Traveling waves in periodic metric graphs via spatial dynamics

TL;DR

This paper introduces a new concept of traveling waves in periodic metric graphs, analyzing their existence and behavior beyond finite time intervals using spatial dynamics, and supports findings with numerical simulations.

Contribution

It develops a novel approach to study traveling waves on periodic metric graphs via spatial dynamics, revealing the existence of modulating pulse solutions with oscillatory tails.

Findings

Existence of traveling modulating pulse solutions with oscillatory tails.

Numerical simulations demonstrate wave propagation and tail formation.

Variational methods do not capture these solutions in the zero wave speed limit.

Abstract

The purpose of this work is to introduce a concept of traveling waves in the setting of periodic metric graphs. It is known that the nonlinear Schr{\"o}dinger (NLS) equation on periodic metric graphs can be reduced asymptotically on long but finite time intervals to the homogeneous NLS equation, which admits traveling solitary wave solutions. In order to address persistence of such traveling waves beyond finite time intervals, we formulate the existence problem for traveling waves via spatial dynamics. There exist no spatially decaying (solitary) waves because of an infinite-dimensional center manifold in the spatial dynamics formulation. Existence of traveling modulating pulse solutions which are solitary waves with small oscillatory tails at very long distances from the pulse core is proven by using a local center-saddle manifold. We show that the variational formulation fails to capture existence of such modulating pulse solutions even in the singular limit of zero wave speeds where true (standing) solitary waves exist. Propagation of a traveling solitary wave and formation of a small oscillatory tail outside the pulse core is shown in numerical simulations of the NLS equation on the periodic graph.