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 equationfor small time at the boundaries of the sector of analyticity. Previous work [7], [5] 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.

Shock waves and compactons for fifth-order non-linear dispersion equations

“…Here, (1.16) indeed belongs to the NDE family, though it seems proper semigroups of its discontinuous solutions (if any) have never been examined. On the other hand, moving blow-up singularities and other types of complex singularities of the modified Harry Dym equation, u t = u 3 u xxx − u x − 1 2 u 3 , have been described in [6] by delicate asymptotic expansion techniques.…”

“…possessing multi-hump, compact solitary solutions [49]. 6 Concerning higher-order in time quasilinear PDEs, let us mention a generalization of the combined dissipative double-dispersive (CDDD) equation (see, e.g., [41])…”

Nonlinear evolution PDEs in $\mathbb R^{+}\times \mathbb C^{d}$: existence and uniqueness of solutions, asymptotic and Borel summability properties

“…Lemma 33. For F ∈ A φ , and ν > 4ρ 0 + α r + 3 large enough, (see Note after Definition (11)), N (F) defined in (25) satisfies the following bounds…”

“…In some cases, the Borel summation approach can be adapted to study short time existence of sectorial solutions and study small time asymptotics. One important application is in the analysis of singularity formation in PDEs [11]. For simplicity, and since some assumptions are less general than in the rest of the paper, we restrict to d = 1 (scalar case) in this section.…”

Existence, uniqueness, analyticity, and Borel summability for Boussinesq equations

“…There has been work on PDEs as well, starting with linear equations [19], [2] followed by general results for a class of nonlinear system of PDEs in complex sectors [7], [8]. A Borel based approach has also led to analysis of complex singularities for a specific PDE [9]. Recent developments include Navier-Stokes initial value problem (see [13], [12], [11]).…”