Complex Singularity Analysis for a nonlinear PDE

Abstract

We introduce a method of rigorous analysis of the location and type of complex singularities for nonlinear higher order PDEs as a function of the initial data. The method is applied to determine rigorously the asymptotic structure of singularities of the modified Harry-Dym equation $$H_t+H_y=-1/2H^3+H^3{H}_{yyy}:H(y,0)=y^{-1/2}$$ for small time at the boundaries of the sector of analyticity. Previous work \cite{CPAM}, \cite{invent03} shows existence, uniqueness and Borel summability of solutions of general PDEs. It is shown that the solution to the above initial value problem is represented convergently by a series in a fractional power of $t$ down to a small annular neighborhood of a singularity of the leading order equation. We deduce that the exact solution has a singularity nearby having, to leading order, the same type.