| Remarks on the Vol'pert theory of travelling-wave solutions for parabolic systems (2007) | |||||||||||||
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| Vol'pert and Vol'pert have shown that the parabolic system of n equations, @u @t = A @ 2 u @x 2 + f(u) has monotone travelling-wave solutions connecting two equilibria, S and T say, under `bistable' and `monostable' conditions on the function f : R n ! R n , when A is a positive-definite diagonal matrix. We observe that this genre of hypotheses on f yields results on the existence and properties of intermediate zeros of f between S and T . These imply the existence of a `chain' of monotone travelling waves from S to T under `unstable' conditions on f for which the Vol'perts have established the non-existence of a direct monotone connection. It is also proved that for wave velocities consistent with their existence theory, there are unstable and stable monotone directions at S and T respectively, and the dimensions of the unstable manifold at S and the stable 1 manifold at T sum to at least 2n, the dimension of the underlying phase space. Some probabilistic appl... | |||||||||||||
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