On the periodic Toda lattice with a self-consistent source (original) (raw)

On the periodic Toda lattice hierarchy with an integral source

Communications in Nonlinear Science and Numerical Simulation, 2017

This work is devoted to the application of inverse spectral problem for integration of the periodic Toda lattice hierarchy with an integral type source. The effective method is presented of constructing the periodic Toda lattice hierarchy with an integral source.

Toda lattice with a special self-consistent source

Theoretical and Mathematical Physics, 2008

We describe a method for integrating the Toda lattice with a self-consistent source using the inverse scattering method for a discrete Sturm-Liouville operator with moving eigenvalues.

The nonabelian Toda lattice: Discrete analogue of the matrix Schrödinger spectral problem

Journal of Mathematical Physics, 1980

We investigate the discrete analog of the matrix Schrodinger spectral problem and derive the simplest nonlinear differential-difference equation associated to such problem solvable by the inverse spectral transform. We also display the one and two soliton solution for this equation and tersely discuss their main features.

Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

Annales de l’institut Fourier, 2013

M. Toda in 1967 (J. Phys. Soc. Japan, 22 and 23) considered a lattice model with exponential interaction and proved, as suggested by the Fermi-Pasta-Ulam experiments in the 1950s, that it has exact periodic and soliton solutions. The Toda lattice, as it came to be known, was then extensively studied as one of the completely integrable (differential-difference) non-linear equations which admit exact solutions in terms of theta functions of hyperelliptic curves. In this paper, we extend Toda's original approach to give hyperelliptic solutions of the Toda lattice in terms of hyperelliptic Kleinian (sigma) functions for arbitrary genus. The key identities are given by generalized addition formulae for the hyperelliptic sigma functions (J.C. Eilbeck et al., J. reine angew. Math. 619, 2008). We then show that periodic (in the discrete variable, a standard term in the Toda lattice theory) solutions of the Toda lattice correspond to the zeros of Kiepert-Brioschi's division polynomials, and note these are related to solutions of Poncelet's closure problem. One feature of our solution is that the hyperelliptic curve is related in a non-trivial way to the one previously used. Contents 1. Introduction 2. Genus-one case 3. Hyperelliptic curve X g and sigma functions 3.1. Geometrical setting for hyperelliptic curves 3.2. Sigma function and its derivatives 4. The addition formulae 4.1. Generalized Frobenius-Stickelberger formula 4.2. The algebraic addition formula 4.3. The analytic addition formula 5. The σ-function solution of the Toda lattice 6. Periodicity of the Toda-lattice solution 6.1. Division polynomials ψ 2N 6.2. A periodic Toda lattice 6.3. Hyperelliptic curveX g,N −1 for the periodic Toda lattice

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.

Integrable boundary problems for 2D Toda lattice

Fuel and Energy Abstracts, 2010

Spatially two-dimensional Toda lattice is examined in the aspect of correct formulation of boundary problems that can be solved within the scheme of the Inverse Scattering Method. It is shown that there exists a large set of integrable boundary problems and various curves can be chosen as boundaries for those problems. Explicit solutions are presented for problems on closed and unclosed curves taken as boundary contours.