Bifurcations of standing localized waves on periodic graphs

TL;DR

This paper investigates bifurcations of standing localized waves in the nonlinear Schrödinger equation on periodic graphs, revealing two families of small-amplitude solutions with specific symmetry, positivity, and decay properties.

Contribution

It introduces a novel analysis of bifurcations on periodic graphs using discrete map techniques and establishes the existence and properties of localized wave families.

Findings

Existence of two distinct symmetric localized wave families.

Proven positivity and exponential decay of solutions.

Asymptotic reduction linking discrete maps to the stationary NLS equation.

Abstract

The nonlinear Schrodinger (NLS) equation is considered on a periodic metric graph subject to the Kirchhoff boundary conditions. Bifurcations of standing localized waves for frequencies lying below the bottom of the linear spectrum of the associated stationary Schrodinger equation are considered by using analysis of two-dimensional discrete maps near hyperbolic fixed points. We prove existence of two distinct families of small-amplitude standing localized waves, which are symmetric about the two symmetry points of the periodic graphs. We also prove properties of the two families, in particular, positivity and exponential decay. The asymptotic reduction of the two-dimensional discrete map to the stationary NLS equation on an infinite line is discussed in the context of the homogenization of the NLS equation on the periodic metric graph.