Multicomponent cnoidal waves in cascade parametric frequency conversion

It is shown that four-mode interaction in quasi-synchronous cascade frequency conversion on a quadratic nonlinearity can be described in terms of an effective cubic nonlinearity, which reduces the problem to solving the system of two coupled nonlinear Schrödinger equations (NSEs) with respect to the amplitudes of waves involved in both nonlinear processes. Analytic solutions of a new type found for this system have the form of cnoidal waves with components representing the sum and difference of the identical fundamental solutions of the NSE with shifted arguments. The obtained solutions cover the entire range of variation of boundary conditions, allowing the optimisation of the conversion efficiency in any particular situation.