Yuri B. Suris

Integrable systems on quad-graphs

Discrete (lattice) systems constitute a well-established part of the theory of integrable systems. They came up already in the early days of the theory (see, e.g. [11, 12]), and took gradually more and more important place in it (cf. a review in [18]). Nowadays many experts in the field agree that discrete integrable systems are in many respects even more fundamental than the continuous ones. They play a prominent role in various applications of integrable systems such as discrete differential geometry (see, e.g., a review in [9]). Traditionally, independent variables of discrete integrable systems are considered as belonging to a regular square lattice Z (or its multidimensional analogs Z). Only very recently, there appeared first hints on the existence of a rich and meaningful theory of integrable systems on nonsquare lattices and, more generally, on arbitrary graphs. The relevant publications are almost exhausted by [2, 3, 5, 6, 16, 20, 21, 22]. We define integrable systems on graphs as flat connections with the values in loop groups. This is very natural definition, and experts in discrete integrable systems will not only immediately accept it, but might even consider it trivial. Nevertheless, it crystallized only very recently, and seems not to appear in the literature before [3, 5, 6]. (It should be noted that a different framework for integrable systems on graphs is being developed by Novikov with collaborators [16, 20, 21].) We were led to considering such systems by our (with Hoffmann) investigations of circle patterns as objects of discrete complex analysis: in [5, 6] we demonstrated that certain classes of circle patterns with the combinatorics of regular hexagonal lattice

New integrable systems related to the relativistic Toda lattice

The bi - Hamiltonian structure of the relativistic Toda lattice is exploited to introduce some new integrable lattice systems. Their integrable discretizations are obtained by means of the general procedure proposed recently by the author. Backlund transformations between the new systems and the relativistic Toda lattice (in both the continuous and discrete time formulations) are established.

Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function

Abstract Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The linear theory is based on the discrete Cauchy-Riemann equations, the nonlinear one is based on the notion of circle patterns. We clarify the role of the rhombic condition in both theories: under this condition the corresponding equations are integrable (in the sense of 3D consistency, which yields also the existense of zero curvature representations, Bäcklund transformations etc.). We demonstrate that in some precise sense the linear theory is a linearization of the nonlinear one: the tangent space to a set of integrable circle patterns at an isoradial point consists of discrete holomorphic functions which take real (imaginary) values on two sublattices. We extend solutions of the basic equations of both theories to ℤ d , where d is the number of different edge slopes of the quasicrystallic tiling. In the linear theory, we give an integral representation of an arbitrary discrete holomorphic function, thus proving the density of discrete exponential functions. We introduce the d-dimensional discrete logarithmic function which is a generalization of Kenyon’s discrete Green’s function, and uncover several new properties of this function. We prove that it is an isomonodromic solution of the discrete Cauchy-Riemann equations, and that it is a tangent vector to the space of integrable circle patterns along the family of isomonodromic discrete power functions.

A discrete-time relativistic Toda lattice

For each of the two simplest Hamiltonian flows from the relativistic Toda hierarchy we introduce two integrable symplectic discretizations. All four discrete-time systems are demonstrated to belong to the same hierarchy and to exemplify the general scheme for symplectic maps on groups equipped with quadratic Poisson brackets. The initial-value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flows are found for all maps.

Discrete Lagrangian Reduction, Discrete Euler–Poincaré Equations, and Semidirect Products

A discrete version of Lagrangian reduction is developed within the context of discrete time Lagrangian systems on G × G, where G is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of G. Within this context, the reduction of the discrete Euler–Lagrange equations is shown to lead to the so-called discrete Euler–Poincaré equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler–Poincaré equations leads to discrete Hamiltonian (Lie–Poisson) systems on a dual space to a semiproduct Lie algebra.

On some integrable systems related to the Toda lattice

We discuss some of the integrable lattices introduced recently by R Yamilov. We demonstrate that they are closely related to the usual Toda lattice by means of a sort of Backlund transformations. We also apply the general procedure of integrable discretization and obtain their integrable finite-difference approximations. These novel integrable discrete-time systems are also related to the discrete-time Toda lattice by means of the Backlund transformations. The whole construction exploits the tri-Hamiltonian structure of the Toda lattice.

INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEM: LOCAL EQUATIONS OF MOTION AND THEIR HAMILTONIAN PROPERTIES

We develop the approach to the problem of integrable discretization based on the notion of r-matrix hierarchies. One of its basic features is the coincidence of Lax matrices of discretized systems with the Lax matrices of the underlying continuous time systems. A common feature of the discretizations obtained in this approach is non-locality. We demonstrate how to overcome this drawback. Namely, we introduce the notion of localizing changes of variables and construct such changes of variables for a large number of examples, including the Toda and the relativistic Toda lattices, the Volterra and the relativistic Volterra lattices, the second flows of the Toda and of the Volterra hierarchies, the modified Volterra lattice, the Belov–Chaltikian lattice, the Bogoyavlensky lattices, the Bruschi–Ragnisco lattice. We also introduce a novel class of constrained lattice KP systems, discretize all of them, and find the corresponding localizing change of variables. Pulling back the differential equations of motion under the localizing changes of variables, we find also (sometimes novel) integrable one-parameter deformations of integrable lattice systems. Poisson properties of the localizing changes of variables are also studied: they produce interesting one-parameter deformations of the known Poisson algebras.

Lax Matrices for Yang-Baxter Maps

Abstract It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quad-graphs has been recently discovered by A. Bobenko and one of the authors, and by F. Nijhoff.

A note on an integrable discretization of the nonlinear Schrödinger equation

We revisit integrable discretizations for the nonlinear Schrodinger equation due to Ablowitz and Ladik. We demonstrate how their main drawback, the non-locality, can be overcome. Namely, we factorize the non-local difference scheme into the product of local ones. This must improve the performance of the scheme in the numerical computations dramatically. Using the equivalence of the Ablowitz - Ladik and the relativistic Toda hierarchies, we find the interpolating Hamiltonians for the local schemes and show how to solve them in terms of matrix factorizations.

Bi-Hamiltonian structure of the qd algorithm and new discretizations of the Toda lattice

Abstract We introduce two new discretizations of the Toda lattice related to the qd algorithm. They are demonstrated to belong to the same hierarchy as the continuous-time system, and to exemplify the general scheme for symplectic maps on Lie algebras with r -matrix Poisson brackets. The initial value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flows are found for both maps.