On the transverse stability of smooth solitary waves in a two-dimensional Camassa–Holm equation
Anna Geyer (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Yue Liu (University of Texas at Arlington)
Dmitry E. Pelinovsky (McMaster University, Nizhny Novgorod State Technical University)
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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.