Volterra Lattice Equation Research Articles

Infinitely many conservation laws and integrable discretizations for some lattice soliton equations

In this paper, by means of the Lax representations, we demonstrate the existence of infinitely many conservation laws for the general Toda-type lattice equation, the relativistic Volterra lattice equation, the Suris lattice equation and some other lattice equations. The conserved density and the associated flux are given formulaically. We also give an integrable discretization for a lattice equation with n dependent coefficients.

A Nonlocal Kac-van Moerbeke Equation Admitting N-Soliton Solutions

Using our previous work on reflectionless analytic difference operators and a nonlocal Toda equation, we introduce analytic versions of the Volterra and Kac-van Moerbeke lattice equations. The real-valued N-soliton solutions to our nonlocal equations correspond to self-adjoint reflectionless analytic difference operators with N bound states. A suitable scaling limit gives rise to the N-soliton solutions of the Korteweg-de Vries equation.

Integrable hierarchies of nonlinear difference-difference equations and symmetries

We construct the hierarchy of nonlinear difference-difference equations associated with the discrete Schrodinger spectral problem. As examples of equations contained in this hierarchy we obtain the discrete-time Toda and Volterra lattice equations. In the case of the time-discrete Toda lattice, we construct its Lie point and generalized symmetries. Finally, we present its Backlund transformations and relate it to the already constructed symmetries.

Bäcklund Transformation for Solutions of the Modified Volterra Lattice Equation

We present new expressions to a pair of Backlund transformations for the modified Volterra lattice equation. We derive a superposition formula to generate multi-soliton solutions. In an appropriate continuum limit, the Backlund transformations coincide with those for the modified Korteweg-de Vries equation.

Toda and Volterra lattice equations from discrete symmetries of KP hierarchies

The discrete models of the Toda and Volterra chains are being constructed out of the continuum two-boson KP hierarchies. The main tool is the discrete symmetry preserving the Hamiltonian structure of the continuum models. The two-boson currents of KP hierarchy are being associated with sites of the corresponding chain by successive actions of discrete symmetry.