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

A special solution of the ordinary differential equation $u''_{xx}=u^3-tu-x$ 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.