Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations

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Abstract

This paper is devoted to study a class of systems of nonlinear Schrödinger equations: $\left\{\begin{array}{rcl} -\Delta u+u-u^{3}=\epsilon v, \\ -\Delta v+v-v^{3}=\epsilon u, \end{array}\right.$ in $\mathbb{R}^{n}$ with dimension n = 1,2,3. Our main result states that if $\mathcal{P}$ denotes a regular polytope centered at the origin of $\mathbb{R}^{n}$ such that its side is greater than the radius, then there exists a solution with one multi-bump component having bumps located near the vertices of $\xi\mathcal{P}$ , where ${\xi\sim \log(1/\varepsilon)}$ , while the other component has one negative peak.

A. Ambrosetti was Supported by M.U.R.S.T within the PRIN 2004 “Variational methods and nonlinear differential equations”.
E. Colorado was Partially supported by Secretaría de Estado de Ministerio de Educación y Ciencia, Spain (Grant Ref. EX2005-0112) and by Research Project of MEC of Spain (Ref. BMF2003-03772).
D. Ruiz was Partially supported by S.I.S.S.A., by the Research Project of MEC, Spain (Ref. MTM2005-01331) and by Research Group of Junta de Andalucia (Grant Ref. FQM-116).