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

TL;DR

This paper applies canonical perturbation theory to analyze stationary nonlinear Schrödinger equations on basic quantum graph structures, revealing how nonlinear effects influence spectral properties and scattering phenomena.

Contribution

It extends previous methods to derive asymptotic solutions for nonlinear quantum graphs, connecting nonlinear spectral features to linear graph characteristics at low intensities.

Findings

Spectral curves reduce to linear characteristic functions at low intensity.

Nonlinear scattering approaches linear scattering amplitudes in the low intensity limit.

Bifurcation and multistability phenomena emerge in the short-wave asymptotics.

Abstract

We consider exact and asymptotic solutions of the stationary cubic nonlinear Schr\"odinger equation (NLSE) 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 \cite{paper1}. 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-wave length asymptotics we discuss how genuine nonlinear effects such as 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.