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Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations. (English) Zbl 1123.35015

The paper deals with the class of systems of nonlinear Schrödinger equations \[ \left\{ \begin{align*}{ -\Delta u+ u-u^3=\epsilon v\,,\cr -\Delta v+ v-v^3=\epsilon u\,,}\end{align*} \right. \] in \(\mathbb R^n\) with dimensions \(n=1,\,2,\,3\). The main result states that if \(\mathcal P\) is 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 a bump located near the vertices of \(\xi {\mathcal P}\), where \(\xi\sim\log(1/\epsilon)\), while the other component has one negative peak.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
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