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

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