Variational principle for nonlinear wave propagation in dissipative systems

Area of Science:

• Physics
• Applied Mathematics
• Complex Systems

Background:

• Many natural systems exhibit dynamics governed by nonlinear waves.
• Understanding wave front propagation is crucial in diverse scientific fields.

Purpose of the Study:

• To demonstrate that nonlinear wave front dynamics in extended systems with positive surface tension can be formulated as a gradient system.
• To identify the variational potential governing this asymptotic behavior.

Main Methods:

• Analysis of nonlinear wave front propagation in extended systems.
• Formulation of asymptotic dynamics as a gradient system.
• Identification of the variational potential based on geometric properties.

Main Results:

• The asymptotic dynamics of wave fronts with positive surface tension can be consistently described by a gradient system.
• The variational potential is a linear combination of the wave front's occupied volume and surface area.
• This potential changes monotonically over time, indicating a predictable evolution.

Conclusions:

• The gradient system formulation provides a powerful simplification for studying nonlinear wave front dynamics.
• This approach is applicable even when the underlying field equations lack a gradient structure.
• The identified variational potential offers insights into the fundamental drivers of wave front evolution.