Asymptotic existence theorem for formal solutions with singularities of nonlinear partial differential equations via multisummability

Abstract

In this paper, we consider the summability of formal solutions with singularities (such as logarithmic singularities, functional power singularities, etc.) of nonlinear partial differential equations in the complex domain. The main result is as follows: when a formal solution with singularities is given, under appropriate assumptions related to the formal solution, the equation has a true solution that admits the given formal solution as an asymptotic expansion. The proof is done by constructing a new formal solution that is equivalent to the given formal solution in the asymptotic sense and is multisummable in a suitable direction. The assumptions are stated in terms of the Newton polygon associated with the given formal solution.