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Felmer, Patricio; Martínez, Salomé; Tanaka, Kazunaga

High frequency solutions for the singularly-perturbed one-dimensional nonlinear Schrödinger equation. (English) Zbl 1116.34067

Arch. Ration. Mech. Anal. 182, No. 2, 333-366 (2006).
Summary: This article is devoted to the nonlinear Schrödinger equation \[ \varepsilon^2 u'' - V(x)u + | u|^{p-1}u = 0 \] when the parameter \(\varepsilon\) approaches zero, where \(p>1\) and potential \(V(x)\) is positive and of class \(C^1\). All possible asymptotic behaviors of bounded solutions can be described by means of envelopes, or alternatively by adiabatic profiles. We prove that for every envelope, there exists a family of solutions reaching that asymptotic behavior, in the case of bounded intervals. We use a combination of the Nehari finite dimensional reduction together with degree theory. Our main contribution is to compute the degree of each cluster, which is a key piece of information in order to glue them.

Cited in 6 Documents

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34E15 Singular perturbations for ordinary differential equations

Keywords:

asymptotic behavior of solution; Nehar’s method; positive simple clusters; sign-changing simple clusters; degree theoretic approach
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References:

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