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Variational and Stability Properties of Constant Solutions to the NLS Equation on Compact Metric Graphs

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Abstract

We consider the nonlinear Schrödinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the the graph is compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional: local and global minimality, and (orbital) stability. We consider both the subcritical regime and the critical one, in which the features of the graph become relevant. We describe how the above properties change according to the topology and the metric properties of the graph.

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Acknowledgements

The authors are grateful to Prof. Diego Noja for enlightening discussions on the issue of the stability of stationary solutions.

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Authors and Affiliations

  1. Dipartimento di Scienza ed Alta Tecnologia, Università degli Studi dell’Insubria, Via Valleggio 11, 22100, Como, Italy

    Claudio Cacciapuoti

  2. Dipartimento di Scienze Matematiche “G.L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Simone Dovetta & Enrico Serra

Authors
  1. Claudio Cacciapuoti
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  2. Simone Dovetta
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  3. Enrico Serra
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Correspondence to Enrico Serra.

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Cacciapuoti, C., Dovetta, S. & Serra, E. Variational and Stability Properties of Constant Solutions to the NLS Equation on Compact Metric Graphs. Milan J. Math. 86, 305–327 (2018). https://doi.org/10.1007/s00032-018-0288-y

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  • DOI: https://doi.org/10.1007/s00032-018-0288-y

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