Generalized Casorati Determinant and Positon–Negaton-Type Solutions of the Toda Lattice Equation (original) (raw)

Complexiton solutions of the Toda lattice equation

Physica A: Statistical Mechanics and its Applications, 2004

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 forming complexitons. Moreover, a feasible way is presented to compute the required eigenfunctions, along with examples of real complexitons of lower order.

Rational solutions of the Toda lattice equation in Casoratian form

Chaos, Solitons & Fractals, 2004

A recursive procedure is presented for constructing rational solutions to the Toda lattice equation through the Casoratian formulation. It allows us to compute a broad class of rational solutions directly, without computing long wave limits in soliton solutions. All rational solutions arising from the Taylor expansions of the generating functions of soliton solutions are special ones of the general class, but only a Taylor expansion containing even or odd powers leads to non-constant rational solutions. A few rational solutions of lower order are worked out.

The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras

Ergodic Theory and Dynamical Systems, 1981

We associate to each complex simple Lie algebra g a hierarchy of evolution equations; in the simplest case g = sl(2) they are the modified KdV equations. These new equations are related to the two-dimensional Toda lattice equations associated with g in the same way that the modified KdV equations are related to the sinh-Gordon equation.

On the Integrable Generalization of the 1D Toda Lattice

2010

A generalized Toda Lattice equation is considered. The associated linear problem (Lax representation) is found. For simple case N = 3 the τ -function Hirota form is presented that allows to construct an exast solutions of the equations of the 1DGTL. The corresponding hierarchy and its relations with the nonlinear Schrodinger equation and Hersenberg ferromagnetic equation are discussed.

Spectrum and Generation of Solutions of the Toda Lattice

Discrete Dynamics in Nature and Society, 2009

Sufficient conditions for constructing a set of solutions of the Toda lattice are analyzed. First, under certain conditions the invariance of the spectrum ofJ(t)is established in the complex case. Second, given the tri-diagonal matrixJ(t)defining a Toda lattice solution, the dynamic behavior of zeros of polynomials associated toJ(t)is analyzed. Finally, it is shown by means of an example how to apply our results to generate complex solutions of the Toda lattice starting with a given solution.