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Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schrodinger equation

Roger J. Thelwell·John D. Carter·Bernard Deconinck
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Abstract

The two-dimensional cubic nonlinear Schrodinger equation admits a large family of one-dimensional bounded traveling-wave solutions. All such solutions may be written in terms of an amplitude and a phase. Solutions with piecewise constant phase have been well studied previously. Some of these solutions were found to be stable with respect to one-dimensional perturbations. No such solutions are stable with respect to two-dimensional perturbations. Here we consider stability of the larger class of solutions whose phase is dependent on the spatial dimension of the one-dimensional wave form. We study the spectral stability of such nontrivial-phase solutions numerically, using Hill's method. We present evidence which suggests that all such nontrivial-phase solutions are unstable with respect to both one- and two-dimensional perturbations. Instability occurs in all cases: for both the elliptic and hyperbolic nonlinear Schrodinger equations, and in the focusing and defocusing case.

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Roger J. Thelwell, John D. Carter, Bernard Deconinck. 2005-09-23. Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schrodinger equation. https://doi.org/10.1088/0305-4470%2F39%2F1%2F006

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