On the susceptibility of bright nonlinear Schrödinger solitons to long–wave transverse instability

Abstract

A new theory for transverse instability of bright solitons of equations of nonlinear Schrödinger (NLS) type is presented, based on a natural deformation of the solitons into a four–parameter family. This deformation induces a set of four diagnostic functionals which encode information about transverse instability. These functionals include the deformed power, the deformed momentum and two new functionals. The main result is that a sufficient condition for long–wave transverse instability is completely determined by these functionals. Whereas longitudinal instability is determined by a single partial derivative (the Vakhitov–Kolokolov criterion), the condition for transverse instability requires 10 partial derivatives. The theory is illustrated by application to scalar NLS equations with general potential, and vector NLS equations for optical media with ξ(2) nonlinearity.