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 ${\omega }_{\ast }\in (0,{\omega }_p]$ such that the ground states coincide with the line standing waves for $\omega \in (0,{\omega }_{\ast }]$ and are different from the line standing waves for $\omega \in ({\omega }_{\ast },\infty )$.

To appear in JDDE