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Transverse linear stability of line solitons for 2D Toda

Tetsu Mizumachi

TL;DR

This work addresses the problem of transverse linear stability for $1$-line solitons of the fully discrete $2$D Toda lattice by leveraging Darboux transformations and Jost-function techniques. The authors map the linearized dynamics to a zero-potential problem via a Darboux correspondence, and prove stability in an exponentially weighted space, with the dominant secular component described by a time-derivative of the soliton modulated by a function solving a $1$D damped wave equation. They provide a detailed spectral and functional-analytic framework (Lax pair, Jost/dual Jost functions, and weighted Sobolev-type spaces) to control both high- and low-frequency contributions and to obtain precise long-time asymptotics. The results not only establish linear stability but also yield an explicit asymptotic profile for the perturbations, laying groundwork toward nonlinear stability analyses in this transversely extended Toda setting.

Abstract

The $2$-dimensional Toda lattice ($2$D Toda) is a completely integrable semi-discrete wave equation with the KP-II equation in its continuous limit. Using Darboux transformations, we prove the linear stability of $1$-line solitons for $2$D Toda of any size in an exponentially weighted space. We prove that the dominant part of solutions to the linearized equation around a $1$-line soliton is a time derivative of the $1$-line soliton multiplied by a function of time and transverse variables. The amplitude is described by a $1$-dimensional damped wave equation in the transverse variable, as is the case with the linearized KP-II equation.

Transverse linear stability of line solitons for 2D Toda

TL;DR

This work addresses the problem of transverse linear stability for -line solitons of the fully discrete D Toda lattice by leveraging Darboux transformations and Jost-function techniques. The authors map the linearized dynamics to a zero-potential problem via a Darboux correspondence, and prove stability in an exponentially weighted space, with the dominant secular component described by a time-derivative of the soliton modulated by a function solving a D damped wave equation. They provide a detailed spectral and functional-analytic framework (Lax pair, Jost/dual Jost functions, and weighted Sobolev-type spaces) to control both high- and low-frequency contributions and to obtain precise long-time asymptotics. The results not only establish linear stability but also yield an explicit asymptotic profile for the perturbations, laying groundwork toward nonlinear stability analyses in this transversely extended Toda setting.

Abstract

The -dimensional Toda lattice (D Toda) is a completely integrable semi-discrete wave equation with the KP-II equation in its continuous limit. Using Darboux transformations, we prove the linear stability of -line solitons for D Toda of any size in an exponentially weighted space. We prove that the dominant part of solutions to the linearized equation around a -line soliton is a time derivative of the -line soliton multiplied by a function of time and transverse variables. The amplitude is described by a -dimensional damped wave equation in the transverse variable, as is the case with the linearized KP-II equation.
Paper Structure (8 sections, 24 theorems, 215 equations)

This paper contains 8 sections, 24 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Jost functions and secular modes
  3. Daroboux transformations
  4. Linear stability of the null solution
  5. Daroboux transformations for the linearized equation around $1$-soliton
  6. Proof of Theorems \ref{['thm:main']} and \ref{['thm:profile']}
  7. Miscellaneous results
  8. Orthogonality relation of $g^\pm(\eta)$ and $g^{\pm,*}(\eta)$

Key Result

Theorem 1.1

Let $\alpha\in(0,2\kappa)$ and $c=\sinh\kappa/\kappa$. Suppose that $\eta_0\in(0,\eta_*(\alpha))$ and $t_0\in\mathbb{R}$. If $\mathbf{R'}(t)$ is a solution of eq:linear-1 in the class $C(\mathbb{R};\ell^2_\alpha H^1(\mathbb{R}))\cap C^1(\mathbb{R};\ell^2_\alpha L^2(\mathbb{R}))$ and if for every $\e holds at $t=t_0$, then for every $t\ge t_0$, where $K$ and $b$ are positive constants that are ind

Theorems & Definitions (58)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • ...and 48 more