Behaviour of the extended Toda lattice

Abstract

We consider the first member of an extended Toda lattice hierarchy. This system of equations is differential with respect to one independent variable and differential-delay with respect to a second independent variable. We use asymptotic analysis to consider the long wavelength limits of the system. By considering various magnitudes for the parameters involved, we derive reduced equations related to the Korteweg-de Vries and potential Boussinesq equations. Highlights:• we analyse the behaviour of solutions of the extended Toda lattice • we derive PDEs which are asymptotic approximations of the lattice • we find similarity solutions of these limiting PDEs • we show that in certain cases the PDEs can be transformed to the Boussinesq and/or KdV equations

The second Painlevé equation, a related nonautonomous semidiscrete equation, and a limit to the first Painlevé equation: Scalar and matrix cases

“…We have also given in Section 4 a more general discussion of our asymptotic analysis, and in Section 5 a component-wise approach. This then extends our earlier papers, where we have explored connections between various integrable systems [20,21,22]. In addition, we have given Hamiltonian formulations of the matrix first and second Painlevé equations.…”

Behaviour of the extended modified Volterra lattice–Reductions to generalised mKdV and NLS equations

“…In this paper we consider various asymptotic reductions of this extended modified Volterra equation for small amplitude solutions, which reveal connections between this equation and generalisations of the modified Korteweg-de Vries equation (mKdV) and the nonlinear Schrödinger equations (NLS). Similar analyses of the extended Toda lattice [6] and extended Volterra system [4] in [7,8] demonstrated connections between these extended systems and the Korteweg-de Vries equation (KdV). The modified Volterra system is potentially much more interesting, since it supports breather solutions as well as travelling waves.…”

Non-Classical Symmetry Analysis of a Class of Nonlinear Lattice Equations

“…In 1993, Levi [8] proposed the classical Lie symmetry method for differential-difference equations. The Toda equation is a classical model of differential-difference equation [9][10][11][12], whose symmetries have been studied [13] and the differential-difference Lie symmetry method was applied to solve a class of Toda-like lattice equations [14]. In 2022, a survey of the connection between orthogonal polynomials, Toda lattices and related lattices, and Painlevé equations (discrete and continuous) was given [15].…”