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

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.

On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

“…Yet, one can see an additional curve that does not connect to the linear spectrum as N → 0. This has originally been found in previous work [15] by coincidence, as the the numerical method jumped from one curve to another where they almost touch in the diagram. We stress that in a numerical approach it is very hard to make sure that all solutions of interest are found, even if one restricts the search to a restricted region in parameter space.…”

“…While this is mathematically sound, fixing the deformation parameter m is not a very useful approach in an applied setting. A more physical approach (and one that is useful when we consider star graphs) is to fix the L 2 -norm N By direct calculation (see [15]) we express the L 2 -norms in terms of elliptic integrals (see Appendix A) as…”

“…Among the physical applications of nonlinear wave equations on metric graphs is light transmission through a network of optical fibres or Bose-Einstein condensates in quasi one-dimensional traps. We refer to [14,15] where a detailed overview of the recent literature and some applications is given and just summarise here the relevant work related to the nodal counting. In a previous work some of us have shown that Sturm's oscillation theorem is generically broken for nonlinear quantum stars, apart from the special case of an interval [16].…”

NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph

“…Gnutzmann and Waltner have approached this numerical problem differently [20,21]. Rather than solve equation (1.9), they use the exact Jacobi elliptic function solutions to define solutions on each edge of the graph.…”

Existence of the ground state for the NLS with potential on graphs

“…The existence of the ground state for the tadpole-graph for any m > 0 has been proved in [9], see also [11]. A general approach to the study of the stationary solutions has been recently proposed in [24,25]. The stationary solutions on a compact star-graph, in a setting in which the nonlinear term changes from edge to edge has been studied in [35,40].…”