arXiv subjects
Publications and source records attributed to A. N. Leznov.
The exactly integrable systems connected with semisimple series $A$ for arbitrary grading are presented in explicit form. Their general solutions are expressed in terms of the matrix elements of various fundamental representations of $A_n$ groups. The simplest example of such systems is the generalized Toda chain with the matrices of arbitrary dimensions in each point of the lattice.
All exactly integrable systems connected with the semisimple algebras of the second rank with an arbitrary choice of the grading in them are presented in explicit form. General solution of such systems are expressed in terms of the matrix elements of two fundamental representations of corresponding semisimple groups.
A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in terms of matrix elements of fundamental representations of semisimple $A_n$ algebras for a given group element. The possibility of generalizing this construction to multi-dimensional case is discussed.
The problem of construction of irreducible representations of quantum $A^q_n$ algebras is solved at the level of explicit integration of the linear (inhomogeneous) system in finite differences in the n-dimensional space. The general solution of this system is given explicitly and particular ones, which correspond to the irreducible representations are selected.
The general solution of the two-dimensional integrable generalization of the f-Toda chain with fixed ends is explicitly presented in terms of matrix elements of various fundamental representations of the SL(n|n-1) supergroup. The dominant role of the representation theory of graded Lie algebras in the problem of constructing integrable mappings and lattices is demonstrated.
It is shown that the Gel'fand-Tsetlin realization of irreducible representations of the $A_n$ algebra is directly connected with a linear exactly integrable system in the n-dimensional space. General solution for this system is explicitly given.
The new class of integrable mappings and chains is introduced. Corresponding (1+2) integrable systems invariant with respect to such discrete transformations are represented in explicit form. Soliton like solutions of them are represented in terms of matrix elements of fundamental representations of semisimple A_n algebras for a given group element.
A new integrable N=2 supersymmetric f-Toda mapping in (1|2) superspace, acting like the symmetry transformation of N=2 supersymmetric NLS hierarchy, is proposed. The first two Hamiltonian structures and the recursion operator connecting all evolution systems and Hamiltonian structures of the N=2 super-NLS hierarchy are constructed in explicit form using only invariance conditions with respect to the f-Toda mapping. A new representation for its Hamiltonians is observed.
The integrability of the recently introduced N=2 supersymmetric f-Toda chain, under appropriate boundary conditions, is proven. The recurrent formulae for its general solutions are derived. As an example, the solution for the simplest case of boundary conditions is presented in explicit form.
The formalism of integrable mappings is applied to the problem of constructing hierarchies of $(1+2)$ dimensional integrable systems in the $(2|2)$ superspace. We find new supersymmetric integrable mappings and corresponding to them new hierarchies of integrable systems which, at the reduction to the $(1|2)$ superspace, possess $N=2$ supersymmetry. The general formulae obtained for the hierarchies are used to explicitly derive their first nontrivial equations possessing a manifest $N=2$ supersymmetry. New bosonic substitutions and hierarchies are obtained from the supersymmetric counterparts in the bosonic limit.
A sequence of solutions to the equation of symmetry for the continuous Toda chain in $1+2$ dimensions is represented in explicit form. This fact leads to the supposition that this equation is completely integrable.
We establish an explicit form of the Backlund transformation for the most known integrable systems.
It is shown that the general solution of a homogeneous Monge-Ampère equation in $n$-dimensional space is closely connected with the exactly (but only implicitly) integrable system \frac {\partial ξ_{j}}{\partial x_0}+\sum_{k=1}^{n-1} ξ_{k} \frac {\partial ξ_{j}}{\partial x_{k}}=0 \label{1} Using the explicit form of solution of this system it is possible to construct the general solution of the Monge-Ampère equation.
A general method for the construction of solutions of the d'Alambertian and double d'Alambertian (harmonic and bi-harmonic) equations with local dependence of arbitrary functions upon two independent arguments is proposed. The connection between solutions of this kind and self-dual configurations of gauge fields having no singularities is established.
The properties of discrete nonlinear symmetries of integrable equations are investigated. These symmetries are shown to be canonical transformations. On the basis of the considered examples, it is concluded, that the densities of the conservation laws are changed under these transformations by spatial divergencies.
It is shown how the canonical symmetry is used to look for the hierarchy of the Hamiltonian operators relevant to the system under consideration. It appears that only the invariance condition can be used to solve the problem.
The properties of the canonical symmetry of the nonlinear Schrödinger equation are investigated. The densities of the local conservation laws for this system are shown to change under the action of the canonical symmetry by total space derivatives.
It is shown that each integrable mapping is connected with a hierarchical completely integrable sytem of equations of evolution type which are invariant with respect to the transformation described by this mapping.