A comprehensive study of bound-states for the nonlinear Schr\"odinger equation on single-knot metric graphs

TL;DR

This paper investigates the existence, properties, and symmetry of bound-states for the nonlinear Schrödinger equation on single-knot metric graphs, using nonvariational methods like phase-plane analysis and asymptotic estimates.

Contribution

It provides the first comprehensive analysis of action ground-states on single-knot graphs, including existence, characterization, and symmetry-breaking results, using novel nonvariational techniques.

Findings

Existence of action ground-states for generic graphs.

Complete analysis of positive monotone bound-states for regular graphs.

Characterization of bound-states for small and large graph lengths.

Abstract

We study the existence and qualitative properties of action ground-states (that is, bound-states with minimal action) {of the nonlinear Schr\"odinger equation} over single-knot metric graphs -- which are made of half-lines, loops and pendants, all connected at a single vertex. First, we prove existence of action ground-state for generic single-knot graphs, even in the absence of an associated variational problem. Second, for regular single-knot graphs of length $\ell$, we perform a complete analysis of positive monotone bound-states. Furthermore, we characterize all positive bound-states when $\ell$ is small and prove some symmetry-breaking results for large $\ell$. Finally, we apply the results to some particular graphs to illustrate the complex relation between action ground-states and the topological {and metric} features of the underlying metric graph. The proofs are nonvariational, using a careful phase-plane analysis, the study of sections of period functions, asymptotic estimates and blowup arguments. We show, in particular, how nonvariational techniques are complementary to variational ones in order to deeply understand bound-states of the nonlinear Schr\"odinger equation on metric graphs.