Short-wave transverse instabilities of line solitons of the 2-D hyperbolic nonlinear Schr\"odinger equation

TL;DR

This paper demonstrates that line solitons in the 2D hyperbolic nonlinear Schrödinger equation are unstable to small-period transverse perturbations, providing detailed asymptotic growth rates for these instabilities.

Contribution

It introduces a rigorous analysis of short-wave transverse instabilities of line solitons using spectral and asymptotic methods, with explicit growth rate formulas.

Findings

Line solitons are unstable to short-wave transverse perturbations.

Explicit asymptotic expressions for instability growth rates are derived.

The analysis employs Jost functions, Sommerfeld conditions, and Lyapunov--Schmidt decomposition.

Abstract

We prove that line solitons of the two-dimensional hyperbolic nonlinear Schr\"odinger equation are unstable with respect to transverse perturbations of arbitrarily small periods, {\em i.e.}, short waves. The analysis is based on the construction of Jost functions for the continuous spectrum of Schr\"{o}dinger operators, the Sommerfeld radiation conditions, and the Lyapunov--Schmidt decomposition. Precise asymptotic expressions for the instability growth rate are derived in the limit of short periods.