Pattern propagation in nonlinear dissipative systems
The paper, motivated in a large part by a similarity of pattern propagation in a wide class of continuous systems described by nonlinear parabolic equations, is devoted to a generalization of marginal-stability hypothesis that has been suggested in the theory of dendritic crystal growth. Starting with a nonlinear diffusion problem and ''amplitude'' equation the authors examine different kinds of propagating solutions. They arrive at a conjecture on the validity of marginal-stability principle also away from the threshold for the pattern-forming models. Finally, three different pattern-forming models (the Hohenberg-Swift- Pomeau-Manneville equation, the reaction diffusion equations and the fourth order nonlinear parabolic equation derived in the theory of crystal growth) are solved numerically to support the hypothesis.
Hohenberg-Swift-Pomeau- Manneville equation
fourth order nonlinear parabolic equation
scientific article; zbMATH DE number 4010064