  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.
