Abstract
We study the Stokes phenomenon for the solutions of general homogeneous linear moment partial differential equations with constant coefficients in two complex variables under condition that the Cauchy data are holomorphic on the complex plane but finitely many singular or branching points with the appropriate growth condition at the infinity. The main tools are the theory of summability and multisummability, and the theory of hyperfunctions. Using them, we describe Stokes lines, anti-Stokes lines, jumps across Stokes lines, and a maximal family of solutions.
Highlights
In this article, we generalise our results from [18] concerning summability and Stokes phenomenon for the formal solutions of the Cauchy problem for the complex heat equation
The relations between solutions are studied in the context of the Stokes phenomenon. It means that we find the Stokes lines, which separate different actual solutions constructed from the same multisummable formal power series solution
We get the main result of the paper about the maximal family of solutions and the Stokes phenomenon for Eq 1, which is given in Theorem 3
Summary
We generalise our results from [18] concerning summability and Stokes phenomenon for the formal solutions of the Cauchy problem for the complex heat equation. We describe such family of solutions of Eq 1 in the case when formal solution u is multisummable (Theorem 1). We return to the equation (1), and using the theory of multisummability, we get the main result of the paper, i.e. the description of a maximal family of solution, Stokes lines, and jumps across them for the equation (1), which is given in Theorem 3. We present a few examples of special cases of moment partial differential equations with constant coefficients, where by using hyperfunctions we derive the form of jumps across obtained Stokes lines
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