Standing waves on a flower graph

Abstract

A flower graph consists of a half line and N symmetric loops connected at a single vertex with N≥2 (it is called the tadpole graph if N=1). We consider positive single-lobe states on the flower graph in the framework of the cubic nonlinear Schrödinger equation. The main novelty of our paper is a rigorous application of the period function for second-order differential equations towards understanding the symmetries and bifurcations of standing waves on metric graphs. We show that the positive single-lobe symmetric state (which is the ground state of energy for small fixed mass) undergoes exactly one bifurcation for larger mass, at which point (N−1) branches of other positive single-lobe states appear: each branch has K larger components and (N−K) smaller components, where 1≤K≤N−1. We show that only the branch with K=1 represents a local minimizer of energy for large fixed mass, however, the ground state of energy is not attained for large fixed mass if N≥2. Analytical results obtained from the period function are illustrated numerically.