On new types of integrable 4-wave interactions

TL;DR

This paper introduces a novel method for constructing integrable nonlinear evolution equations using Riemann-Hilbert problems with variable-dependent sewing functions, exemplified by a new 4-wave interaction model.

Contribution

It develops a new approach to generate integrable equations via RHPs with variable-dependent sewing functions, leading to new 4-wave interaction types with quadratic spectral operators.

Findings

Constructed a family of commuting differential operators from RHPs.

Derived a new class of integrable 4-wave interaction equations.

Presented a Lax pair with quadratic spectral dependence in so(5) algebra.

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.