Transverse stability of line soliton and characterization of ground state for wave guide Schr\"{o}dinger equations

TL;DR

This paper investigates the transverse stability of line solitons and characterizes ground states for wave guide Schrödinger equations on cylindrical domains, identifying stability thresholds and the relation between ground states and standing waves.

Contribution

It establishes stability criteria for line solitons depending on frequency and characterizes the ground states in relation to these solitons for the wave guide Schrödinger equation.

Findings

Existence of a critical frequency _p for stability.

Stability for frequencies below _p, instability above.

Ground states coincide with line standing waves below a certain frequency _*.

Abstract

In this paper, we study the transverse stability of the line Schr\"{o}dinger soliton under a full wave guide Schr\"{o}dinger flow on a cylindrical domain $\mathbb R\times\mathbb T$. When the nonlinearity is of power type $|\psi|^{p-1}\psi$ with $p>1$, we show that there exists a critical frequency $\omega_{p} >0$ such that the line standing wave is stable for $0<\omega < \omega_{p}$ and unstable for $\omega > \omega_{p}$. Furthermore, we characterize the ground state of the wave guide Schr\"{o}dinger equation. More precisely, we prove that there exists $\omega_{*} \in (0, \omega_{p}]$ such that the ground states coincide with the line standing waves for $\omega \in (0, \omega_{*}]$ and are different from the line standing waves for $\omega \in (\omega_{*}, \infty)$.