On new types of integrable 4-wave interactions

Abstract

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 λ and take values in the so(5) algebra.

IntegrableG-strands on semisimple Lie groups

“…We consider the Lie algebra valued functions on an open connected set U Ă R 2 given by Lpt, sq :" λ 2 a `λµpt, sq ´γpt, sq P g, Mpt, sq :" λ 2 b ´λπpt, sq ´ξpt, sq P g, where a, b P g are constant Lie algebra elements. The associated zero curvature representation (ZCR) is defined as [10,11] B t L ´Bs M `rL, Ms " 0 .…”

“…It is also possible to obtain Hamiltonian structures and Lax equations for the principal chiral model generated by using matrix differential operators with polynomial dependence on a parameter [3]. For example, a recent paper [11] derives new types of integrable 4-wave equations via a Lie algebra approach for sop5, Cq. These equations were derived directly from a zero curvature representation (ZCR) that is quadratic in a spectral parameters and were shown to admit solutions that may be constructed via the Riemann-Hilbert problem.…”