M. Leo

Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system

The stability of the one-mode nonlinear solutions of the Fermi-Pasta-Ulam beta system is numerically investigated. No external perturbation is considered for the one-mode exact analytical solutions, the only perturbation being that introduced by computational errors in the numerical integration of motion equations. The threshold energy for the excitation of the other normal modes and the dynamics of this excitation are studied as a function of the parameter micro characterizing the nonlinearity, the energy density epsilon and the number N of particles of the system. The results achieved confirm in part previous ones, obtained with a linear analysis of the problem of the stability, and clarify the dynamics by which a one-mode exchanges energy with the other modes with increasing energy density. In a range of energy density near the threshold value and for various values of the number of particles N, the nonlinear one-mode exchanges energy with the other linear modes for a very short time, immediately recovering all its initial energy. This sort of recurrence is very similar to Fermi recurrences, even if in the Fermi recurrences the energy of the initially excited mode changes continuously and only periodically recovers its initial value. A tentative explanation for this intermittent behavior, in terms of Floquets theorem, is proposed. Preliminary results are also presented for the Fermi-Pasta-Ulam alpha system which show that there is a stability threshold, for large N, independent of N.

On the Relation between Lie Symmetries and Prolongation Structures of Nonlinear Field Equations : Non-Local Symmetries

An algebraic method is devised to look for non-local symmetries of the pseudopotential type of nonlinear field equations. The method is based on the use of an infinite-dimensional subalgebra of the prolongation algebra L associated with the equations under consideration. Our approach, which is applied by way of example to the Dym and the Korteweg-de Vries equation, allows us to obtain a general formula for the infinitesimal operator of non-local symmetries expressed in terms of elements of L. The method could be exploited to investigate the symmetry properties of other nonlinear field equations possessing nontrivial prolongations.

Thermostatistics in the neighbourhood of the π-mode solution for the Fermi–Pasta–Ulam β system: from weak to strong chaos

We consider a π-mode solution of the Fermi–Pasta–Ulam β system. By perturbing it, we study the system as a function of the energy density from a regime where the solution is stable to a regime where it is unstable, first weakly and then strongly chaotic. We introduce, as an indicator of stochasticity, the ratio ρ (when it is defined) between the second and the first moment of a given probability distribution. We will show numerically that the transition between weak and strong chaos can be interpreted as the symmetry breaking of a set of suitable dynamical variables. Moreover, we show that in the region of weak chaos there is numerical evidence that the thermostatistic is governed by the Tsallis distribution.

On the isospectral-eigenvalue problem and the recursion operator of the Harry-Dym equation

SummaryThe isospectral-eigenvalue problem for the Herry-Dym equation is derived using a prolongation technique. The eigenvalue equation is then exploited to find a recursion operator.

An analysis of the Transverse Optical Klystron (TOK) and the Free Electron Laser (FEL) through the exact solution of their evolution equation

Abstract We give the exact analytical solution of the evolution equation of an electron beam interacting in a wiggler with an electromagnetic wave. The initial energy distribution is assumed both monochromatic and gaussian. Furthermore, the evolution in a drift space is considered. Some basic characteristics of the FEL and TOK are derived from the mathematical properties of the solution.

Integrable nonlinear field equations and loop algebra structures

Abstract We apply the (direct and inverse) prolongation method to a couple of nonlinear Schrodinger equations. These are taken as a laboratory field model for analyzing the existence of a connection between the integrability property and loop algebras. Exploiting a realization of the Kac-Moody type of the incomplete prolongation algebra associated with the system under consideration, we develop a procedure which allows us to generate a new class of integrable nonlinear field equations containing the original ones as a special case.

Symmetry properties and exact patterns in birefringent optical fibers.

We carry out a group-theoretical study of the pair of nonlinear Schrödinger equations describing the propagation of waves in nonlinear birefringent optical fibers. We exploit the symmetry algebra associated with these equations to provide examples of specific exact solutions. Among them, we obtain the soliton profile, which is related to the coordinate translations and the constant change of phase.

Nonlinear evolution equations and nonabelian prolongations

A systematic analysis of the class of nonlinear evolution equations ut +uxxx +φ(u, ux)=0 is carried out within the Estabrook–Wahlquist prolongation scheme.

Prolongation theory of the three-wave resonant interaction

SummaryThe three-wave resonant interaction in 1+1 dimensions is studied systematically within the prolongation scheme, which turns out to be a convenient framework for handling this process. Our analysis is carried out via a method which exploits both theSL3,c algebra associated with the three-wave equations and integrability requirements. Our procedure, which may be applied to other nonlinear evolution equations, works without fixing from the start representations of the algebra, and in the case of pseudopotentials with an arbitrary number of components. Bäcklund transformations which provide known and new explicit solutions are found, and inherent symmetry properties are discovered.RiassuntoSi studia sistematicamente l’interazione di tre onde risonanti in dimensioni 1+1 nello schema di prolungamento, che si rivela molto conveniente per la trattazione di questo processo. La nostra analisi si basa su una procedura che utilizza l’algebraSL3,c associata all’equazioni delle tre onde e condizioni d’integrabilità. Il nostro metodo, che può essere applicato ad altre equazioni di evoluzione non lineari, funziona senza fissare in partenza rappresentazioni dell’algebra, e nel caso di pseudopotenziali con un numero arbitrario di componenti. Sono determinate trasformazioni di Bäcklund che forniscono sia soluzioni note che soluzioni nuove, e si scoprono alcune proprietà di simmetria interne delle equazioni in esame.

On the use of closed non-Abelian prolongation algebras to find Bäcklund transformations of nonlinear evolution equations

SummaryWe describe a simple method to find Bäcklund transformations of a given nonlinear evolution equation, exploiting the closed non-Abelian prolongation algebra related to the equation, within the Estabrook-Wahlquist prolongation scheme, without using explicit representations from the start. This procedure works generally forN-dimensional pseudopotentials.