Alexander P. Veselov

Discrete versions of some classical integrable systems and factorization of matrix polynomials

Discrete versions of several classical integrable systems are investigated, such as a discrete analogue of the higher dimensional force-free spinning top (Euler-Arnold equations), the Heisenberg chain with classical spins and a new discrete system on the Stiefel manifold. The integrability is shown with the help of a Lax-pair representation which is found via a factorization of certain matrix polynomials. The complete description of the dynamics is given in terms of Abelian functions; the flow becomes linear on a Prym variety corresponding to a spectral curve. The approach is also applied to the billiard problem in the interior of anN-dimensional ellipsoid.

Commutative rings of partial differential operators and Lie algebras

We give examples of finite gap Schrödinger operators in the two-dimensional case.

Yang-Baxter maps and integrable dynamics

Abstract The hierarchy of commuting maps related to a set-theoretical solution of the quantum Yang–Baxter equation (Yang–Baxter map) is introduced. They can be considered as dynamical analogues of the monodromy and/or transfer-matrices. The general scheme of producing Yang–Baxter maps based on matrix factorisation is discussed in the context of the integrability problem for the corresponding dynamical systems. Some examples of birational Yang–Baxter maps coming from the theory of the periodic dressing chain and matrix KdV equation are discussed.

Growth and integrability in the dynamics of mappings

The growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed. In the general case such characteristics as complexity by Arnold or the number of the different images for the multiple-valued mappings are growing exponentially. It is shown that the integrability is closely related with thepolynomial growth. The analogies with quantum integrable systems are discussed.

Two-dimensional Schro¨dinger operator: inverse scattering transform and evolutional equations

Abstract The inverse problem for the two-dimensional Schrodinger operator on the data from one energy level is solved in a special class of “finite-zone” or “algebraic” operators. This class seems to be dense among all smooth periodic Schrodinger operators. Evolutional equations associated with this problem are constructed.

Deformed quantum Calogero-Moser problems and Lie superalgebras

The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized root systems suggested by V. Serganova. For the classical series a recurrent formula for the quantum integrals is found, which implies the integrability of these problems. The corresponding algebras of the quantum integrals are investigated, the explicit formulas for their Poincare series for generic values of the deformation parameter are presented.

New integrable generalizations of Calogero–Moser quantum problem

A one-parameter deformation of Calogero–Moser quantum problem is introduced. It is shown that corresponding Schrodinger operator is integrable for any value of the parameter and algebraically integrable in case of integer value.

Yang-Baxter maps and symmetries of integrable equations on quad-graphs

A connection between the Yang-Baxter relation for maps and the multidimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It is shown that the Yang-Baxter variables can be chosen as invariants of the multiparameter symmetry groups of the equations. We use the classification results by Adler, Bobenko, and Suris to demonstrate this method. Some new examples of Yang-Baxter maps are derived in this way from multifield integrable equations.

Integrable Schrödinger operators with magnetic fields: Factorization method on curved surfaces

The factorization method for Schrodinger operators with magnetic fields on a two-dimensional surface M2 with nontrivial metric is investigated. This leads to the new integrable examples of such operators and brings a new look at some classical problems such as the Dirac magnetic monopole and the Landau problem. The global geometric aspects and related spectral properties of the operators from the factorization chains are discussed in detail. We also consider the Laplace transformations on a curved surface and extend the class of Schrodinger operators with two integrable levels introduced in the flat case by S. P. Novikov and one of the authors.

Deformations of the root systems and new solutions to generalised WDVV equations

A special class of solutions to the generalised WDVV equations related to a finite set of covectors is investigated. Some geometric conditions on such a set which guarantee that the corresponding function satisfies WDVV equations are found (check-conditions). These conditions are satisfied for all root systems and their special deformations discovered in the theory of the Calogero-Moser systems by O.Chalykh, M.Feigin and the author. This leads to the new solutions for the generalized WDVV equations.Abstract A special class of solutions to the generalised WDVV equations related to a finite set of covectors is investigated. Some geometric conditions on such a set which guarantee that the corresponding function satisfies WDVV equations are found (∨-conditions). These conditions are satisfied for all root systems and their special deformations discovered in the theory of the Calogero–Moser systems by Chalykh, Feigin and the author. This leads to the new solutions for the generalized WDVV equations.