Uniqueness of the Stationary Wave for the Extended Fisher–Kolmogorov Equation

AbstractThe extended Fisher–Kolmogorov equation, ut=−βuxxxx+uxx+u−u3, 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. We prove 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. Our methods 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. We also prove uniform a priori bounds on all bounded stationary solutions, valid for any β>0