Parametric frequency conversion, nonlinear Schrödinger equation, and multicomponent cnoidal waves

Abstract

It is shown that, in a three-coupled-mode approximation, exact analytical solution of a system of equations, that describes a steady-state parametric frequency conversion, can be obtained as a solution of three closed nonlinear Schrödinger equations, coupled only through their boundary conditions. A reason for such a possibility lies in the description of the competition of two simultaneous second-order nonlinear processes in terms of an effective cascade cubic nonlinearity. Specific features and nontrivial asymptotes of complex self-consistent periodic solutions of the nonlinear Schrödinger equation are discussed.