Asymptotic behaviour of a special solution of Abel's equation relating to a cusp catastrophe

A special solution of Abel's ordinary differential equation of the first kind $u'_{x}+u^3-tu-x=0$ is considered, which describes the behaviour of a broad spectrum of solutions of partial differential equations with a small parameter in the neighbourhood of cusp points of their slowly varying equilibrium positions. The existence of this special solution is demonstrated; an asymptotic formula for it as $|x|\to\infty$, $t\to-\infty$ is constructed and substantiated.
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