Instabilities in the two-dimensional cubic nonlinear Schrödinger equation.
The two-dimensional cubic nonlinear Schrödinger equation (NLS) can be used as a model of phenomena in physical systems ranging from waves on deep water to pulses in optical fibers. In this paper, we establish that every one-dimensional traveling wave solution of NLS with linear phase is unstable with respect to some infinitesimal perturbation with two-dimensional structure. If the coefficients of the linear dispersion terms have the same sign (elliptic case), then the only unstable perturbations have transverse wavelength longer than a well-defined cutoff. If the coefficients of the linear dispersion terms have opposite signs (hyperbolic case), then there is no such cutoff and as the wavelength decreases, the maximum growth rate approaches a well-defined limit.