Birth of step-like contrast structures connected with a cusp catastrophe

A special solution of the ordinary differential equation u''xx=u3-tu-x is considered which relates to solutions of a broad spectrum of partial differential equations with a small parameter. The function u(x,t) is the dominant term of asymptotic expressions with respect to the small parameter for these solutions near cusp points of the limiting solution. The existence of this special function u(x,t) is proved; its uniform asymptotics at infinity are constructed and substantiated.