A note on the nonlinear derived Cauchy problem

TL;DR

This paper generalizes the analytic Cauchy problem to the derived nonlinear PDE context, extending Kashiwara's formulation to D-algebraic and derived analytic cases, and introduces the characteristic variety for such systems.

Contribution

It adapts Kashiwara's formulation of the Cauchy problem to the derived nonlinear PDE setting and defines the characteristic variety in this new context.

Findings

Generalization of the Cauchy problem to derived nonlinear PDEs

Extension of Kashiwara's formulation to D-algebraic and derived analytic cases

Introduction of the characteristic variety for derived nonlinear PDE systems

Abstract

We define and study a generalization of the analytic Cauchy problem, that specializes to the Cauchy-Kowaleskaya-Kashiwara problem in the linear case. The main leitmotive of this text is to adapt Kashiwara's formulation of this problem both to the relatively D-algebraic case and to the derived analytic situation. Along the way, we define the characteristic variety of a derived nonlinear partial differential system.