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Modified wave operators for the fourth-order nonlinear Schrödinger-type equation with cubic nonlinearity. (English) Zbl 1123.35072

The author considers the fourth-order nonlinear Schrödinger equation
\[ i\partial_tu-{1\over4} \partial_x^4u=\lambda| u| ^2u \] with cubic nonlinearity, a long range scattering problem where the solutions are not expected to be asymptotically free. Following the basic results in T. Ozawa [Commun. Math. Phys. 139, No. 3, 479–493 (1991; Zbl 0742.35043)], for final data \(v(t)\) that are small in a weighted Sobolev space with mean zero, a unique global solution \(u \in C(\mathbb R,L^2(\mathbb R)) \cap L^8_{loc} (\mathbb R,L^\infty(\mathbb R))\) is constructed such that \[ \| u(t)-v(t)\| _{L^2_x(\mathbb R)}=O(t^{-\alpha}) \] as \(t\to\infty\), for all \(\alpha\in (3/8,1)\). The proof consists of two steps: The first step is an application of the contraction mapping principle to find the solution \(u\) near \(\infty\), using Strichartz type estimates for the free evolution group \(W(t)\). In contrast to the second-order problem, here no “MDFM”-decomposition is known for \(W(t)\). This difficulty is overcome by using the method of stationary phase. In the second step global existence as \(t\to-\infty\) is shown.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
35B40 Asymptotic behavior of solutions to PDEs
81U05 \(2\)-body potential quantum scattering theory

Citations:

Zbl 0742.35043
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References:

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