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 \(\omega \in (\omega _{*}, \infty )\).

在本文中，我们研究了在圆柱形区域\（{\ mathbb {R}} \ times {\ mathbb {T}} \）上的全波导管Schrödinger流下Schrödinger孤子线的横向稳定性。当非线性具有\（p> 1 \）的幂类型\（| \ psi | ^ {p-1} \ psi \）时，我们表明存在一个临界频率\（\ omega _ {p}> 0 \）使得线驻波对于\（0 <\ omega <\ omega _ {p} \）是稳定的，而对于\（\ omega> \ omega _ {p} \）则不稳定。此外，我们表征了波导Schrödinger方程的基态。更确切地说，我们证明存在\（\ omega _ {*} \ in（0，\ omega _ {p}] \）使得基态与\（\ omega \ in（0，\ omega _ {*}] \）的线驻波重合，并且与\（\ omega \ in（\ omega _ { *}，\ infty）\）。