Effective cubic nonlinearity and cnoidal waves in the degenerate parametric frequency conversion

It is shown that under the degenerate conditions (SHG and type-I degenerate parametric generation), the increase in the order of a system of truncated wave equations reduces the initial problem to the solution of two independent nonlinear Schrödinger equations, which allows one to write down the analytic solutions of the corresponding problems by using the standard methods for constructing the searched-for solutions in the form of cnoidal waves. It is established that in realising of cascade processes (quasi-synchronism), one of which is degenerate, this approach based on the introduction of the concept of the effective cascade cubic nonlinearity is not very convenient because the latter in the general case is no longer the Kerr one.