Construction of solitary wave solution of the nonlinear focusing schr\"odinger equation outside a strictly convex obstacle for the $L^2$-supercritical case - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2022

Construction of solitary wave solution of the nonlinear focusing schr\"odinger equation outside a strictly convex obstacle for the $L^2$-supercritical case

Résumé

We consider the focusing $L^2$-supercritical Schr\"odinger equation in the exterior of a smooth, compact, strictly convex obstacle. We construct a solution behaving asymptotically as a solitary waves on $R^3$, as large time. 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 solution of NLS blowing up at several blow-up point together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at 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 whole Euclidean space.

Dates et versions

hal-03916581 , version 1 (30-12-2022)

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Oussama Landoulsi. Construction of solitary wave solution of the nonlinear focusing schr\"odinger equation outside a strictly convex obstacle for the $L^2$-supercritical case. 2022. ⟨hal-03916581⟩
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