Instability of solitary-wave solutions of the 3-dimensional Kadomtsev-Petviashvili equation

Abstract

The generalized Kadomtsev-Petviashvili system of equations in three space dimensions, $$ \begin{cases} u_t + u^p u_x + u_{xxx} - v_y - w_z = 0, \\ v_x = u_y, \\ w_x = u_z, \end{cases} \tag*{(*)} $$ has been shown by de Bouard and Saut to possess solitary-wave solutions if and only if $ 1 \le < 4/3. $ It is demonstrated here that these localized traveling-waves, when considered as solutions of the initial-value problem for $ (*) $, are dynamically unstable to perturbations.