On new types of integrable 4-wave interactions

Abstract

We start with a Riemann-Hilbert Problems (RHP) with canonical normalization whose sewing functions depends on two or more additional variables. Using Zakharov-Shabat theorem we are able to construct a family of ordinary differential operators for which the solution of the RHP is a common fundamental analytic solution. This family of operators obviously commute provided their coefficients satisfy certain nonlinear evolution equations. Thus we are able to construct new classes of integrable nonlinear evolution equations. We illustrate the method with an example of a new type 4-wave interactions. Its Lax pair consists of operators which are both quadratic in the spectral parameter $\lambda$ and take values in the so(5) algebra.