Faces of Relativistic Toda Chain

Abstract

We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda Lattice hierarchy (2DTL). This reduction implies that the RTC is gauge equivalent to the discrete AKNS hierarchy and, which is the same, to the two-component Volterra hierarchy while its forced (semi-infinite) variant is described by the unitary matrix integral. The integrable properties of the RTC hierarchy are revealed in different frameworks of: Lax representation, orthogonal polynomial systems, and $\tau$-function approach. Relativistic Toda molecule hierarchy is also considered, along with the forced RTC. Some applications to biorthogonal polynomial systems are discussed.

51 pages, LaTeX, no figures (some misprints are corrected)

Cited by in corpus (29)

• Multiscale N=2 SUSY field theories, integrable systems and their stringy/brane origin -- I
• A survey of Hirota's difference equations
• The Ruijsenaars-Schneider Model in the Context of Seiberg-Witten Theory
• Toda hierarchies and their applications
• Casorati Determinant Form of Dark Soliton Solutions of the Discrete Nonlinear Schrödinger Equation
• Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle
• Direct and inverse spectral transform for the relativistic Toda lattice and the connection with Laurent orthogonal polynomials
• Comment on integrability in Dijkgraaf-Vafa beta-ensembles
• Coupled Nonlinear Schrödinger equation and Toda equation (the Root of Integrability)
• Inverse moment problem for elementary co-adjoint orbits
• Modified melting crystal model and Ablowitz-Ladik hierarchy
• Fermionic Quantum Gravity
• Isospectral flows related to Frobenius-Stickelberger-Thiele polynomials
• Direct Connection between the R_{II} Chain and the Nonautonomous Discrete Modified KdV Lattice
• Orbifold melting crystal models and reductions of Toda hierarchy
• Generalized Ablowitz-Ladik hierarchy in topological string theory
• Nonautonomous ultradiscrete hungry Toda lattice and a generalized box-ball system
• Modified melting crystal model and Ablowitz-Ladik hierarchy
• Tau-functions for the Ablowitz--Ladik hierarchy: the matrix-resolvent method
• Explicit solutions for a (2+1)-dimensional Toda-like chain
• Relativistic Toda chain at root of unity III. Relativistic Toda chain hierarchy
• Squared eigenfunction symmetry of the D$\Delta$mKP hierarchy and its constraint
• A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy
• Inverse moment problem for non-Abelian Coxeter double Bruhat cells
• Five-Dimensional Gauge Theories and Whitham-Toda Equation
• Lax matrices for a 1-parameter subfamily of van Diejen--Toda chains
• Poisson brackets on rational functions and multi-Hamiltonian structure for integrable lattices
• An isospectral transformation between Hessenberg matrix and Hessenberg-bidiagonal matrix pencil without using subtraction
• On an integrable system related to the relativistic Toda lattice -Bäcklund transformation and integrable discretization