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Direct Scattering for the KdV Equation with a Step-like Finite-Gap Potential: A Riemann--Hilbert Approach

Xiaodong Zhu

Abstract

We develop the direct scattering theory for the KdV equation with step-like finite-gap backgrounds under perturbations. More precisely, we consider initial data that asymptotically approach two distinct one-gap periodic travelling wave solutions as \(x \to \pm \infty\). Under suitable assumptions on the perturbation, we formulate the direct scattering problem and establish the analytic structure of the associated scattering data. In particular, we reformulate the problem in terms of a vector Riemann--Hilbert problem, which provides a foundation for the study of long-time asymptotics of perturbed finite-gap potentials. This formulation highlights the connection between step-like finite-gap scattering theory and the Riemann--Hilbert framework arising in soliton-gas type settings.

Direct Scattering for the KdV Equation with a Step-like Finite-Gap Potential: A Riemann--Hilbert Approach

Abstract

We develop the direct scattering theory for the KdV equation with step-like finite-gap backgrounds under perturbations. More precisely, we consider initial data that asymptotically approach two distinct one-gap periodic travelling wave solutions as . Under suitable assumptions on the perturbation, we formulate the direct scattering problem and establish the analytic structure of the associated scattering data. In particular, we reformulate the problem in terms of a vector Riemann--Hilbert problem, which provides a foundation for the study of long-time asymptotics of perturbed finite-gap potentials. This formulation highlights the connection between step-like finite-gap scattering theory and the Riemann--Hilbert framework arising in soliton-gas type settings.
Paper Structure (7 sections, 16 theorems, 192 equations, 3 figures)

This paper contains 7 sections, 16 theorems, 192 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Direct scattering
  4. RH problem for the step-like KdV finite gap potentials
  5. Time Evolution and Inverse Scattering
  6. The periodic travelling wave solution of KdV equation
  7. The Lax spectrum of periodic finite gap solution

Key Result

Lemma 2.1

Suppose that the initial datum $u_0$ satisfies the condition initial and Assumption assumption. The boundary values $r_{1}(\lambda_{\pm})$ are $m_0$--times differentiable on $\Sigma_1^l \setminus \{ i\eta_1^l, i\eta_2^l \},$ and, as $\lambda \to i\eta_j^l$, $j=1,2$, they exhibit square--root behavio For $\lambda\in\mathbb{R}$, the spectral function $\rho(\lambda)$ is regular at $\lambda=0$ and bel

Figures (3)

  • Figure 1: The Lax spectrum associated with the left background $u_0^l(x)$ (blue) and the right background $u_0^r(x)$ (red), which share the common endpoint $0$. The endpoints of the spectral bands for $u_0^l$ are given by $-(\eta_1^l)^2$ and $-(\eta_2^l)^2$, while the endpoints of the spectral bands for $u_0^r$ are given by $-(\eta_1^r)^2$ and $-(\eta_2^r)^2$.
  • Figure 2: The Riemann surface $\mathscr{X}(\eta_1,\eta_2)$ and its canonical homology basis $\mathscr{A}, \mathscr{B}$.
  • Figure 3: The six configurations (i)--(vi) of step-like finite-gap potential bands on the upper half plane. The red ($\Sigma_1^r$) and blue ($\Sigma_1^l$) solid lines represent the spectral bands of $u_0^r(x)$ and $u_0^l(x)$, respectively.

Theorems & Definitions (39)

  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Lemma 3.3
  • ...and 29 more