57 research outputs found
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
Reductions of Multicomponent mKdV Equations on Symmetric Spaces of DIII-Type
New reductions for the multicomponent modified Korteveg-de Vries (MMKdV)
equations on the symmetric spaces of {\bf DIII}-type are derived using the
approach based on the reduction group introduced by A.V. Mikhailov. The
relevant inverse scattering problem is studied and reduced to a Riemann-Hilbert
problem. The minimal sets of scattering data , which
allow one to reconstruct uniquely both the scattering matrix and the potential
of the Lax operator are defined. The effect of the new reductions on the
hierarchy of Hamiltonian structures of MMKdV and on are
studied. We illustrate our results by the MMKdV equations related to the
algebra and derive several new MMKdV-type equations
using group of reductions isomorphic to , ,
.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras
The construction of a family of real Hamiltonian forms (RHF) for the special
class of affine 1+1-dimensional Toda field theories (ATFT) is reported. Thus
the method, proposed in [1] for systems with finite number of degrees of
freedom is generalized to infinite-dimensional Hamiltonian systems. The
construction method is illustrated on the explicit nontrivial example of RHF of
ATFT related to the exceptional algebras E_6 and E_7. The involutions of the
local integrals of motion are proved by means of the classical R-matrix
approach.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
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