Recursion operator and dispersionless rational Lax representation

On construction of recursion operators from Lax representation

In this work we develop a general procedure for constructing the recursion operators for non-linear integrable equations admitting Lax representation. Several new examples are given. In particular we find the recursion operators for some KdV-type systems of integrable equations.

A Strange Recursion Operator Demystified

We show that a new integrable two-component system of KdV type studied by Karasu (Kalkanlı) et al. (arXiv: nlin.SI/0203036) is bihamiltonian, and its recursion operator, which has a highly unusual structure of nonlocal terms, can be written as a ratio of two compatible Hamiltonian operators found by us. Using this we prove that the system in question possesses an infinite hierarchy of local com...

Lax Operator for the Quantised Orthosymplectic

Lax operator for Macdonald symmetric functions

This is a joint work with Evgeny Sklyanin. Using the Lax operator formalism, we construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables and of two parameters q,t are their eigenfunctions. We express our operators in terms of the Hall-Littlewood symmetric functions of the same variables and of the parameter t corresponding to th...

Effective inverse spectral problem for rational Lax matrices and applications

We reconstruct a rational Lax matrix of size R+ 1 from its spectral curve (the desingularization of the characteristic polynomial) and some additional data. Using a twisted Cauchy–like kernel (a bi-differential of bi-weight (1−ν, ν)) we provide a residue-formula for the entries of the Lax matrix in terms of bases of dual differentials of weights ν, 1−ν respectively. All objects are described in...