463 research outputs found
On the Dressing Method for the Generalised Zakharov--Shabat System
The dressing procedure for the Generalised Zakharov--Shabat system is well
known for systems, related to sl(N) algebras. We extend the method,
constructing explicitly the dressing factors for some systems, related to
orthogonal and symplectic Lie algebras. We consider 'dressed' fundamental
analytical solutions with simple poles at the prescribed eigenvalue points and
obtain the corresponding Lax potentials, representing the soliton solutions for
some important nonlinear evolution equations.Comment: 17 pages, LaTe
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
Riemann-Hilbert Problems with canonical normalization and families of commuting operators
We start with a Riemann-Hilbert Problems (RHP) with canonical normalization
whose sewing functions depends on several 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. Thus
we are able to construct new classes of integrable nonlinear evolution
equations.Comment: 14 pages, Submitted to Pliska Stud. Math. Bulga
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