Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

Gnutzmann, Sven; Waltner, Daniel

Abstract

We consider exact and asymptotic solutions of the stationary cubic nonlinear Schrödinger equation on metric graphs. We focus on some basic example graphs. The asymptotic solutions are obtained using the canonical perturbation formalism developed in our earlier paper [S. Gnutzmann and D. Waltner, Phys. Rev. E 93, 032204 (2016)]. For closed example graphs (interval, ring, star graph, tadpole graph), we calculate spectral curves and show how the description of spectra reduces to known characteristic functions of linear quantum graphs in the low-intensity limit. Analogously for open examples, we show how nonlinear scattering of stationary waves arises and how it reduces to known linear scattering amplitudes at low intensities. In the short-wavelength asymptotics we discuss how genuine nonlinear effects may be described using the leading order of canonical perturbation theory: bifurcation of spectral curves (and the corresponding solutions) in closed graphs and multistability in open graphs.

Citation

Gnutzmann, S., & Waltner, D. (in press). Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures. Physical Review E, 94(6), Article 062216. https://doi.org/10.1103/PhysRevE.94.062216

• Journal Article Type: Article
• Acceptance Date: Dec 2, 2016
• Online Publication Date: Dec 22, 2016
• Deposit Date: Feb 21, 2017
• Publicly Available Date: Feb 21, 2017
• Journal: Physical Review E
• Print ISSN: 2470-0045
• Electronic ISSN: 2470-0053
• Publisher: American Physical Society
• Peer Reviewed: Peer Reviewed
• Volume: 94
• Issue: 6
• Article Number: 062216
• DOI: https://doi.org/10.1103/PhysRevE.94.062216
• Keywords: quantum graphs, nonlinear waves
• Public URL: https://nottingham-repository.worktribe.com/output/832512
• Publisher URL: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.062216
• Contract Date: Feb 21, 2017