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On the transverse stability of smooth solitary waves in a two-dimensional Camassa-Holm equation (2307.12821v1)
Published 24 Jul 2023 in math.AP, math-ph, math.DS, math.MP, nlin.PS, and nlin.SI
Abstract: We consider the propagation of smooth solitary waves in a two-dimensional generalization of the Camassa--Holm equation. We show that transverse perturbations to one-dimensional solitary waves behave similarly to the KP-II theory. This conclusion follows from our two main results: (i) the double eigenvalue of the linearized equations related to the translational symmetry breaks under a transverse perturbation into a pair of the asymptotically stable resonances and (ii) small-amplitude solitary waves are linearly stable with respect to transverse perturbations.