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+u3-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|→∞, t→-∞ is constructed and substantiated.