Stability of traveling waves of nonlinear Schr\"odinger equation with nonzero condition at infinity

TL;DR

This paper analyzes the stability of traveling wave solutions in nonlinear Schr"odinger equations with nonzero conditions at infinity, providing sharp instability criteria, constructing unstable waves, and exploring transversal instability across various models and dimensions.

Contribution

It establishes sharp instability criteria for 3D Gross-Pitaevskii waves, extends results to higher dimensions, constructs unstable waves for cubic-quintic equations, and analyzes transversal instability in 2D.

Findings

Sharp instability criterion for 3D GP traveling waves

Construction of unstable traveling waves in cubic-quintic models

Identification of transversal instability in 2D GP waves

Abstract

We study the stability of traveling waves of nonlinear Schr\"odinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models are Gross-Pitaevskii (GP) equation and cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type sub-critical or critical nonlinearity, we construct slow traveling waves and prove their nonlinear instability in any dimension. For traveling waves without vortices (i.e. nonvanishing) of general nonlinearity in any dimension, we find the sharp condition for linear instability. Third, we prove that any 2D traveling wave of (GP) is transversally unstable and find the sharp interval of unstable transversal wave numbers. Near unstable traveling waves of above cases, we construct unstable and stable invariant manifolds.