Mario Bruschi

Recursion operator and bäcklund transformations for the Ruijsenaars-Toda lattice

Abstract We exhibit the recursion operator and the whole class of Backlund transformations for a relativistic version of the Toda lattice recently introduced by Ruijsenaars. These results allow us to prove the complete integrability of the system.

LAX REPRESENTATION AND COMPLETE-INTEGRABILITY FOR THE PERIODIC RELATIVISTIC TODA LATTICE

Abstract Three different Lax representations for the periodic relativistic Toda lattice are exhibited. The complete integrability of the system is also proven.

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

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.

Integrable cellular automata

Abstract We present a general procedure to associate hierarchies of cellular automata with given spectral problems choosing, as illustrative example, the discrete Schrodinger problem. For these cellular automata we construct a countable number of constants of motion using standard spectral techniques and we perform numerical experiments showing interesting dynamical features and particle content. In the second part of the paper we introduce other cellular automata characterized by a very rich particle content and by the existence of constants of motion.

The periodic relativistic Toda lattice: direct and inverse problem

The direct and inverse spectral problem for the periodic relativistic Toda lattice is solved. The time evolution of the spectral variables can be calculated explicitly. A set of new canonically conjugated variables is constructed, in terms of which the motion is completely separated.

On a new integrable Hamiltonian system with nearest-neighbour interaction

A new integrable system with nearest-neighbour interaction is derived. It is shown that this system is both completely integrable and explicitly solvable by a suitable change of variables. A spectral transform approach is also presented, and the infinite limit is briefly investigated.

Existence of a Lax pair for any member of the class of nonlinear evolution equations associated to the matrix Schrödinger spectral problem

In recent years, th rough a genera l ized Wronskian technique, a large class of nonl inear evo lu t ion equat ions (hereafter referred to as CD class) associated to the m a t r i x SchrSdinger spect ra l p rob lem has been discovered, which can be solved by the inverse spectral t r ans fo rm (IST) (1). I n th is l e t te r we show tha t a L a x pa i r can be cons t ruc ted for any equa t ion of this class. This allows us to find out, for a g iven t ime evolu t ion of the (~ potent ia l ~), no t only the t ime evo lu t ion of the spectral data , b u t also the t i m e evolut ion of the e igenfunet ions of the associated l inear operator . We recal l t ha t s imilar findings, bu t for a subclass of CD, were prev ious ly achieved by iV~IODEK (2), while CASE and CHIC (3) ob ta ined the t ime evolu t ion of the e igenfunct ions in the scalar case. L e t us consider the ma t r ix SchrSdinger equa t ion

Nonlinear evolution equations associated with the chiral-field spectral problem

SummaryIn this paper we derive and investigate the class of non-linear evolution equations (NEEs) associated with the linear problemϕx=λAψ. It turns out that many physically interesting NEEs pertain to this class: for instance, the chiral-field equation, the nonlinear Klein-Gordon equations, the Heisenberg and Papanicolau spin chain models, the modified Boussinesq equation, the Wadati-Konno-Ichikawa equations, etc. We display also the Bäcklund transformations for such a class and exploit them to derive in a special case the one-soliton solution.RiassuntoIn questo lavoro si deriva e studia la classe di equazioni nonlineari di evoluzione associate con il problema lineareϕx=λAψ. A questa classe appartengono molte equazione interessanti: per esempio l’equazione del campo chirale, le equazioni di Klein-Gordon non lineari, i modelli di spin di Heisenberg e Papanicolau, l’equazione di Boussinesq modificata, le equazioni di Wadati-Konno-Ichikawa, ecc. Per questa classe di equazioni sono anche mostrate le trasformate di Bäcklund, utilizzate per derivare, in un caso particolare, la soluzione ad un solitone.

AN INVERTIBLE TRANSFORMATION AND SOME OF ITS APPLICATIONS

Several applications of an explicitly invertible transformation are reported. This transformation is elementary and therefore all the results obtained via it might be considered trivial; yet the findings highlighted in this paper are generally far from appearing trivial until the way they are obtained is revealed. Various contexts are considered: algebraic and Diophantine equations, nonlinear Sturm–Liouville problems, dynamical systems (with continuous and with discrete time), nonlinear partial differential equations, analytical geometry, functional equations. While this transformation, in one or another context, is certainly known to many, it does not seem to be as universally known as it deserves to be, for instance it is not routinely taught in basic University courses (to the best of our knowledge). The main purpose of this paper is to bring about a change in this respect; but we also hope that some of the findings reported herein — and the multitude of analogous findings easily obtainable via this technique — will be considered remarkable by the relevant experts, in spite of their elementary origin.

On the integrability of certain matrix evolution equations

Abstract The matrix equation U =2 U 3 +AU+UA is integrable (here U=U(t) is a n×n-matrix, with n an arbitrary positive integer, and A is an arbitrary constant n×n-matrix). The matrix evolution equation U =U 2 +a is also integrable (a arbitrary scalar constant). The matrix evolution equation U =f(U) , where f U is an arbitrary function of U (and of no other matrix, so that the commutator U ,f U vanishes) possesses at least n2−n (scalar) constants of motion. Lax pairs are exhibited for all these second-order n×n-matrix evolution ODEs.