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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
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