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Wave collapse and instability of solitary waves of a generalized Kadomtsev-Petviashvili equation. (English) Zbl 0824.35116

Summary: The solutions of a family of generalized Kadomtsev-Petviashvili equation in two dimensions, \[ (u_ t + u^ mu_ x + u_{xxx})_ x = \sigma^ 2 u_{yy}, \] are studied with appropriate boundary conditions and constraints on the initial data for \(m\) in a dense set \(m \geq 0\) and for \(\sigma^ 2 = 1\). If \(m \geq 4\), then the solution can blow up in finite time. We show that if \(1 \leq m < 4\), a solitary wave solution exists which is unstable if \(m > 4/3\). In this and related problems, the instability of solitary waves is associated with onset of wave collapse. We give numerical evidence to support this.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
76E30 Nonlinear effects in hydrodynamic stability
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