Stability of Solitary Waves for Trapped Dipolar Quantum Gases in the Limit Case of the Cigar-Shaped Model

Abstract

This paper concerns the existence and stability of solitary waves for the nonlinear Schrödinger equation with a partial confinement, which describes the limit case of the cigar-shaped model in Bose-Einstein condensate of dipolar quantum gases. More precisely, we applied a compactness argument, which comes from the confining potential x ¹₂ +x ²₂ , to overcome the lack of compactness caused by the translation invariance with respect to x₃. Then, since the mass supercritical character of this equation, we construct orbitally stable solutions by adapting a suitable localized minimization problem. Finally, the stability of solitary waves is obtained.