On the relationship between nonlinear equations integrable by the method of characteristics and equations associated with commuting vector fields

Abstract

It was shown recently that Frobenius reduction of the matrix fields reveals interesting relations among the nonlinear partial differential equations (PDEs) integrable by the inverse spectral transform method (S-integrable PDEs), linearizable by the Hopf–Cole substitution (C-integrable PDEs), and integrable by the method of characteristics (Ch-integrable PDEs). However, only two classes of S-integrable PDEs have been involved: soliton equations (such as Korteweg–de Vries, nonlinear Schrödinger, Kadomtsev–Petviashvili, and Davey–Stewartson equations) and GL(N,C) self-dual-type PDEs (such as Yang–Mills equation). In this paper we consider the simple five-dimensional nonlinear PDE from another class of S-integrable PDEs, namely, scalar nonlinear PDE which is the commutativity condition of the pair of vector fields. We show its origin from the (1+1)-dimensional hierarchy of Ch-integrable PDEs after certain composition of Frobenius type and differential reductions imposed on the matrix fields. Matrix generalization of the above scalar nonlinear PDE will be derived as well.