THE GENERAL KOVALEVSKAYA THEOREM FOR A CAUCHY PROBLEM

Abstract

The Kovalevskaya theorem is a well-known mathematical result that provides a powerful tool for solving systems of ordinary differential equations (ODEs). However, there has been a longstanding question regarding whether the theorem can be extended to more general systems of partial differential equations (PDEs). In this paper, we address this question by presenting a discussion of the only general Kovalevskaya theorem that can be extended from the theory of ODEs to systems of PDEs. Our discussion begins by reviewing the classical Kovalevskaya theorem for ODEs and its various applications in mathematics and physics. We then introduce the concept of Frobenius manifold, which provides a framework for extending the Kovalevskaya theorem to PDEs. We show how the Frobenius manifold structure can be used to transform a system of PDEs into a system of ODEs, which can then be solved using the classical Kovalevskaya theorem. Overall, our discussion provides a novel perspective on the Kovalevskaya theorem and its potential applications in PDE theory. By extending the theorem to this broader class of equations, we open up new possibilities for solving complex problems in mathematics, physics, and engineering.