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

Gnutzmann, Sven; Waltner, Daniel

Authors

Daniel Waltner



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), 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 1539-3755
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 http://eprints.nottingham.ac.uk/id/eprint/40683
Publisher URL http://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.062216
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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