Instabilities of one-dimensional trivial-phase solutions of the two-dimensional cubic nonlinear Schrِdinger equation

The two-dimensional cubic nonlinear Schrödinger equation (NLS) is used as a model of a wide variety of physical phenomena. In this paper, we study the stability of a class of its one-dimensional, periodic, traveling-wave solutions. First, we establish that all such solutions are unstable with respect to two-dimensional perturbations with long wavelengths in the transverse dimension. Second, we establish that all such solutions are unstable with respect to two-dimensional perturbations with arbitrarily short wavelengths if the coefficients of the linear dispersion terms in the NLS have opposite signs. Both arguments rely on formal perturbation methods. Third, we use the Fourier–Floquet–Hill numerical method to examine the spectral stability problem. We present detailed spectra for twelve different solutions and demonstrate strong agreement between the numerical and asymptotic results.