Instabilities in the two-dimensional cubic nonlinear Schrodinger equation

Abstract

The two-dimensional cubic nonlinear Schrodinger 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 trivial 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 then the only unstable perturbations have transverse wavelength longer than a well-defined cut-off. If the coefficients of the linear dispersion terms have opposite signs, then there is no such cut-off and as the wavelength decreases, the maximum growth rate approaches a well-defined limit.

4 pages, 4 figures

References in corpus (2)

• Stationary solutions of the one-dimensional nonlinear Schroedinger equation: I. Case of repulsive nonlinearity
• Stationary solutions of the one-dimensional nonlinear Schroedinger equation: II. Case of attractive nonlinearity

Cited by in corpus (4)

• Stabilities of one-dimensional stationary states of Bose-Einstein condensates
• Elliptic finite-band potentials of a non-self-adjoint Dirac operator
• Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schrodinger equation
• On the variation of of bi-periodic waves in the transverse direction