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

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.