Nonlinear Schrödinger equation and multicomponent cnoidal waves in parametric frequency conversion

It is shown that the exact analytic solution of the problem of stationary parametric frequency conversion, including second harmonic generation and parametric amplification in a medium with quadratic nonlinearity, in the approximation of three interacting modes is reduced to the solution of three independent systems of nonlinear equations. Each of these systems consisting of two nonlinear Schrödinger equations is related to other systems only by the boundary conditions and describes a multicomponent cnoidal wave containing two noninterfering components. The problem can be represented in this way because the competition of two processes (the merging and decomposition of quanta) proceeding simultaneously on the second-order nonlinearity can be described through the effective cascade cubic nonlinearity.