On the Whitham system for the (2+1)-dimensional nonlinear Schr\"odinger equation

We derive the Whitham modulation equations for the nonlinear Schr\"odinger equation in the plane (2d NLS) with small dispersion. The modulation equations are derived in terms of both physical and Riemann 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 case, the traveling waves are found to be unstable. This result is consistent with all previous investigations of such stability by other methods. These results are supported by direct numerical calculations.