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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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