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Uniqueness of the stationary wave for the extended Fisher-Kolmogorov equation. (English) Zbl 0965.34039

Summary: The extended Fisher-Kolmogorov equation

u t =-βu xxxx +u xx +u-u 3 ,β>0,

models a binary system near the Lifshitz critical point and is known to exhibit a stationary heteroclinic solution joining the equilibria ±1. For the classical case, β=0, the heteroclinic is u(x)=tanh(x/2) and is unique up to the obvious symmetries.

The author proves the conjecture that the uniqueness persists all the way to β=1/8, where the onset of spatial chaos associated with the loss of monotonicity of the stationary wave is known to occur. The method used are non-perturbative and employ a global cross-section to the Hamiltonian flow of the stationary fourth-order equation on the energy level of ±1. He also proves uniform a priori bounds on all bounded stationary solutions, valid for any β>0.

MSC:
34C37Homoclinic and heteroclinic solutions of ODE
34C28Complex behavior, chaotic systems (ODE)
34F05ODE with randomness