Stability of two-dimensional finite-amplitude envelope solitons

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Abstract
The stability properties of stationary, cylindrically symmetric solutions of the scalar and vectorial exponential nonlinear Schrödinger equation are analyzed in two spatial dimensions. Both equations are fundamental models describing envelope solitons, like Langmuir cavitons in plasma physics and laser pulses in nonlinear optics. The scalar as well as the vectorial equations have a saturating nonlinearity which allows us to treat finite amplitudes. Using a Liapunov functional for stability, we prove that scalar envelope solitons are two dimensionally completely stable. This has to be seen in contrast to previous results for the three-dimensional case where only large-amplitude solitons have been found to be stable. Localized solutions of the vectorial equation are stable to perturbations which only depend on the radius (longitudinal stability). Thus, no radial collapse of initially well-behaved plasma cavitons occurs in two spatial dimensions. For azimuthally dependent electrostatic perturbations we find instability. This indicates that vectorial envelope solitons suffer a symmetry-disturbing transversal collapse.