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Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph
Sven Gnutzmann1, Daniel Waltner2
1School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom.
Physical Review. E
|January 14, 2017
Summary
We study nonlinear Schrödinger equations on metric graphs, finding that low intensities simplify to linear quantum graph behavior. Higher intensities reveal nonlinear effects like spectral curve bifurcation and multistability.
Area of Science:
- Mathematical Physics
- Quantum Mechanics
- Nonlinear Dynamics
Background:
- The nonlinear Schrödinger equation models various physical phenomena.
- Metric graphs provide a framework for studying systems with complex connectivity.
- Understanding nonlinear behavior on graphs is crucial for diverse applications.
Purpose of the Study:
- To analyze exact and asymptotic solutions of the stationary cubic nonlinear Schrödinger equation on metric graphs.
- To investigate the transition from linear to nonlinear behavior in these systems.
- To explore phenomena like spectral curve bifurcation and multistability.
Main Methods:
- Utilizing canonical perturbation formalism for asymptotic solutions.
- Analyzing spectral curves for closed graphs (interval, ring, star, tadpole).
- Investigating nonlinear scattering for open graphs.
Main Results:
- Low-intensity limit shows spectral descriptions reduce to linear quantum graph characteristic functions.
- Nonlinear scattering in open graphs reduces to linear scattering amplitudes at low intensities.
- Short-wavelength asymptotics reveal genuine nonlinear effects: spectral curve bifurcation and multistability.
Conclusions:
- Canonical perturbation theory effectively describes nonlinear effects on metric graphs.
- The study bridges linear and nonlinear regimes in quantum graph systems.
- Findings offer insights into complex phenomena arising from nonlinearity in structured systems.
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