Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation

Liu, Yue

doi:10.1090/S0002-9947-00-02465-X

Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation
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Trans. Amer. Math. Soc. 353 (2001), 191-208 Request permission

Abstract:

In this paper we consider a generalized Kadomtsev-Petviashvili equation in the form \begin{equation*}( u_{t} + u_{xxx} + u^{p} u_{x} )_{x} = u_{yy} \quad (x, y) \in R^{2}, t \ge 0. \end{equation*} It is shown that the solutions blow up in finite time for the supercritical power of nonlinearity $p \ge 4/3$ 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.

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