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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

${u}_{t}=-\beta {u}_{xxxx}+{u}_{xx}+u-{u}^{3},\phantom{\rule{1.em}{0ex}}\beta >0,$

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, $\beta =0$, the heteroclinic is $u\left(x\right)=tanh\left(x/\sqrt{2}\right)$ 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 $±1$. He also proves uniform a priori bounds on all bounded stationary solutions, valid for any $\beta >0$.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions of ODE 34C28 Complex behavior, chaotic systems (ODE) 34F05 ODE with randomness