Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions
The author investigates unstable solitary-wave solutions of the generalised Kadomtsev-Petviash\-vili (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 \textit{J. Bona} and \textit{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 \textit{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).
scientific article; zbMATH DE number 1797539
scientific article