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 $\lambda$ 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 $\mathcal{T}_i$, $i=1,2$ 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 $\mathcal{T}_i$ are studied. We illustrate our results by the MMKdV equations related to the algebra $\mathfrak{g}\simeq so(8)$ and derive several new MMKdV-type equations using group of reductions isomorphic to ${\mathbb Z}_{2}$, ${\mathbb Z}_{3}$, ${\mathbb Z}_{4}$.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

Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice

We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus $k$. We also discuss particular cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal potential ($k\to 0$) our solutions model a quasi-one dimensional quantum degenerate Bose-Fermi mixture trapped in optical lattice. In the limit $k\to 1$ the solutions are expressed by hyperbolic function solutions (vector solitons). Thus we are able to obtain in an unified way quasi-periodic and periodic waves, and solitons. The precise conditions for existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA