Exact expression for the propagating front velocity in nonlinear discrete systems under nonreciprocal coupling

TL;DR

This paper derives an exact formula for the velocity of propagating fronts in nonlinear discrete systems with nonreciprocal coupling, linking shape and speed, validated by numerical simulations.

Contribution

It introduces a novel approach treating fronts as rigid objects, enabling exact velocity computation in complex discrete systems with nonreciprocal coupling.

Findings

The formula accurately predicts front velocity in various discrete systems.

Numerical simulations confirm perfect agreement with the theoretical predictions.

The approach provides new insights into the properties of nonlinear wave fronts.

Abstract

Nonlinear waves are a robust phenomenon observed in complex systems ranging from mechanics to ecology. Fronts are fundamental due to their robustness against perturbations and capacity to propagate one state over another. Controlling and understanding these waves is then fundamental to make use of their properties. Their velocity is one of the most important properties, which can be theoretically computed only in limited conditions of the dynamical system, and it becomes elusive in the presence of spatial discreteness and nonreciprocal coupling. This work reveals that fronts in discrete systems can be treated as rigid objects when analyzing their whole trajectory instead of the instantaneous one. Then, a relationship between the front velocity and its found shape is given. The formula provides insight into fronts' long-observed properties and agrees with the approximative and parameterized methods described in the literature. Numerical simulations show perfect agreement with the theory.