On small stationary localized solutions for the generalized 1-D Swift–Hohenberg equation

Abstract

We prove the existence of small localized stationary solutions for the generalized Swift-Hohenberg equation and find under some assumption a part of a boundary of their existence in the parameter plane. The related stationary equation creates a reversible Hamiltonian system with two degrees of freedom that undergoes the Hamiltonian-Hopf bifurcation with an additional degeneracy. We investigate this bifurcation in a two-parameter unfolding by means of the sixth-order normal form for the related Hamiltonian. The region where no localized solutions exist has been pointed out as well. (c) 1995 American Institute of Physics.