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Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions. (English) Zbl 1061.35115
The author investigates unstable solitary-wave solutions of the generalised Kadomtsev-Petviashvili (KP) equation \[ (u_t+u^pu_x+u_{xxx})_x=u_{yy}+u_{zz},\quad (x,y,z)\in{\mathbb R}^2,\;\;t\geq 0, \] with \(p\geq 1.\) Here, the solitary waves are travelling waves of the form \(\phi(x-ct,y,z)\) with \(u\rightarrow 0\) as \(x^2+y^2+z^2\rightarrow \infty\), propagating in the \(x\)-direction.
The paper extends the results of a previous work by J. Bona and Y. Lui (cited in the references as a preprint). Using a methodology developed by the author [Trans. Am. Math. Soc. 353, 191–208 (2001; Zbl 0949.35120)], it is shown that, provided the initial conditions are close to an unstable solitary wave, the solution blows up in finite time if \(1\leq p<4/3\). This result improves on the prior result of J. Saut [Acta Appl. Math. 39, 477–487 (1995; Zbl 0839.35121)] which established a corresponding result for \(p\geq 2\) (and for negative energy only).

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35Q51 Soliton equations
35B35 Stability in context of PDEs
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