Large data wave operator for the generalized Korteweg-de Vries equations

Abstract

We consider the generalized Korteweg-de Vries equations : \[ u_t + (u_{xx} + u^p)_x =0, \quad t,x \in \mathbb R, \] for $p \in (3,\infty)$. Let $U(t)$ be the associated linear group. Given $V$ in the weighted Sobolev space $H^{2,2} = \{ f \in L^2 : (1+|x|)^2(1-\partial_x^2)f \|_{L^2} < \infty\}$, possibly large, we construct a solution $u(t)$ of the generalized Korteweg-de Vries equation such that : \[ \lim_{t \to \infty} \| u(t) - U(t) V \|_{H^1} =0. \] We also prove uniqueness of such a solution in an adequate space. In the $L^2$-critical case ($p=5$), this result can be improved to any possibly large function $V$ in $L^2$ (with convergence in $L^2$).