Complex Singularity Analysis for a nonlinear PDE

Details Complex Singularity Analysis for a nonlinear PDE Complex Singularity Analysis for a nonlinear PDE

2006-08-12

arxiv.org/abs/math/0608305v1

Mathematics

Analysis of PDEs

Scientific paper

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 $$ H_t + H_y = - {1/2} H^3 + H^3 H_{yyy} : H(y, 0) = y^{-1/2} $$ 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.

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