Discrete dynamics in cluster integrable systems from geometric \(R\)-matrix transformations

Bäcklund Transformations for Finite-dimensional Integrable Systems: a Geometric Approach

We present a geometric construction of Bäcklund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Bäcklund transformations, which are naturally parametrized by the points on the spectral curve(s) of the system. The key idea is that a point on the curve determines, through the Abel-Jacobi map, a vector o...

Generalized r-matrix structure and algebro-geometric solution for integrable systems

The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a generalized Lax matrix instead of usual Lax pair. The generalized r-matrix structure and Hamiltonian functions are presented on the basis of fundamental Pois...

Bäcklund Transformations for Integrable Geometric Curve Flows

We study the Bäcklund transformations of integrable geometric curve flows in certain geometries. These curve flows include the KdV and Camassa-Holm flows in the two-dimensional centro-equiaffine geometry, the mKdV and modified Camassa-Holm flows in the two-dimensional Euclidean geometry, the Schrödinger and extended Harry-Dym flows in the three-dimensional Euclidean geometry and the Sawada-Kote...

From Discrete Integrable Systems to Cellular Automata

Integrable Discrete Linear Systems and One - Matrix Random Models

In this paper we analyze one–matrix models by means of the associated discrete linear systems. We see that the consistency conditions of the discrete linear system lead to the Virasoro constraints. The linear system is endowed with gauge invariances. We show that invariance under time–independent gauge transformations entails the integrability of the model, while the double scaling limit is con...