Afendikov A.
Pulse solutions of some hydrodynamical problems in unbounded domains
We consider several hydrodynamic problems in unbounded domains where in the vicinity of the instability threshold the dynamics is governed by the generalized Cahn-Hilliard equation. For time independent solutions of this equation we recover Bogdanov-Takens bifurcation without parameter in the 3-dimensional reversible system with a line of equilibria. This line of equilibria is neither induced by symmetries, nor by first integrals. At isolated points, normal hyperbolicity of the line fails due to a transverse double eigenvalue zero. In case of bi-reversible problem the complete set ℬ of all small bounded solutions consists of periodic profiles, homoclinic pulses and a heteroclinic front-back pair (Asymptot. Anal. 60(3,4) (2008), 185–211). Later the small perturbation of the problem where only one symmetry is left was studied. Then ℬ consist entirely of trivial equilibria and multipulse heteroclinic pairs (Asymptotic Analysis, Volume 72, Number 1-2 , 2011, pp. 31-76). Our aim is to discuss hydrodynamic problems, where the reversibility breaking perturbation can’t be considered as small. We obtain the existence of a pair of heteroclinic solutions and partial results on their stability.
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