initial Data

Abstract

In this manuscript we set up the direct and inverse scattering problems for step-like traveling-wave solutions of the nonlinear Schrödinger equation. Specifically, we consider initial data satisfying as and as , where and are elliptic traveling waves. Under suitable assumptions on the initial data we formulate the direct scattering problem and establish analytic properties of the scattering data. We then formulate the inverse problem as a Riemann--Hilbert problem and prove its solvability. Finally, we observe that this Riemann--Hilbert formulation is a special case of the one arising for full soliton-gas initial data.
Paper Structure (14 sections, 22 theorems, 235 equations, 6 figures)

This paper contains 14 sections, 22 theorems, 235 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Statement of the result.
  3. Elliptic travelling waves
  4. Genus one Riemann--Hilbert problem
  5. Perturbative Argument
  6. Case 1. $z_2^\ell \in \Sigma_1^\ell\setminus\Sigma_1^r$.
  7. Case 2. $z_2^\ell\in\Sigma_1^\ell\cap\Sigma_1^r$.
  8. Time evolution
  9. Proof of Proposition \ref{['proposition_elliptic']}
  10. Proof of Proposition \ref{['propm']}
  11. Asymptotics of $W(x,t;z)$ as $z\to\infty$
  12. Expansion of $b(z)$ as $z\to\infty$
  13. Behavior of Cauchy transform near endpoints of integration for a continuous density
  14. An example of initial data with pure step elliptic functions

Key Result

Theorem 2

Let $u(x)-u_0^\ell(x)\in L^1(\mathbb{R}^-)$, $u(x)-u_0^r(x)\in L^1(\mathbb{R}^+)$, and $a(z)$, $b(z)$, $b_1(z)$ and $b_2(z)$ be the scattering data in e:S, then

Figures (6)

  • Figure 1: Two typical configurations of the spectral curve $\Sigma$: (a) $\Sigma$ is disjoint from the real axis; (b) $\Sigma$ intersects the real axis.
  • Figure 2: Relative configurations of $\Sigma^\ell$ and $\Sigma^r$: (a) disjoint; (b) intersecting at a single point; (c) overlapping along a segment.
  • Figure 3: The homology basis for the Riemann surface $\mathcal{X}$ associated with $R^2=(z-z_1)(z-\overline{z_1})(z-z_2)(z-\overline{z_2})$.
  • Figure 4: When $(m,s)$ is in the white region the spectrum \ref{['spectrum1']} of the ZS operator has two arcs $\Sigma_1$ and $\Sigma_2$ that do not intersect the real axis ${\mathbb R}$ (Figure \ref{['fig:spectralcurves']}(a)). When $(m,s)$ is in the blue region the arcs $\Sigma_1$ and $\Sigma_2$ intersect the real axis ${\mathbb R}$ (Figure \ref{['fig:spectralcurves']}(b)).
  • Figure 5: Schematic of the cuts in ${\mathbb C}^+$ for the case in which: (a) $\Sigma_1^r$ is on top of $\Sigma_1^\ell$, and the two cuts overlap on the segment connecting $z_1^r$ and $z_2^\ell$; (b) $\Sigma_1^\ell$ lies completely in the interior of $\Sigma_1^r$. In both (a) and (b), $\Sigma_1^\ell$ and $\Sigma_1^r$ can be interchanged, yielding four configurations in total.
  • ...and 1 more figures

Theorems & Definitions (44)

  • Theorem 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Lemma 6
  • proof
  • Proposition 8
  • proof
  • Proposition 9
  • Remark 10
  • ...and 34 more