Transverse linear stability of line solitons for 2D Toda

TL;DR

This paper proves the linear stability of 1-line solitons in the 2D Toda lattice, showing that solutions behave like damped waves in the transverse direction, extending stability results from KP-II to a semi-discrete integrable system.

Contribution

It establishes the linear stability of 1-line solitons for the 2D Toda lattice using Darboux transformations, a novel result for this integrable semi-discrete wave equation.

Findings

Solutions' dominant part is a time derivative of the soliton times a function of time and transverse variables.

Amplitude dynamics follow a 1D damped wave equation in the transverse direction.

Stability analysis extends KP-II results to the 2D Toda lattice.

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.