Transverse Stability of Line Soliton and Characterization of Ground State for Wave Guide Schrödinger Equations

Abstract

In this paper, we study the transverse stability of the line Schrödinger soliton under a full wave guide Schrödinger 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 Schrödinger 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 \(\omega \in (\omega _{*}, \infty )\).