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Large data wave operator for the generalized Korteweg-de Vries equations. (English) Zbl 1212.35408
Summary: We consider the generalized Korteweg-de Vries equation $$u_t+(u_{xx}+u^p)_x=0$$, $$t,x\in \mathbb R,$$ for $$p\in (3,\infty )$$. Let $$U(t)$$ be the associated linear group. Given $$V$$ in the weighted Sobolev space $$H^{2,2}=\{f\in L^2:\| (1+| x| )^2(1-\partial ^2_x)f\| _{L^2}<\infty \}$$, 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 $$L^2$$).

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35B40 Asymptotic behavior of solutions to PDEs