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On new types of integrable 4-wave interactions
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 and take values in the so(5) algebra.Comment: 8 pages, reported at AMITANS-4 conference, June 11-16, 2012, St.St.
Constantine and Helena, Varna, Bulgari
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