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Long range scattering for nonlinear Schrödinger equations in one space dimension. (English) Zbl 0742.35043
Summary: We consider the scattering problem for the nonlinear Schrödinger equation in $$1+1$$ dimensions: $i\partial_ tu+(1/2)\partial^ 2u=\lambda| u|^ 2u+\mu| u| ^{p- 1}u,\quad(t,x)\in\mathbb{R}\times\mathbb{R}, (*)$ where $$\partial=\partial/\partial x$$, $$\lambda\in\mathbb{R}\backslash\{0\}$$, $$\mu\in\mathbb{R}$$, $$p>3$$. We show that modified wave operators for (*) exist on a dense set of a neighborhood of zero in the Lebesgue space $$L^ 2(\mathbb{R})$$ or in the Sobolev space $$H^ 1(\mathbb{R})$$. The modified wave operators are introduced in order to control the long range nonlinearity $$\lambda| u|^ 2u$$.

##### MSC:
 35P25 Scattering theory for PDEs 35Q55 NLS equations (nonlinear Schrödinger equations)
##### Keywords:
scattering; nonlinear Schrödinger equation; wave operators
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##### References:
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