## 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

Zbl 0742.35043
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