Exactly Solvable And Integrable Systems

The algebraic matrix hierarchy approach based on affine Lie sl(n) algebras leads to a variety of 1+1 soliton equations. By varying the rank of the underlying sl(n) algebra as well as its gradation in the affine setting, one encompasses the set of the soliton equations of the constrained KP hierarchy. The soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine sl(n) algebras realized in the unconventional gradations. Such soliton solutions exhibit non-trivial dependence on the KdV (odd) time flows and KP (odd and even) time flows which distinguishes them from the conventional structure of the Darboux-Bäcklund Wronskian solutions of the constrained KP hierarchy.

Equations of motion for a classical 3d discrete model, whose auxialiary system is a linear system, are investigated. The Lagrangian form of the equations of motion is derived. The Lagrangian variables are a triplet of "tau functions". The equations of motion for the Triplet of Tau functions are Three Trilinear equations. Simple solitons for the trilinear equations are given. Both the dispersion relation and the phase shift reflect the triplet structure of equations.

The briefly review on the common spin description of the nonlinear evolution equations.

Matrix Riccati equations and other nonlinear ordinary differential equations with superposition formulas are, in the case of constant coefficients, shown to have the same exact solutions as their group theoretical discretizations. Explicit solutions of certain classes of scalar and matrix Riccati equations are presented as an illustration of the general results.

Local (or modified) Yang -- Baxter equation (LYBE) gives the functional map from the parameters of the weights in the left hand side to the parameters of the correspondent weights in the right hand side of LYBE. Such maps solve the functional tetrahedron equation. In this paper all the maps associated with LYBE of the ferro-electric type with single parameter in each weight matrix are classified.

Group theoretical methods are used to study the equations describing \chi^{(2)}:\chi^{(2)} cascading. The equations are shown not to be integrable by inverse scattering techniques. On the other hand, these equations do share some of the nice properties of soliton equations. Large families of explicit analytical solutions are obtained in terms of elliptic functions. In special cases, these periodic solutions reduce to localized ones, i.e., solitary waves. All previously known explicit solutions are recovered, and many additional ones are obtained

It is shown that the 3-body trigonometric G_2 integrable system is exactly-solvable. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary values of the coupling constants, the Hamiltonian can be expressed as a quadratic polynomial in the generators of some Lie algebra of differential operators in a finite-dimensional representation. Four infinite families of eigenstates, represented by polynomials, and the corresponding eigenvalues are described explicitly.

This paper adds two observations to the work solv-int/9701016 where some eigenstates for a model based on tetrahedron equation have been constructed. The first observation is that there exists a more "algebraic" construction of one-particle states, resembling the 1+1-dimensional algebraic Bethe ansatz. The second observation is that the strings introduced in solv-int/9701016 are symmetries of a transfer matrix, rather than just eigenstates.

This paper continues the series begun with works solv-int/9701016 and solv-int/9702004. Here we show how to construct eigenstates for a model based on tetrahedron equation using the tetrahedral Zamolodchikov algebras. This yields, in particular, new eigenstates for the model on infinite lattice -- `perturbed', or `broken', strings.

This paper continues the series begun with works solv-int/9701016, solv-int/9702004 and solv-int/9703010. Here we construct more sophisticated strings, combining ideas from those papers and some considerations involving solutions of tetrahedron equation due to Sergeev, Mangazeev and Stroganov.