On some linear integral equations generating solutions of nonlinear partial differential equations

Two types of linear inhomogeneous integral equations, which yield solutions of a broad class of nonlinear evolution equations, are investigated. One type is characterized by a two-fold integration with an arbitrary measure and contour over a complex variable k, and thier complex conjugates, whereas the other one has a two-fold integration over one and the same contour. The inhomogeneous term, which may contain an arbitrary function of k, makes it possible to define a matrix structure on the solutions of the integral equations. The elements of these matrices are shown to obey a system of partial differential equations, the special form of which depends on the choice of the dispersion relation occurring in the integral equations. For special elements of the matrices closed partial differential equations can be derived, such as e.g. the nonlinear Schrödinger equation and the (real and complex) modified Korteweg-de Vries and sine-Gordon equations. The relations between the matrix elements are shown to lead to Miura transformations between the various partial differential equations.