An elementary construction of complex patterns in nonlinear schrödinger equations

Manuel Del Pino, Patrido Felmer, Kazunaga Tanaka

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equation -ε2u″ + V(x)u = |u|p-1u in R, p > 1, when s is a small parameter. Given any finite set of points x1 < X2 < ⋯ < xm constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n i, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each xi as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.

Original languageEnglish
Pages (from-to)1653-1671
Number of pages19
JournalNonlinearity
Volume15
Issue number5
DOIs
Publication statusPublished - 2002 Sep 1

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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