On the Whitham system for the (2+1)-dimensional nonlinear Schrodinger equation

Whitham modulation equations are derived for the nonlinear Schrodinger equation in the plane ((2+1)-dimensional nonlinear Schrodinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann-type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one-dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations.