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Symmetry 2019, 11(2), 185; https://doi.org/10.3390/sym11020185

On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

1
Department of Mathematics, Technion—Israel Institute of Technology, Haifa 3200003, Israel
2
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
3
Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA
*
Author to whom correspondence should be addressed.
Received: 14 January 2019 / Revised: 31 January 2019 / Accepted: 1 February 2019 / Published: 5 February 2019
(This article belongs to the Special Issue Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks)
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Abstract

We consider stationary waves on nonlinear quantum star graphs, i.e., solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e., the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting, such as the classic Sturm oscillation theorem. View Full-Text
Keywords: quantum graphs; nonlinear Schrödinger equation; nodal structure quantum graphs; nonlinear Schrödinger equation; nodal structure
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Band, R.; Gnutzmann, S.; Krueger, A.J. On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs. Symmetry 2019, 11, 185.

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