Complex periodic solutions of the nonlinear Schrödinger equation and nondegenerate multicomponent cnoidal waves in parametric frequency conversion

A new type of complex periodic solutions of the nonlinear Schrödinger equation is described which can be obtained in the collinear interaction of three plane monochromatic waves (modes) in a quadratic nonlinear medium. On passing to real variables (quadrature components), the solutions of this new type describe nondegenerate two-component cnoidal waves consisting of two 'incoherent' (noninterfering) components. The amplitudes of these components perform additional (with respect to the modulus oscillations) intricate nonlinear oscillations phase-shifted by π/2, which are consistent with oscillations of the solution modulus described by an elliptic function.