Exactly Solvable And Integrable Systems

Models generalizing the su(2) XX spin-chain were recently introduced. These XXC models also have an underlying su(2) structure. Their construction method is shown to generalize to the chains based on the fundamental representations of the A_m Lie algebras. Integrability of the new models is shown in the context of the quantum inverse scattering method. Their R-matrix is found and shown to yield a representation of the Hecke algebra. The diagonalization of the transfer matrices is carried out using the algebraic Bethe Ansatz. I comment on eventual generalizations and possible links to reaction-diffusion processes.

The method of multiscale analysis is constructed for dicrete systems of evolution equations for which the problem is that of the far behavior of an input boundary datum. Discrete slow space variables are introduced in a general setting and the related finite differences are constructed. The method is applied to a series of representative examples: the Toda lattice, the nonlinear Klein-Gordon chain, the Takeno system and a discrete version of the Benjamin-Bona-Mahoney equation. Among the resulting limit models we find a discrete nonlinear Schroedinger equation (with reversed space-time), a 3-wave resonant interaction system and a discrete modified Volterra model.

Integrable lattice equations arising in the context of singular manifold equations for scalar, multicomponent KP hierarchies and 2D Toda lattice hierarchy are considered. These equation generate the corresponding continuous hierarchy of singular manifold equations, its Bäcklund transformations and different forms of superposition principles. They possess rather special form of compatibility representation. The distinctive feature of these equations is invariance under the action of Möbius transformation. Geometric interpretation of these discrete equations is given.

We study the integrable systems in higher dimensions which can be written not by the Hirota's bilinear form but by the trilinear form. We explicitly discuss about the Bogoyavlenskii-Schiff(BS) equation in (2 + 1) dimensions. Its analytical proof of multi soliton solution and a new feature are given. Being guided by the strong symmetry, we also propose a new equation in (3 + 1) dimensions.

We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation, ϕ xt + ϕ xxxz /4+ ϕ x ϕ xz + ϕ xx ϕ z /2+ ∂ −1 x ϕ zzz /4=0 . This equation is obtained by unifying two directional generalization of the KdV equation, composing the closed ring with the KP equation and Bogoyavlenskii-Schiff equation. We also find the Miura transformation which yields the same ring in the corresponding modified equations.

Two dimensional classical integrable systems and different integrable deformations for them are derived from phase space realizations of classical sl(2) Poisson coalgebras and their q− deformed analogues. Generalizations of Morse, oscillator and centrifugal potentials are obtained. The N=2 Calogero system is shown to be sl(2) coalgebra invariant and the well-known Jordan-Schwinger realization can be also derived from a (non-coassociative) coproduct on sl(2) . The Gaudin Hamiltonian associated to such Jordan-Schwinger construction is presented. Through these examples, it can be clearly appreciated how the coalgebra symmetry of a hamiltonian system allows a straightforward construction of different integrable deformations for it.

A N=4 supersymmetric matrix KP hierarchy is proposed and a wide class of its reductions which are characterized by a finite number of fields are described. This class includes the one-dimensional reduction of the two-dimensional N=(2|2) superconformal Toda lattice hierarchy possessing the N=4 supersymmetry -- the N=4 Toda chain hierarchy -- which may be relevant in the construction of supersymmetric matrix models. The Lax pair representations of the bosonic and fermionic flows, corresponding local and nonlocal Hamiltonians, finite and infinite discrete symmetries, the first two Hamiltonian structures and the recursion operator connecting all evolution equations and the Hamiltonian structures of the N=4 Toda chain hierarchy are constructed in explicit form. Its secondary reduction to the N=2 supersymmetric alpha=-2 KdV hierarchy is discussed.

The local Sugawara constructions of the "small" N=4 SCA in terms of supercurrents of N=2 extensions of the affinization of the sl(2|1) and sl(3) algebras are investigated. The associated super mKdV type hierarchies induced by N=4 SKdV ones are defined. In the sl(3) case the existence of two inequivalent Sugawara constructions is found. The long one involves all the affine sl(3)-valued currents, while the "short" one deals only with those from the subalgebra sl(2)\oplus u(1). As a consequence, the sl(3)-valued affine superfields carry two inequivalent mKdV type super hierarchies induced by the correspondence between "small" N=4 SCA and N=4 SKdV hierarchy. However, only the first hierarchy posseses genuine global N=4 supersymmetry. We discuss peculiarities of the realization of this N=4 supersymmetry on the affine supercurrents.

In this paper we introduce a system of nonlinear ordinary differential equations which in a particular case reduces to Volterra's system. We found in two simplest cases the complete sets of the integrals of motion using Nambu--Poisson reformulation of the Hamiltonian dynamics. In these cases we have solved the systems by quadratures.

Two types of integrable coupled nonlinear Schrodinger (NLS) equations are derived by using Zakharov-Shabat (ZS) dressing method.The Lax pairs for the coupled NLS equations are also investigated using the ZS dressing method. These give new types of the integrable coupled NLS equations with certain additional terms. Then, the exact solutions of the new types are obtained. We find that the solution of these new types do not always produce a soliton solution even they are the kind of the integrable NLS equations.