ncts

The Camassa-Holm equation models the unidirectional propagation of waves in shallow water. The stability of its solitary traveling wave solutions with respect to localised perturbations has been extensively studied. In this talk, I will focus on transverse stability, that is, stability with respect to perturbations transverse to the direction of propagation. To this end we consider a two-dimensional generalisation of the Camassa-Holm equation, which is similar to the way the KP equation extends the famous KdV equation to two dimensions. To prove transverse stability we study the spectrum of an operator which arises after linearisation around the perturbation in suitably weighted spaces. We show that 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 that the continuous spectrum is located in the left half-plane. Moreover, we prove that small-amplitude solitary waves are linearly stable with respect to transverse perturbations by performing careful resolvent estimates and making use of an asymptotic reduction of CH to KdV. The talk is based on joint work with Yue Liu and Dmitry Pelinovsky.