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Annals of Physics 02/2013; 333:12-18. · 2.86 Impact Factor
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ABSTRACT: In a recent paper [M. Leo, R. A. Leo, and P. Tempesta, J. Stat. Mech. (2011) P03003], it has been shown that the π/2-mode exact nonlinear solution of the Fermi-Pasta-Ulam β system, with periodic boundary conditions, admits two energy density thresholds. For values of the energy density ε below or above these thresholds, the solution is stable. Between them, the behavior of the solution is unstable, first recurrent and then chaotic. In this paper, we study the chaotic behavior between the two thresholds from a statistical point of view, by analyzing the distribution function of a dynamical variable that is zero when the solution is stable and fluctuates around zero when it is unstable. For mesoscopic systems clear numerical evidence emerges that near the second threshold, in a large range of the energy density, the numerical distribution is fitted accurately with a q-Gaussian distribution for very large integration times, suggesting the existence of a quasistationary state possessing a weakly chaotic behavior. A normal distribution is recovered in the thermodynamic limit.
Physical Review E 03/2012; 85(3 Pt 1):031149. · 2.26 Impact Factor
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ABSTRACT: We consider a $\pi$-mode solution of the Fermi-Pasta-Ulam $\beta$ 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 is unstable, first weakly and then strongly chaotic. We introduce, as indicator of stochasticity, the ratio $\rho$ (when 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. Comment: 15 pages, 5 figures
Journal of Statistical Mechanics Theory and Experiment 03/2010; 2010:P04021. · 1.73 Impact Factor
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ABSTRACT: We present a detailed numerical and analytical study of the stability properties of the N/4 (pi/2-mode) one-mode nonlinear solution of the Fermi-Pasta-Ulam-beta system. The numerical analysis is made as a function of the number N of the particles of the system and of the product lambda=epsilonbeta , where epsilon is the energy density and beta is the parameter characterizing the nonlinearity. It is shown that, both for beta>0 and beta<0 , the instability threshold value |lambda(t)(N)| converges, with increasing N , to the same value 2pi(2)(3N(2)) , that for beta>0 |lambda(t)N(2)| is a decreasing function of N as in the pi-mode, whereas, for beta<0 , it is an increasing one. The asymptotic behavior of |lambda(t)| for large values of N is analytically obtained in both cases with a Floquet analysis of the stability.
Physical Review E 07/2007; 76(1 Pt 2):016216. · 2.26 Impact Factor
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ABSTRACT: We apply the Bogoliubov-Krilov method of averaging to the study of the stability of the pi -mode solution (N/2 one-mode nonlinear solution) of the Fermi-Pasta-Ulam- beta system, with negative values of the nonlinearity parameter beta in the Hamiltonian of the system. The analysis is made as a function of the number N of the particles and of the product lambda = epsilon|beta|, where epsilon is the energy density. The results of this application are in excellent agreement with those obtained by the direct integration of motion equation.
Physical Review E 11/2006; 74(4 Pt 2):047201. · 2.26 Impact Factor
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ABSTRACT: 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 Floquet's 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.
Physical Review E 05/2004; 69(4 Pt 2):046604. · 2.26 Impact Factor
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ABSTRACT: We provide an exact regular solution of an operator system arising as the prolongation structure associated with the heavenly equation. This solution is expressed in terms of operator Bessel coefficients.
12/2003;
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ABSTRACT: We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator which provide, at the classical level, the same equation of motion as the conventional Hamiltonian. These Hamiltonians, say $K_{1}$ and $K_{2}$, result to be explicitly time-dependent and can be expressed as a formal rotation of two cubic polynomial functions, $H_{1}$ and $H_{2}$, of the canonical variables (q,p). We investigate the role of these fouled Hamiltonians at the quantum level. Adopting a canonical quantization procedure, we construct some quantum models and analyze the related eigenvalue equations. One of these models is described by a Hamiltonian admitting infinite self-adjoint extensions, each of them has a discrete spectrum on the real line. A self-adjoint extension is fixed by choosing the spectral parameter $\epsilon$ of the associated eigenvalue equation equal to zero. The spectral problem is discussed in the context of three different representations. For $\epsilon =0$, the eigenvalue equation is exactly solved in all these representations, in which square-integrable solutions are explicity found. A set of constants of motion corresponding to these quantum models is also obtained. Furthermore, the algebraic structure underlying the quantum models is explored. This turns out to be a nonlinear (quadratic) algebra, which could be applied for the determination of approximate solutions to the eigenvalue equations.
Journal of Mathematical Physics 04/2002; · 1.29 Impact Factor
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Physica A Statistical and Theoretical Physics 01/2002; 305:371-380. · 1.37 Impact Factor
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ABSTRACT: The Navier-Stokes-Fourier model for a 3D thermoconducting viscous fluid, where the evolution equation for the temperature T contains a term proportional to the rate of energy dissipation, is investigated analitically at the light of the rotational invariance property. Two cases are considered: the Couette flow and a flow with a radial velocity between two rotating impermeable and porous coaxial cylinders, respectively. In both cases, we show the existence of a maximum value of T, T_max, when the difference of temperature Delta T=T_2-T_1 on the surfaces of the cylinders is assigned. The role of T_max is discussed in the context of different physical situations. Comment: 11 pages, 7 figures
07/2001;
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ABSTRACT: We study a (2+1)-dimensional model of an incompressible thermoconducting fuid named Navier–Stokes–Fourier system. We apply a group-theoretical analysis. In correspondence of the generators ofthe symmetry group allowed by this model, exact solutions are found. Some of them show possible interesting physical interpretations. In our first exploration, this feature is illustrated by dealing with special cases.
Physica A Statistical and Theoretical Physics 01/2001; 293:421-434. · 1.37 Impact Factor
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ABSTRACT: 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.
Progress of Theoretical Physics 01/2001; 105:77-97. · 2.29 Impact Factor
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ABSTRACT: We study a class of one-dimensional nonlinear lattices with nearest-neighbour interactions described by a potential of the
binomial type. This potential contains a free parameter which can be chosen to reproduce a variety of models, such as the
Toda, the Fermi-Pasta-Ulam and the Coulomb-like lattices. Carrying out essentially numerical experiments, the effects of soliton
propagation on a lattice with defects are investigated. In particular, the properties of the localized mode, generated by
the propagation of the soliton through the defect, are discussed with respect to the defect mass and the potential parameter,
in the light of a simple theoretical model. Furthermore, an interesting phenomenon is observed: the amplitude of the speed
of the mass defect shows a sequel of resonance peaks in terms of the mass defect. The positions of these peaks appear to be
independent of the potential parameter.
The European Physical Journal D 07/2000; 11(3):327-334. · 1.48 Impact Factor
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ABSTRACT: We show that certain infinitesimal operators of the Lie-point symmetries of the incompressible 3D Navier-Stokes equations give rise to vortex solutions with different characteristics. This approach allows an algebraic classification of vortices and throws light on the alignment mechanism between the vorticity and the vortex stretching vector. The symmetry algebra associated with the Navier-Stokes equations turns out to be infinite- dimensional. New vortical structures, generalizing in some cases well-known configurations such as, for example, the Burgers and Lundgren solutions, are obtained and discussed in relation to the value of the dynamic angle. A systematic treatment of the boundary conditions invariant under the symmetry group of the equations under study is also performed, and the corresponding invariant surfaces are recognized. Comment: 40 pages, no figures
Physica A Statistical and Theoretical Physics 07/2000; 286:79-108. · 1.37 Impact Factor
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ABSTRACT: We show that the use of the main characteristics of the circle map leads naturally to establish a few statements on primes and pseudoprimes. In this way a Fermat's theorem on primes and some interesting properties of pseudoprimes are obtained.
01/2000;
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ABSTRACT: 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 equations, allows us to obtain a general formula for the infinitesimal operator of the 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.
12/1999;
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ABSTRACT: A systematic analysis of a continuous version of a binomial lattice, containing a real parameter $\gamma$ and covering the Toda field equation as $\gamma\to\infty$, is carried out in the framework of group theory. The symmetry algebra of the equation is derived. Reductions by one-dimensional and two-dimensional subalgebras of the symmetry algebra and their corresponding subgroups, yield notable field equations in lower dimensions whose solutions allow to find exact solutions to the original equation. Some reduced equations turn out to be related to potentials of physical interest, such as the Fermi-Pasta-Ulam and the Killingbeck potentials, and others. An instanton-like approximate solution is also obtained which reproduces the Eguchi-Hanson instanton configuration for $\gamma\to\infty$. Furthermore, the equation under consideration is extended to $(n+1)$--dimensions. A spherically symmetric form of this equation, studied by means of the symmetry approach, provides conformally invariant classes of field equations comprising remarkable special cases. One of these $(n=4)$ enables us to establish a connection with the Euclidean Yang-Mills equations, another appears in the context of Differential Geometry in relation to the socalled Yamabe problem. All the properties of the reduced equations are shared by the spherically symmetric generalized field equation. Comment: 30 pages, LaTeX, no figures. Submitted to Annals of Physics
02/1998;
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ABSTRACT: A system of equations of the reaction-diffusion type is studied in the framework of both the direct and the inverse prolongation structure. We find that this system allows an incomplete prolongation Lie algebra, which is used to find the spectral problem and a whole class of nonlinear field equations containing the original ones as a special case. Comment: 16 pages, LaTex. submitted to Inverse Problems
10/1997;
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ABSTRACT: The Estabrook-Wahlquist prolongation method is applied to the (compact and noncompact) continuous isotropic Heisenberg model in 1 + 1 dimensions. Using a special realization (an algebra of the Kac-Moody type) of the arising incomplete prolongation Lie algebra, a whole family of nonlinear field equations containing the original Heisenberg system is generated. Comment: Tex file, 10 pages
04/1996;
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ABSTRACT: We apply the (direct and inverse) prolongation method to a couple of nonlinear Schr{\"o}dinger 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 with allows us to generate a new class of integrable nonlinear field equations containing the original ones as a special case. Comment: 13 pages, latex, no figures,
01/1995;
