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

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.