On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

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...

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Bibliographic Details
Published in:Symmetry (Basel) 2019-02, Vol.11 (2), p.185
Main Authors: Band, Ram, Gnutzmann, Sven, Krueger, August J.
Format: Article
Language:English
Subjects:
Eigenvalues
Graph matching
Graphs
Schrodinger equation
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Summary: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.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym11020185