Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations

摘要

We consider the long time asymptotics of solutions that are close to a solitary wave solution to the generalized Korteweg de Vries equation u(t) + u(p)u(x) + u(xxx) = 0 for x is an element of R, t > 0. If 1 less than or equal top< 4 and the spectrum of the linearized equation around the initial solitary wave has the simplest possible structure, the solitary wave turns out to be asymptotically stable with respect to finite energy perturbations with polynomial decay as x --> infinity. Furthermore, we show that the asymptotics of the solution for large time is given by a sum of a solitary wave with slightly displaced parameters and a small dispersion if 2 < p< 4. [References: 37]