Abstract
The problem of finding heteroclinic orbits for the quintic Ginzburg-Landau equation in the case that the imaginary parts of the coefficients are O (ϵ), with 0 < ϵ ⪡ 1, is considered. When χ = χ c , where χ is a distinguished parameter in the equation, and ϵ = 0 there exists a singular heteroclinic orbit connecting a stable finite amplitude state to the stable zero state. It is shown that for a certain region of parameter space this orbit persists for ϵ small, with a wave speed c( ϵ) which is O (ϵ). Furthermore, using an approach reminiscent of the Melnikov method, the sign of c′(0) is calculated. This calculation is crucial in settling the persistence question. Finally, it is shown that for this wave speed it is possible for additional heteroclinic orbits to exist. These additional orbits connect two finite amplitude states.
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