*(English)*Zbl 0965.34039

Summary: The extended Fisher-Kolmogorov equation

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 $u\left(x\right)=tanh(x/\sqrt{2})$ 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:

34C37 | Homoclinic and heteroclinic solutions of ODE |

34C28 | Complex behavior, chaotic systems (ODE) |

34F05 | ODE with randomness |