Complex Singularity Analysis for a nonlinear PDE

الأصل · EN

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 Hₜ + Hy = - 1/2 H³ + H³ Hyyy: H(y, 0) = y⁻¹/² for small time at the boundaries of the sector of analyticity. Previous work CPAM, 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.