Algebraically simple involutive differential systems and the Cauchy problem

Abstract

We consider systems of polynomial nonlinear partial differential equations (PDEs) possessing certain properties. Such systems were studied by the American mathematician Thomas in the 1930s, and he called them (algebraically) simple. Thomas gave a constructive procedure to split an arbitrary system of PDEs into a finite number of simple subsystems. The class of simple involutive systems of PDEs includes normal, or Kovalevskaya-type, systems and Riquier orthonomic passive systems. Systems of this class admit well-posed Cauchy problems. We discuss the basic features of the splitting algorithm, completion of simple systems to involution, and the well-posedness of the Cauchy problem. Two illustrative examples are given. Bibliography: 17 titles.