Variational principle for non-linear wave propagation in dissipative systems

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

Dynamics of dissipative structures in reaction-diffusion equations

The authors investigate the dynamics of dissipative structures in a reaction-diffusion system. They propose an analytical theory for the behavior of dissipative structures and show that a dissipative structure (DS) that is stable in a one-dimensional homogeneous medium can be induced to drift by slow variation of the diffusion coefficients, or by curvature of the DS in higher dimensions. In one spatial dimension, this motion can be in the direction of increasing or decreasing diffusion coefficient, depending on properties of the DS which can be determined analytically. In two and three dimensions this drift is proportional to the sum of the curvature of the DS and the gradient of the diffusion coefficient of the medium. The analysis of this motion uses standard ideas from perturbation theory to find an equation of motion for the location of the DS. Numerical simulations in one and two dimensions show good quantitative agreement with the theoretical results