16 research outputs found
Reductions of integrable equations on A.III-type symmetric spaces
We study a class of integrable non-linear differential equations related to
the A.III-type symmetric spaces. These spaces are realized as factor groups of
the form SU(N)/S(U(N-k) x U(k)). We use the Cartan involution corresponding to
this symmetric space as an element of the reduction group and restrict generic
Lax operators to this symmetric space. The symmetries of the Lax operator are
inherited by the fundamental analytic solutions and give a characterization of
the corresponding Riemann-Hilbert data.Comment: 14 pages, 1 figure, LaTeX iopart styl
On the Caudrey-Beals-Coifman System and the Gauge Group Action
The generalized Zakharov-Shabat systems with complex-valued Cartan elements
and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their
gauge equivalent are studies. This includes: the properties of fundamental
analytical solutions (FAS) for the gauge-equivalent to CBC systems and the
minimal set of scattering data; the description of the class of nonlinear
evolutionary equations solvable by the inverse scattering method and the
recursion operator, related to such systems; the hierarchies of Hamiltonian
structures.Comment: 12 pages, no figures, contribution to the NEEDS 2007 proceedings
(Submitted to J. Nonlin. Math. Phys.
N-wave interactions related to simple Lie algebras. Z_2- reductions and soliton solutions
The reductions of the integrable N-wave type equations solvable by the
inverse scattering method with the generalized Zakharov-Shabat systems L and
related to some simple Lie algebra g are analyzed. The Zakharov- Shabat
dressing method is extended to the case when g is an orthogonal algebra.
Several types of one soliton solutions of the corresponding N- wave equations
and their reductions are studied. We show that to each soliton solution one can
relate a (semi-)simple subalgebra of g. We illustrate our results by 4-wave
equations related to so(5) which find applications in Stockes-anti-Stockes wave
generation.Comment: 18 pages, 1 figure, LaTeX 2e, IOP-style; More clear exposition.
Introduction and Section 5 revised. Some typos are correcte
The Generalised Zakharov-Shabat System and the Gauge Group Action
The generalized Zakharov-Shabat systems with complex-valued non-regular
Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC
systems) and their gauge equivalent are studied. This study includes: the
properties of fundamental analytical solutions (FAS) for the gauge-equivalent
to CBC systems and the minimal set of scattering data; the description of the
class of nonlinear evolutionary equations, solvable by the inverse scattering
method, and the recursion operator, related to such systems; the hierarchies of
Hamiltonian structures. The results are illustrated on the example of the
multi-component nonlinear Schrodinger (MNLS) equations and the corresponding
gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type models,
related to so(5;C) algebra.Comment: 17 pages, 1 figure, LaTeX. arXiv admin note: substantial text overlap
with arXiv:0710.330