Transverse instability of the CH-KP-I equation. (English) Zbl 1499.35068
Summary: The Camassa-Holm-Kadomtsev-Petviashvili-I equation (CH-KP-I) is a two dimensional generalization of the Camassa-Holm equation (CH). In this paper, we prove transverse instability of the line solitary waves under periodic transverse perturbations. The proof is based on the framework of [F. Rousset and N. Tzvetkov, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 2, 477–496 (2009; Zbl 1169.35374)]. Due to the high nonlinearity, our proof requires necessary modification. Specifically, we first establish the linear instability of the line solitary waves. Then through an approximation procedure, we prove that the linear effect actually dominates the nonlinear behavior.
MSC:
| 35B35 | Stability in context of PDEs |
| 35C07 | Traveling wave solutions |
| 35G25 | Initial value problems for nonlinear higher-order PDEs |
Keywords:
Camassa-Holm-Kadomtsev-Ketviashvili-I equation; line solitary waves; transverse instabilityCitations:
Zbl 1169.35374References:
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