In memoriam: Boris Dubrovin

Letters in Mathematical Physics -     

Quantum R-matrices and factorization problems

A relation between quantum R-matrices and certain factorization problem in Hopf algebras is established. A definition of dressinf transformation in the quantum case is also given.     

Poisson Groups and Differential Galois Theory of Schroedinger Equation on the Circle

We combine the projective geometry approach to Schroedinger equations on the circle and differential Galois theory with the theory of Poisson Lie groups to construct a natural Poisson structure on the space of wave functions (at the zero energy level). Applications to KdV-like nonlinear equations are discussed. The same approach is applied to 2^nd order difference operators on a one-dimensional lattice, yielding an extension of the lattice Poisson Virasoro algebra.     

Monodromy map and classicalr-matrices

A regularization of the Poisson brackets of the monodromy matrices based on the method of zero range potentials is proposed. Classification of the regularized brackets is reduced to the classical Yang-Baxter identity for the square of the initial Lie algebra. Bibliography: 11 titles. Respectfully dedicated to O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 156–166. Translated by M. A. Semenov-Tian-Shansky.     

Dual parametrization of GPDs versus the double distribution Ansatz

We establish a link between the dual parametrization of GPDs and a popular parametrization based on the double distribution Ansatz, which is in prevalent use in phenomenological applications. We compute several first forward-like functions that express the double distribution Ansatz for GPDs in the framework of the dual parametrization and show that these forward-like functions make the dominant contribution into the GPD quintessence function. We also argue that the forward-like functions with 1 contribute to the leading singular small-x_Bj behavior of the imaginary part of DVCS amplitude. This makes the small-x_Bj behavior of independent of the asymptotic behavior of PDFs. Assuming analyticity of Mellin moments of GPDs in the Mellin space we are able to fix the value of the D -form factor in terms of the GPD quintessence function N(x, t) and the forward-like function Q ₀(x, t) .     

Classical r-matrices and compatible Poisson brackets for coupled KdV systems

The formalism of classical r-matrices is used to construct families of compatible Poisson brackets for some nonlinear integrable systems connected with Virasoro algebras. We recover the coupled KdV [1] and Harry Dym [2] systems associated with the auxiliary linear problem 1 $$\sum\limits_{i = 0}^N {\lambda '\left( {a_i \frac{{{\text{d}}^{\text{2}} }}{{{\text{dx}}^2 }} + {\text{u}}_{\text{i}} } \right)} \psi = 0$$ .     

The Kowalewski top 99 years later: A Lax pair,generalizations and explicit solutions

A natural Lax pair for the Kowalewski top is derived by using a general group-theoretic approach. This gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite-band integration theory.     

A new integrable case of the motion of the 4-dimensional rigid body

A Lax pair for a new family of integrable systems on SO(4) is presented. The construction makes use of a twisted loop algebra of theG ₂ Lie algebra. We also describe a general scheme producing integrable cases of the generalized rigid body motion in an external field which have a Lax representation with spectral parameter. Several other examples of multi-dimensional tops are discussed.