This paper is devoted to a numerical study of the familiar α+β FPU model. Precisely, we here discuss, revisit and combine together two main ideas on the subject: (i) In the system, at small specific energy ε = E/N , two well separated time–scales are present: in the former one a kind of metastable state is produced, while in the second much larger one, such an intermediate state evolves and rea...

Three inequivalent real forms of the complex N=4 supersymmetric Toda chain hierarchy (Nucl. Phys. B558 (1999) 545, solv-int/9907004) in the real N = 2 superspace with one even and two odd real coordinates are presented. It is demonstrated that the first of them possesses a global N = 4 supersymmetry, while the other two admit a twisted N = 4 supersymmetry. A new superfield basis in which supers...

A q-discrete version of the two-dimensional Toda molecule equation is proposed through the direct method. Its solution, Bäcklund transformation and Lax pair are discussed. The reduction to the q-discrete cylindrical Toda molecule equation is also discussed. 1 On leave from Department of Applied Mathematics, Faculty of Engineering, Hiroshima University. 1

The recurrent relations between the eigenfunctions for GL(N,R) and GL(N−1,R) quantum Toda chains is derived. As a corollary, the Mellin-Barnes integral representation for the eigenfunctions of a quantum open Toda chain is constructed for the N -particle case. E-mail: [email protected] E-mail: [email protected]

We study the Toda equations in the continuous level, discrete level and ultradiscrete level in terms of elliptic and hyperelliptic σ and ψ functions of genera one and two. The ultradiscrete Toda equation appears as a discrete-valuation of recursion relations of ψ functions.

We provide a comprehensive treatment of the single and double commutation method as a tool for constructing soliton solutions of the Toda and Kac-van Moerbeke hierarchy on arbitrary background. In addition, we present a novel construction based on the single commutation method. As an illustration we compute the N -soliton solution of the Toda and Kac-van Moerbeke hierarchy.

We apply an analogue of the Zakharov-Shabat dressing method to obtain infinite matrix solutions to the Toda lattice hierarchy. Using an operator transformation we convert some of these into solutions in terms of integral operators and Fredholm determinants. Others are converted into a class of operator solutions to the l-periodic Toda hierarchy.

In this paper we present multidimensional analogues of both the continuousand discrete-time Toda lattices. The integrable systems that we consider here have two or more space coordinates. To construct the systems, we generalize the orthogonal polynomial approach for the continuous and discrete Toda lattices to the case of multiple orthogonal polynomials.

We find explicit (multisoliton) solutions for nonabelian integrable systems such as periodic Toda field equations, Langmuir equations, and Schrödinger equations for functions with values in any associative algebra. The solution for nonabelian Toda field equations for root systems of types A, B, C was expressed by the authors in [EGR] using quasideterminants introduced and studied in [GR1-GR4]. ...