We consider the nonlinear Schrödinger equation with focusing power-type nonlinearity on compact graphs with at least one terminal edge, i.e., an edge ending with a vertex of degree 1. On the one hand, we introduce the associated action functional, and we provide a profile description of positive low action solutions at large frequencies, showing that they concentrate on one terminal edge, where they coincide with suitable rescaling of the unique solution to the corresponding problem on the half-line. On the other hand, a Lyapunov-Schmidt reduction procedure is performed to construct one-peaked and multipeaked positive solutions with sufficiently large frequency, exploiting the presence of one or more terminal edges.

Peaked and low action solutions of NLS equations on graphs with terminal edges / Dovetta, S.; Ghimenti, M.; Micheletti, A. M.; Pistoia, A.. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 52:3(2020), pp. 2874-2894. [10.1137/19M127447X]

Peaked and low action solutions of NLS equations on graphs with terminal edges

Dovetta S.;Ghimenti M.;Micheletti A. M.;Pistoia A.
2020

Abstract

We consider the nonlinear Schrödinger equation with focusing power-type nonlinearity on compact graphs with at least one terminal edge, i.e., an edge ending with a vertex of degree 1. On the one hand, we introduce the associated action functional, and we provide a profile description of positive low action solutions at large frequencies, showing that they concentrate on one terminal edge, where they coincide with suitable rescaling of the unique solution to the corresponding problem on the half-line. On the other hand, a Lyapunov-Schmidt reduction procedure is performed to construct one-peaked and multipeaked positive solutions with sufficiently large frequency, exploiting the presence of one or more terminal edges.
2020
Least action; Lyapunov-Schmidt reduction; nonlinear Schrödinger; Peaked solutions; quantum graphs; terminal edges
01 Pubblicazione su rivista::01a Articolo in rivista
Peaked and low action solutions of NLS equations on graphs with terminal edges / Dovetta, S.; Ghimenti, M.; Micheletti, A. M.; Pistoia, A.. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 52:3(2020), pp. 2874-2894. [10.1137/19M127447X]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1466884
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