Singularities of $A$ and $B$ Types in Asymptotic Analysis of Solutions of a Parabolic Equation

The Cauchy problem for a quasi-linear parabolic equation with a small parameter multiplying a higher derivative is considered in two cases where the solution of the limit problem has a point of gradient catastrophe. The integrals determining the leading approximation correspond to the Lagrange singularity of type $A_3$ and the boundary singularity of type $B_3$. For another choice of the initial function, singular points corresponding to $A_{2n+1}$ and $B_{2n+1}$ with arbitrary $n\ge 1$ are obtained.