Power and Exponential-power Series Solutions of Evolution Equations

1.1. Brief overview. The theory of partial differential equations when one, or more variables, is in the complex domain, and approaches a characteristic variety has only recently started to develop. In their paper [11], generalized in [12], O. Costin and S. Tanveer proved existence and uniqueness of solutions with given initial conditions, for quasilinear systems of evolution equations in a large enough sector of C. Borel summability of divergent solutions of the heat equation was proved by Lutz, Miyake, and Schäfke [13], and more generally, Borel summability of series solutions of linear equations with constant coefficients was proved, in a general setting, by Balser (see [1], and the references therein). A natural question is to find what formal objects lie beyond formal power series solutions, and what is their connection to power series. The present paper contains initial results in this direction. For ordinary differential equations a comprehensive and general theory of formal solutions (transseries), in a one-to-one correspondence with true solutions, is presented in the fundamental work of Écalle [3]-[5]. The correspondence between transseries and solutions was later proved under nonresonance assumptions by O. Costin, who constructed a generalized Borel transform [6], [7]. O. Costin and Kruskal showed how formal solutions can be used to produce the Stokes constants [9], [8]. Transseries solutions can be used to find the type and location of movable arrays of singularities toward the irregular singular point [8], [10]. Braaksma has recently extended the theory of transseries representations to nonlinear difference equations [2]. The structure of singularities of solutions of difference equations has been obtained by Kuik [15].