Hamiltonian and gradient structures in the Toda flows

In this paper we consider gradient structures in the dynamics and geometry of the asymmetri nonperiodic tridiagonal and full Toda flow equations. We compare and contrast a number of formulations of the nonperiodic Toda equations. In the case of the full Kostant (asymmetric) Toda flow we explain the role of noncommutative integrability in its qualitative behavior. We describe the relationship between the asymmetric Toda flows and the symmetric and indefinite Toda flows, and prove in particular that one may conjugate from the full Kostant Toda flows to the full symmetric Toda flows via a Poisson map.     

From the Toda lattice to the Volterra lattice and back

We discuss the relationship between the multiple Hamiltonian structures of the generalized Toda lattices and that of the generalized Volterra lattices.     

Lax pairs, symmetries and recursion operators for Toda lattices

With the exception of some minor results and some conjectures, this paper is a survey of the finite nonperiodic Toda lattices and some of their generalizations. The areas investigated include Lax pairs, master symmetries, recursion operators, higher Poisson brackets, invariants, and group symmetries for such systems.     

Iso-spectral deformations of general matrix and their reductions on Lie algebras

We study an iso-spectral deformation of the general matrix which is a natural generalization of the nonperiodic Toda lattice equation. This deformation is equivalent to the Cholesky flow, a continuous version of the Cholesky algorithm, introduced by Watkins. We prove the integrability of the deformation and give an explicit formula for the solution to the initial value problem. The formula is obtained by generalizing the orthogonalization procedure of Szegö. Using the formula, the solution to the LU matrix factorization can the constructed explicitly. Based on the root spaces for simple Lie algebras, we consider several reductions of the equation. This leads to generalized Toda equations related to other classical semi-simple Lie algebras which include the integrable systems studied by Bogoyavlensky and Kostant. We show these systems can be solved explicitly in a unified way. The behaviors of the solutions are also studied. Generically, there are two types of solutions, having either sorting property or blowing up to infinity in finite time.     

The finite Toda lattices

Connection is established between one-dimensional Toda lattices, constructed on the basis of the systems of simple roots of classical and affine Lie algebras, and other integrable systems of interacting particles. That connection allows us to find new lattices differing from the known ones by the interaction of particles near the ends. Some of the new lattices admit non-Abelian generalizations.     

Relativistic Toda systems

We present and study Poincaré-invariant generalizations of the Galilei-invariant Toda systems. The classical nonperiodic systems are solved by means of an explicit action-angle transformation.Work supported by the Netherlands Organisation for the Advancement of Research (NWO)     

Two-dimensional generalized Toda lattice

The zero curvature representation is obtained for the two-dimensional generalized Toda lattices connected with semisimple Lie algebras. The reduction group and conservation laws are found and the mass spectrum is calculated.     

Thermodynamics of one-dimensional nonlinear lattices

Analytic equations were obtained for the thermodynamic parameters of one-dimensional lattices of particles with the Toda and Morse interaction potentials in a canonical Gibbs ensemble. For the same systems, equations were derived for molecular dynamics simulations of thermodynamic processes. Stochastic differential equations were solved with simulating the thermostat by Langevin sources with random forced. Analytic equations for thermodynamic parameters (energy, temperature, and pressure) excellently coincided with molecular dynamics simulation results. The kinetics of system relaxation to the thermodynamic equilibrium state was analyzed. The advantages of simulating the physical properties of systems in a canonical compared with microcanonical ensemble were demonstrated.     

Mass transport theory for the Toda lattices, dispersive and dissipative

To establish mass transport theory on nonlinear lattices, we formulate the Korteweg–deVries (KdV) equation and the Burgers equation using the flow variable representation so as to facilitate comparison with the Boltzmann equation and with the Cahn–Hilliard equation in classical statistical mechanics. We also study Toda lattice microdynamics using the Flaschka representation, and compare with the Liouville equation. Like the linear diffusion equation, the Boltzmann equation and the Liouville equation are to be solved for a distribution function, which is intrinsically probabilistic. Transport theory in linear systems is governed by the isotropic motions of the kinetic equations. In contrast, the KdV perturbation equation derived from the Toda lattice microdynamics expresses hydrodynamic mass transport. The KdV equation in hydrodynamics and the Burgers equation in thermodynamics do not involve a probability distribution function. The nonlinear lattices do not retain isotropy of the mass transport equations. In consequence, it is proposed that in the presence of hydrodynamic flows to the left, KdV wave propagation proceeds to the right. This basic property of the KdV system is extended to thermodynamics in the Burgers system. These features arise because linear systems are driven towards an equilibrium by molecular collisions, whereas the inhomogeneities of the nonlinear lattices are generated by the potential energy of interaction. Diffusion as expressed by the Burgers equation is governed not only by a chemical potential, but also by the Toda lattice potential energy.     

Integrability,multi-soliton and rational solutions,and dynamical analysis for a relativistic Toda lattice system with one perturbation parameter

Under investigation in this paper is a relativistic Toda lattice system with one perturbation parameter α abbreviated as RTL_(α) system by Suris, which may describe the motions of particles in lattices interacting through an exponential interaction force. First of all, an integrable lattice hierarchy associated with an RTL_(α) system is constructed, from which some relevant integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws are investigated. Secondly, the discrete generalized(m, 2 N-m)-fold Darboux transformation is constructed to derive multi-soliton solutions, higher-order rational and semirational solutions, and their mixed solutions of an RTL_(α) system. The soliton elastic interactions and details of rational solutions are analyzed via the graphics and asymptotic analysis. Finally, soliton dynamical evolutions are investigated via numerical simulations,showing that a small noise has very little effect on the soliton propagation. These results may provide new insight into nonlinear lattice dynamics described by RTL_(α) system.     

Asymptotic Stability of Lattice Solitons in the Energy Space

Orbital and asymptotic stability for 1-soliton solutions of the Toda lattice equations as well as for small solitary waves of the FPU lattice equations are established in the energy space. Unlike analogous Hamiltonian PDEs, the lattice equations do not conserve the adjoint momentum. In fact, the Toda lattice equation is a bidirectional model that does not fit in with the existing theory for the Hamiltonian systems by Grillakis, Shatah and Strauss. To prove stability of 1-soliton solutions, we split a solution around a 1-soliton into a small solution that moves more slowly than the main solitary wave and an exponentially localized part. We apply a decay estimate for solutions to a linearized Toda equation which has been recently proved by Mizumachi and Pego to estimate the localized part. We improve the asymptotic stability results for FPU lattices in a weighted space obtained by Friesecke and Pego.     

Noether-type theory for discrete mechanico-electrical dynamical systems with nonregular lattices

We investigate Noether symmetries and conservation laws of the discrete mechanico-electrical systems with nonregular lattices.The operators of discrete transformation and discrete differentiation to the right and left are introduced for the systems.Based on the invariance of discrete Hamilton action on nonregular lattices of the systems with the dissipation forces under the infinitesimal transformations with respect to the time,generalized coordinates and generalized charge quantities,we work out the discrete analog of the generalized variational formula.From this formula we derive the discrete analog of generalized Noether-type identity,and then we present the generalized quasi-extremal equations and properties of these equations for the systems.We also obtain the discrete analog of Noether-type conserved laws and the discrete analog of generalized Noether theorems for the systems.Finally we use an example to illustrate these results.     

Hamiltonian Structure of Non-Abelian Toda Lattice

We prove that finite nonperiodic non-Abelian Toda lattice is Liouville completely integrable.     

Hierarchy of Combined TL-RTL Equations and an Associated (2+1)-Dimensional Lattice Equation

A new (2+1)-dimensional lattice equation is presented based upon the first two members in the hierarchy of the combined Toda lattice and relativistic Toda lattice (TL-RTL) equations in (1+1) dimensions. A Darboux transformation for the hierarchy of the combined TL-RTL equations is constructed. Solutions of the first two members in the hierarchy of the combined TL-RTL equations, as well as the new (2+1)-dimensional lattice equation are explicitly obtained by the Darboux transformation.     

Hierarchy of Combined TL-RTL Equations and an Associated (2+1)-Dimensional Lattice Equation

A new (2 1)-dimensional lattice equation is presented based upon the first two members in the hierarchy of the combined Toda lattice and relativistic Toda lattice (TL-RTL) equations in (1 1) dimensions. A Darboux transformation for the hierarchy of the combined TL-RTL equations is constructed. Solutions of the first two members in the hierarchy of the combined TL-RTL equations, as well as the new (2 1)-dimensional lattice equation are explicitly obtained by the Darboux transformation.     

Darboux–Bäcklund transformation and explicit solutions to a hybrid lattice of the relativistic Toda lattice and the modified Toda lattice

The relativistic Toda lattice and the modified Toda lattice are two important discrete integrable equations. In this paper, we investigate a hybrid lattice equation of the two lattice equations. Darboux–Bäcklund transformation and explicit solutions to the hybrid lattice equation are constructed.     

On finite-zone solutions of relativistic Toda lattices

The finite-zone solutions of relativistic Toda lattices are investigated using the recurrence relations method. As a result, a nonlinear bundle of relativistic Toda lattices is with corresponding stationary and dynamical systems. New Poisson and Hamiltonian structures are introduced. Then the problem of integrating the obtained canonical systems are reduced to the Jacobi problem of inversion.     

Combined Wronskian solutions to the 2D Toda molecule equation

By combining two pieces of bi-directional Wronskian solutions, molecule solutions in Wronskian form are presented for the finite, semi-infinite and infinite bilinear 2D Toda molecule equations. In the cases of finite and semi-infinite lattices, separated-variable boundary conditions are imposed. The Jacobi identities for determinants are the key tool employed in the solution formulations.     

Poisson geometry of the analog of the Miura maps and Bäcklund-Darboux transformations for equations of Toda type and periodic Toda flows

We show that the analog of the Miura maps and Bäcklund-Darboux transformations for a general class of equations of Toda type and for a generalized class of periodic Toda flows are isomorphisms of Poisson Lie groups.     

Classical and quantum-mechanical systems of Toda lattice type. I

The structure of the commutant of Laplace operators in the enveloping and Poisson algebra of certain generalized ax +b groups leads (in this article) to a determination of classical and quantum mechanical first integrals to generalized periodic and non-periodic Toda lattices. Certain new Hamiltonian systems of Toda lattice type are also shown to fit in this framework. Finite dimensional Lax forms for the (periodic) Toda lattices are given generalizing results of Flaschke.Research partially supported by NSF grant MCS 79-03223Research partially supported by NSF grant MCS 79-03153