Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the L^2 -supercritical case
Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the L^2 -supercritical case
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L^2 超临界情况下严格凸障碍物外非线性聚焦薛定谔方程孤立波解的构造
DOI：
10.3934/dcds.2020298
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
O. Landoulsi
中科院分区：
文献类型：
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作者：
O. Landoulsi
我们考虑光滑、紧凑、严格凸障碍物外部的聚焦\开始{文档}$L^2 $\结束{文档} -超临界薛定谔方程\开始{文档}$ \Theta \subset \mathbb{R}^3 $\end{文档}。我们构造了一个解，它在\开始{document}$ \mathbb{R}^3，$\end{document}上大时间渐近地表现为孤立波。当孤立波的速度很高时，解的存在性可以用经典的不动点定理来证明。为了构造具有任意非零速度的解，我们使用类似于F.Merle在1990年引入的紧性参数来构造NLS方程在几个点处爆破的解，并使用Brouwer定理来控制线性化算子在孤子处的不稳定方向。这些解决方案是任意接近的散射阈值由以前的工作R。Killip，M.维桑和X. Zhang，它与S. Roundenko和J. Holmer在桡骨病例中以及以前的作者在T.非桡骨病例中的Duyckaerts。
We consider the focusing \begin{document}$ L^2 $\end{document} -supercritical Schrodinger equation in the exterior of a smooth, compact, strictly convex obstacle \begin{document}$ \Theta \subset \mathbb{R}^3 $\end{document} . We construct a solution behaving asymptotically as a solitary wave on \begin{document}$ \mathbb{R}^3, $\end{document} for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solutions of the NLS equation blowing up at several points together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at the soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of R. Killip, M. Visan and X. Zhang, which is the same as the one on the whole Euclidean space given by S. Roundenko and J. Holmer in the radial case and by the previous authors with T. Duyckaerts in the non-radial case.