Variational principle for non-linear wave propagation in dissipative systems

TL;DR

This paper introduces a variational principle for non-linear wave fronts in dissipative systems, showing they can be modeled as gradient systems with a potential based on volume and surface area, and discusses conditions for vortex filament dynamics.

Contribution

It formulates a variational principle for non-linear wave propagation in dissipative systems, linking wave front dynamics to gradient systems and identifying conditions for vortex filament behavior.

Findings

Wave fronts with positive surface tension form gradient systems.

The variational potential combines volume and surface area of the wave front.

Vortex filaments behave as gradient systems only under specific velocity conditions.

Abstract

The dynamics of many natural systems is dominated by non-linear waves propagating through the medium. We show that the dynamics of non-linear wave fronts with positive surface tension can be formulated as a gradient system. The variational potential is simply given by a linear combination of the occupied volume and surface area of the wave front, and changes monotonically in time. Finally, we demonstrate that vortex filaments can be written as a gradient system only if their binormal velocity component vanishes, which occurs in chemical system with equal diffusion of reactants.