The nonabelian Toda lattice: Discrete analogue of the matrix Schrödinger spectral problem (original) (raw)
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.
A modified Toda spectral problem and its hierarchy of bi-Hamiltonian lattice equations
Journal of Physics A: Mathematical and General, 2004
Starting from a modified Toda spectral problem, a hierarchy of generalized Toda lattice equations with two arbitrary constants is constructed through discrete zero curvature equations. It is shown that the hierarchy possesses a bi-Hamiltonian structure and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals. Two cases of the involved constants present two specific integrable sub-hierarchies, one of which is exactly the Toda lattice hierarchy.
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.
Generalized Casorati Determinant and Positon–Negaton-Type Solutions of the Toda Lattice Equation
Journal of the Physical Society of Japan, 2004
A set of conditions is presented for Casorati determinants to give solutions to the Toda lattice equation. It is used to establish a relation between the Casorati determinant solutions and the generalized Casorati determinant solutions. Positons, negatons and their interaction solutions of the Toda lattice equation are constructed through the generalized Casorati determinant technique. A careful analysis is also made for general positons and negatons, the resulting positons and negatons of order one being explicitly computed. The generalized Casorati determinant formulation for the two dimensional Toda lattice (2dTL) equation is presented. It is shown that positon, negaton and complexiton type solutions in the 2dTL equation exist and these solutions reduce to positon, negaton and complexiton type solutions in the Toda lattice equation by the standard reduction procedure.
Elementary introduction to discrete soliton equations
Nonlinear Systems and Their Remarkable Mathematical Structures, 2018
We will give a short introduction to discrete or lattice soliton equations, with the particular example of the Korteweg-de Vries as illustration. We will discuss briefly how Bäcklund transformations lead to equations that can be interpreted as discrete equations on a Z 2 lattice. Hierarchies of equations and commuting flows are shown to be related to multidimensionality in the lattice context, and multidimensional consistency is one of the necessary conditions for integrability. The multidimensional setting also allows one to construct a Lax pair and a Bäcklund transformation, which in turn leads to a method of constructing soliton solutions. The relationship between continuous and discrete equations is discussed from two directions: taking the continuum limit of a discrete equation and discretizing a continuous equation following the method of Hirota.
2021
This paper investigates a relativistic Toda lattice system with an arbitrary parameter that is a very remarkable generalization of the usual Toda lattice system, which may describe the motions of particles in lattices. Firstly, we study some integrable properties for this system such as Hamiltonian structures, Liouville integrability and conservation laws. Secondly, we construct a discrete generalized (m, 2Nm)-fold Darboux transformation based on its known Lax pair. Thirdly, we obtain some exact solutions including soliton, rational and semi-rational solutions with arbitrary controllable parameters and hybrid solutions by using the resulting Darboux transformation. Finally, in order to understand the properties of such solutions, we investigate the limit states of the diverse exact solutions by using graphic and asymptotic analysis. In particular, we discuss the asymptotic states of rational solutions and exponential-and-rational hybrid solutions graphically for the first time, which might be useful for understanding the motions of particles in lattices. Numerical simulations are used to discuss the dynamics of some soliton solutions. The results and properties provided in this paper may enrich the understanding of nonlinear lattice dynamics.