Book Review: Shock waves and reaction-diffusion equations

Generalized travelling waves for reaction-diffusion equations

In this paper, we introduce a generalization of travelling waves for evolution equations. We are especially interested in reaction-diffusion equations and systems in heterogeneous media (with general operators and general geometry). Our goal is threefold. First we give several definitions, for transition waves, fronts, pulses, global mean speed of propagation, etc. Next, we discuss the meaning ...

Thermal diffusion shock waves.

The Ludwig-Soret effect or thermal diffusion, which refers to the separation of liquid mixtures in a temperature gradient, is governed by a nonlinear, partial differential equation in space and time. It is shown here that the solution to the nonlinear differential equation for a binary mixture predicts the existence of shock waves completely analogous to fluid shocks and obeys an expression for...

Bistable travelling waves for nonlocal reaction diffusion equations

We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium u ≡ 1 is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is “above and away” from the intermediate equilibrium. For focusing kernels we ...

Reaction-Diffusion Equations and Learning

Reaction-diffusion waves in biology.

The theory of reaction-diffusion waves begins in the 1930s with the works in population dynamics, combustion theory and chemical kinetics. At the present time, it is a well developed area of research which includes qualitative properties of travelling waves for the scalar reaction-diffusion equation and for system of equations, complex nonlinear dynamics, numerous applications in physics, chemi...