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Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation. (English) Zbl 0949.35120
Summary: We consider a generalized Kadomtsev-Petviashvili equation in the form \[ ( u_{t} + u_{xxx} + u^{p} u_{x})_{x} = u_{yy}, \quad(x, y) \in \mathbb{R}^{2},\;t \geq 0. \] It is shown that the solutions blow up in finite time for the supercritical power of nonlinearity \( p \geq 4/3 \) with \(p\) the ratio of an even to an odd integer. Moreover, it is shown that the solitary waves are strongly unstable if \( 2 < p < 4\); that is, the solutions blow up in finite time provided they start near an unstable solitary wave.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
76B25 Solitary waves for incompressible inviscid fluids
35B40 Asymptotic behavior of solutions to PDEs
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