Generalization of the inversion equations and application to nonlinear partial differential equations. I

The generalization of the inversion equations corresponding to a system of n linear coupled equations is obtained in two different cases : first for the second order differential linear operator μ2(x)[∂2/∂x2−V0(x)] and second for the linear first order operator μ (x) ∂/∂x. For the latter (which is an extension of the 2×2 case of Ablowitz et al.) we consider only a twofold eigenvalue problem for the n×n system, leaving the more general case to a subsequent paper. The equations of inversion are obtained by simple algebraic method, without considering both the direct and inverse scattering problem as in the classical method and without investigating the properties of the Jost solutions in the complex eigenvalue plane. Always using pedestrian algebraic method, and assuming that the kernel of the integral equation satisfies a linear partial differential equation, we deduce, for the solution of the integral equation, the corresponding nonlinear part of the evolution equation. In this way we show that the explicit construction of the classical solvable nonlinear evolution equations can be extended in two cases : coupled equations and substitution of μ (x) ∂/∂x for ∂/∂x.