Patterns, Nonlinear Dynamics and Applications - PANDA

First meeting: Pattern formation in applications I: Monday 17th December 2001, DAMTP, Cambridge

John Norbury Patterns for inhomogeneous real Ginzburg-Landau equations


We extend the work of Rubinstein, Sternberg and Keller (1989 SIAM J. Appl. Math. 49 116-33) and consider chemical reactions, phase transitions or other processes governed by a semilinear reaction-diffusion equation (with Neumann boundary conditions) for u(x, t,
epsilon)  defined for t > 0 and x being an element of <(<Omega>)over bar> , a subset of Rn by

u(t) = epsilon del . (k(x)delu) + epsilon V1(u)

where x is an element of Omega, epsilon is a small parameter and V is a bistable potential for u; here V and k depend on x and V is even in u. Here one of the stable minimizers is pointwise positive, and the fact that V is even in u then gives that the other stable minimizer is negative.

The reaction rate epsilon V1(u) is large, while the diffusion coefficient is small. If the initial condition u(x, 0) = phi (x) is positive in the open domain Omega (1), negative in the open domain Omega (2) (with Omega (1) intersect Omega (2) = empty set), and zero on a
surface Gamma (epsilon), a subset of Omega, with Omega (1) union Omega (2)  union Gamma (epsilon) = Omega, then u rapidly tends to the positive stable state on Omega (1), and to the negative stable state on Omega (2); an interface of width O(epsilon) develops at Gamma (epsilon). Then each interface moves on a longer O(1/epsilon) timescale, either towards a stable equilibrium position Gamma (epsilon) for epsilon small, or away from unstable equilibrium positions. Here Gamma (0) is the limit curve that arises from {Gamma (epsilon)} as epsilon --> 0. The equilibrium locations for Gamma (0) are calculated from a geometric geodesic condition, together with their local stability. Simple formulae for these are derived, which depend only on the x variation in V and k for epsilon small.

J. Norbury, Mathematical Institute, 24-29 St Giles, Oxford, England.

L.C. Yeh, Department of Applied Mathematics, National Dong Hwa University, Hualien, Taiwan