We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV)
equation. Moreover, we modify the T operator in the the Lax pair of the CKdV
equation, in the search of a (2 + 1)-dimensional case and thereby propose a new
equation in (2+1) dimensions. We named this the (2+1)-dimensional...
A simple and general approach for calculating the elliptic finite-gap solutions of the
Korteweg-de Vries (KdV) equation is proposed. Our approach is based on the use of
the finite-gap equations and the general representation of these solutions in the form
of rational functions of the elliptic Weierstrass...
The Bäcklund transformations for the relativistic lattices of the Toda type and their
discrete analogues can be obtained as the composition of two duality transformations.
The condition of invariance under this composition allows to distinguish effectively the
integrable cases. Iterations of the Bäcklund...
We investigate some classical evolution model in the discrete 2+1 space-time. A map,
giving an one-step time evolution, may be derived as the compatibility condition for
some systems of linear equations for a set of auxiliary linear variables. Dynamical
variables for the evolution model are the coefficients...
We give a review of some recent work on generalization of the Bethe ansatz in the case
of 2 + 1-dimensional models of quantum field theory. As such a model, we consider
one associated with the tetrahedron equation, i.e. the 2+1-dimensional generalization
of the famous YangBaxter equation. We construct...