Related papers: Selected Topics in Classical Integrability
We review some essential aspects of classically integrable systems. The detailed outline of the lectures consists of: 1. Introduction and motivation, with historical remarks; 2. Liouville theorem and action-angle variables, with examples…
We consider integrable boundary conditions for both discrete and continuum classical integrable models. Local integrals of motion generated by the corresponding transfer matrices give rise to time evolution equations for the initial Lax…
We introduce a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs, with $N\times N$ matrices, linear in the spectral parameter. We give a classification scheme for such Lax pairs and the associated discrete integrable systems. We present…
Boundary conditions compatible with classical integrability are studied both directly, using an approach based on the explicit construction of conserved quantities, and indirectly by first developing a generalisation of the Lax pair idea.…
In these lecture notes we aim for a pedagogical introduction to both classical and quantum integrability. Starting from Liouville integrability and passing through Lax pair and r-matrix we discuss the construction of the conserved charges…
We study the integrability of two-dimensional theories that are obtained by a dimensional reduction of certain four-dimensional gravitational theories describing the coupling of Maxwell fields and neutral scalar fields to gravity in the…
In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum $R$-matrices. Here we study the simplest case -- the 11-vertex $R$-matrix and related ${\rm gl}_2$ rational…
A fairly complete list of Toda-like integrable lattice systems, both in the continuous and discrete time, is given. For each system the Newtonian, Lagrangian and Hamiltonian formulations are presented, as well as the 2x2 Lax representation…
We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical $R$-matrix even at the classical level,…
We systematically derive the Lax pair formulation for both discrete and continuum integrable classical theories with consistent boundary conditions.
This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases…
We give a short summary of our recent works on the classical integrable structure of two-dimensional non-linear sigma models defined on squashed three-dimensional spheres. There are two descriptions to describe the classical dynamics, 1)…
New classes of classically integrable models in the cosmological theories with a scalar field are obtained by using freedoms of defining time and fields. In particular, some models with the sum of exponential potentials in the flat spatial…
We propose the systematic construction of classical and quantum two dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
Classical integrable impurities associated to high rank (gl_N) algebras are investigated. A particular prototype i.e. the vector non-linear Schr\"{o}dinger (NLS) model is chosen as an example. A systematic construction of local integrals of…
A master equation expressing the classical integrability of two-dimensional non-linear sigma models is found. The geometrical properties of this equation are outlined. In particular, a closer connection between integrability and T-duality…
This paper is devoted to constructing and studying exactly solvable dynamical systems in discrete time obtained from some algebraic operations on matrices, to reductions of such systems leading to classical field theory models in…
We consider a hierarchy of the natural type Hamiltonian systems of $n$ degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of $2\times 2$…