Blow-up and instability of a regularized long-wave-KP equation

摘要

A regularized long-wave-Kadomtsev-Petviashvili equation of the form (u_t-u_(xxt) + u_x + u~q u_x)_x - u_(yy) = 0, is considered. It is shown that if p≥4, certain initial data can lead to a solution that blows up in finite time. More precisely, under the above condition the solution cannot remain in the Sobolev class H~2(R~2) for all time. Also demonstrated here is the solitary-wave solutions u(x,y,t) = φ_c(x-ct,y), which exist if and only if 1 ≤ p < 4 and c > 1, when considered as solutions of the initial-value problem for (*), are nonlinearly unstable to perturbations of the initial data, if 4/3 < p < 4 and 1 < c < (4p)/(4+p).