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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 $$u_t= -\beta u_{xxxx}+ u_{xx}+ u- u^3,\quad \beta> 0,$$ models a binary system near the Lifshitz critical point and is known to exhibit a stationary heteroclinic solution joining the equilibria $\pm 1$. For the classical case, $\beta= 0$, the heteroclinic is $u(x)= \tanh(x/\sqrt 2)$ and is unique up to the obvious symmetries. The author proves the conjecture that the uniqueness persists all the way to $\beta= 1/8$, 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 $\pm 1$. He also proves uniform a priori bounds on all bounded stationary solutions, valid for any $\beta>0$.

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