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首页> 外文期刊>Discrete and continuous dynamical systems >CONSTRUCTION OF A SOLITARY WAVE SOLUTION OF THE NONLINEAR FOCUSING SCHROEDINGER EQUATION OUTSIDE A STRICTLY CONVEX OBSTACLE IN THE L~2-SUPERCRITICAL CASE
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CONSTRUCTION OF A SOLITARY WAVE SOLUTION OF THE NONLINEAR FOCUSING SCHROEDINGER EQUATION OUTSIDE A STRICTLY CONVEX OBSTACLE IN THE L~2-SUPERCRITICAL CASE

机译：在L〜2超临界壳体中严格凸障的非线性聚焦Schroedinger方程的孤立波解

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摘要

We consider the focusing L~2-supercritical Schroedinger equation in the exterior of a smooth, compact, strictly convex obstacle Θ ⊂ R~3. We construct a solution behaving asymptotically as a solitary wave on R~3, 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 topolog-ical 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.
机译：我们考虑了光滑，紧凑，严格凸障θ⊂r〜3的外部的聚焦L〜2超临界斯克罗德格方程。我们构建一个渐近的解决方案，作为R〜3上的孤独波，大次。当孤波的速度很高时，可以通过经典的固定点参数来证明这种解决方案的存在。为了构造具有任意非零速度的解决方案，我们使用与1990年的F.Merle引入的紧凑性参数，以构建NLS方程的解决方案在几个点中吹出几个点，以及使用Brouwer定理来控制的拓扑结构论证孤子的线性化操作员的不稳定方向。这些解决方案是任意接近由R. killip，M.Visan和X. Zhang的先前工作给出的散射阈值，这与S. Roundenko和J. Holmer给出的整个欧几里德空间中的一个相同径向案例和以前的作者在非径向案例中具有T. Duyckaerts。

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