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

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 $\mathrm{\mid }\psi {\mathrm{\mid }}^{p-1}\psi$ with $p>1$, we show that there exists a critical frequency ${\omega }_p>0$ such that the line standing wave is stable for $0<\omega < \omega_{p}$andunstablefor$\omega > \omega_{p}$. Furthermore, we characterize the ground state of the wave guide Schr"{o}dinger equation. More precisely, we prove that there exists ${\omega }_{\ast }\in (0,{\omega }_p]$ such that the ground states coincide with the line standing waves for $\omega \in (0, \omega_{}]$andaredifferentfromthelinestandingwavesfor$\omega \in (\omega_{}, \infty)$.