On the nodal structure of nonlinear stationary waves on star graphs

Band, Ram; Gnutzmann, Sven; Krueger, August

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.

Citation

Band, R., Gnutzmann, S., & Krueger, A. (2019). On the nodal structure of nonlinear stationary waves on star graphs. Symmetry, 11(2), Article 185. https://doi.org/10.3390/sym11020185

• Journal Article Type: Article
• Acceptance Date: Feb 1, 2019
• Online Publication Date: Feb 5, 2019
• Publication Date: Feb 5, 2019
• Deposit Date: Apr 23, 2019
• Publicly Available Date: Apr 25, 2019
• Journal: Symmetry
• Electronic ISSN: 2073-8994
• Publisher: MDPI
• Peer Reviewed: Peer Reviewed
• Volume: 11
• Issue: 2
• Article Number: 185
• DOI: https://doi.org/10.3390/sym11020185
• Keywords: Quantum graphs; Nonlinear Schrödinger equation; Nodal structure
• Public URL: https://nottingham-repository.worktribe.com/output/1596004
• Publisher URL: https://www.mdpi.com/2073-8994/11/2/185
• Contract Date: Apr 23, 2019