Casorati Determinant Solution for the Relativistic Toda Lattice Equation

We study an integrable system related to the relativistic Toda lattice. The bilinear representation of this lattice is given and the B\"ackulund transformation obtained. A fully discrete version is also introduced with its bilinear B\"acklund transformation and Lax pair. One-soliton solution of the discrete system is presented by use of B\"acklund transformation.

A general Casoratian formulation is proposed for the 2D Toda lattice equation, which involves coupled eigenfunction systems. Various Casoratian type solutions are generated, through solving the resulting linear conditions and using a Baecklund transformation.

It is shown that the $N$-dark soliton solutions of the integrable discrete nonlinear Schr\"odinger (IDNLS) equation are given in terms of the Casorati determinant. The conditions for reduction, complex conjugacy and regularity for the Casorati determinant solution are also given explicitly. The relationship between the IDNLS and the relativistic Toda lattice is discussed.

New integrable lattice systems are introduced, their different integrable discretization are obtained. B\"acklund transformations between these new systems and the relativistic Toda lattice (in the both continuous and discrete time formulations) are established.

A set of coupled conditions consisting of differential-difference equations is presented for Casorati determinants to solve the Toda lattice equation. One class of the resulting conditions leads to an approach for constructing complexiton solutions to the Toda lattice equation through the Casoratian formulation. An analysis is made for solving the resulting system of differential-difference equations, thereby providing the general solution yielding eigenfunctions required for...

In this paper, we discuss several concepts of the modern theory of discrete integrable systems, including: - Time discretization based on the notion of B\"acklund transformation; - Symplectic realizations of multi-Hamiltonian structures; - Interrelations between discrete 1D systems and lattice 2D systems; - Multi-dimensional consistency as integrability of discrete systems; - Interrelations between integrable systems of quad-equations and integrable systems of Lapla...

We establish the pluri-Lagrangian structure for families of B\"acklund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional) systems into certain two-dimensional pluri-Lagrangian lattice systems. This embedding allows us to identify the corner equations (which are the main building blocks of the multi-time Euler-Lagrange equations) with local superposition formulae for B\"acklund transformations. These su...

In [1], a generalized type of Darboux transformations defined in terms of a twisted derivation was constructed in a unified form. Such twisted derivations include regular derivations, difference operators, superderivatives and $q$-difference operators as special cases. The formulae for the iteration of Darboux transformations are expressed in terms of quasideterminants. This approach not only enables one to recover the known Darboux transformations and quasideterminant soluti...

A new integrable lattice system is introduced, and its integrable discretizations are obtained. A B\"acklund transformation between this new system and the Toda lattice, as well as between their discretizations, is established.

The B\"acklund 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 B\"acklund transformations can be described in the terms of nonrelativistic lattices of the Toda type. Several multifield generalizations are presented.