Linear integral equations and multicomponent nonlinear integrable systems I

A systematic method for obtaining multicomponent generalizations of integrable nonlinear partial differential equations (PDE's) is developed. The method starts from a general type of linear integral equations, containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of k (the spectral parameter). Special k-dependent factors in the integrand are shown to induce an extra coupling between solutions of these integral equations with a different source term. In this way the direct linearization is obtained of multicomponent generalizations of various nonlinear PDE's, such as the nonlinear Schrödinger equation, the derivative nonlinear Schrödinger equations, the isotropic Heisenberg spin chain equation, the (complex) sine-Gordon equation, and the massive Thirring model equations. In the present paper (I) we present the general framework to derive finite-matrix PDE's, and we also discuss Bäcklund transformations and multicomponent lattice versions. In a subsequent paper (II) we treat a variety of examples of multicomponent PDE's, and discuss Miura transformations and gauge equivalences.