NDLI: Nonlinear Schrödinger equations near an infinite well potential

Content Provider	SpringerLink
Author	Parnet, Mona Bartsch, Thomas
Copyright Year	2013
Abstract	The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation where the potential $$V_\lambda :\mathbb {R}^N\rightarrow \mathbb {R}$$ is close to an infinite well potential $$V_\infty :\mathbb {R}^N\rightarrow \mathbb {R}$$ , i. e. $$V_\infty =\infty $$ on an exterior domain $$\mathbb {R}^N\setminus \Omega $$ , $$V_\infty \|_\Omega \in L^\infty (\Omega )$$ , and $$V_\lambda \rightarrow V_\infty $$ as $$\lambda \rightarrow \infty $$ in a sense to be made precise. The nonlinearity may be of Gross–Pitaevskii type. A standing wave solution of $$(NLS_\lambda )$$ with $$\lambda =\infty $$ vanishes on $$\mathbb {R}^N\setminus \Omega $$ and satisfies Dirichlet boundary conditions, hence it solves We investigate when a standing wave solution $$\Phi _\infty $$ of the infinite well potential $$(NLS_\infty )$$ gives rise to nearby solutions $$\Phi _\lambda $$ of the finite well potential $$(NLS_\lambda )$$ with $$\lambda \gg 1$$ large. Considering $$(NLS_\infty )$$ as a singular limit of $$(NLS_\lambda )$$ we prove a kind of singular continuation type results.
Ending Page	379
Page Count	17
Starting Page	363
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	1-2
Volume Number	51
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2013-10-10
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Analysis Theoretical, Mathematical and Computational Physics Variational methods for second-order elliptic equations Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian Semilinear elliptic equations NLS-like equations (nonlinear SchrÃ¶dinger) Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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