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 . For the classical case, , the heteroclinic is and is unique up to the obvious symmetries.
The author proves the conjecture that the uniqueness persists all the way to , 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 . He also proves uniform a priori bounds on all bounded stationary solutions, valid for any .