Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2005-09-23
J. Phys. A: Math. Gen. 39 (2006) 73-84
Nonlinear Sciences
Pattern Formation and Solitons
Submitted: 13 pages, 3 figures
Scientific paper
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.
Carter John D.
Deconinck Bernard
Thelwell Roger J.
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