# Short-time evolution of nonlinear Klein-Gordon systems

01 Apr 1987-International Journal of Theoretical Physics (Kluwer Academic Publishers-Plenum Publishers)-Vol. 26, Iss: 4, pp 395-399

TL;DR: In this paper, the short-time evolution of a class of nonlinear Klein-Gordon systems is studied and the behavior of the field variable has an inverse-sine spectrum rather than an exponential one.

Abstract: The short-time evolution of a class of nonlinear Klein-Gordon systems is studied. For nonzero mass, the short-time behavior of the field variable has an inverse-sine spectrum rather than an exponential one.

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TL;DR: In this paper, the authors studied the thermalization of the classical Klein-Gordon equation under a u(4) interaction and showed that the local thermodynamic equilibrium state exhibits a weakly nonlinear behavior in a renormalized wave basis.

Abstract: We study the thermalization of the classical Klein-Gordon equation under a u(4) interaction. We numerically show that even in the presence of strong nonlinearities, the local thermodynamic equilibrium state exhibits a weakly nonlinear behavior in a renormalized wave basis. The renormalized basis is defined locally in time by a linear transformation and the requirement of vanishing wave-wave correlations. We show that the renormalized waves oscillate around one frequency, and that the frequency dispersion relation undergoes a nonlinear shift proportional to the mean square field. In addition, the renormalized waves exhibit a Planck-like spectrum. Namely, there is equipartition of energy in the low-frequency modes described by a Boltzmann distribution, followed by a linear exponential decay in the high-frequency modes.

5 citations

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TL;DR: In this article, a numerical method for investigating the possibility of blow-up after a finite time is introduced for a large class of nonlinear evolution problems, including inviscid and viscous Burgers equations.

Abstract: A numerical method for investigating the possibility of blow-up after a finite time is introduced for a large class of nonlinear evolution problems. With initial data analytic in the space variable(s), the solutions have for any t > 0 complex-space singularities at the edge of an analyticity strip of width δ ( t ) Loss of regularity corresponds to the vanishing of δ ( t ). Numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the Fourier transform of the solutions, from which δ ( t ) is readily obtained. Its time evolution can be traced down to about one mesh length. By extrapolation of δ ( t ), such numerical experiments provide evidence suggesting finite-time blow-up or all-time regularity. The method is tested on the inviscid and viscous Burgers equations and is applied to the one-dimensional nonlinear Schrodinger equation with quartic potential and to the two-dimensional incompressible Euler equation, all with periodic boundary conditions. In the latter case evidence is found suggesting that existing all-time regularity results can be substantially sharpened.

207 citations

### "Short-time evolution of nonlinear K..." refers background or methods in this paper

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TL;DR: Etude du modele β de Fermi-Pasta-Ulam, par integration numerique des equations de mouvement pour un systeme de N oscillateurs couples non lineairement, N allant de 64 a 512.

Abstract: Etude du modele β de Fermi-Pasta-Ulam, par integration numerique des equations de mouvement pour un systeme de N oscillateurs couples non lineairement, N allant de 64 a 512. Le nombre Δn de modes initialement excites est tel que le rapport Δn/N est maintenu constant. On considere le systeme comme un gaz de phonons faiblement couples (modes normaux). Avec Δn/N constant, on trouve une analogie avec la limite thermodynamique de la mecanique statistique, ou le rapport M/V est constant quand le volume V et le nombre de particules M deviennent infinis

159 citations

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105 citations

### "Short-time evolution of nonlinear K..." refers background in this paper

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TL;DR: In this paper, the Fermi-Pasta-Ulam model has been studied following the time evolution of the space Fourier spectrum through the numerical integration of the equations of motion for a system of 128 non-linearly coupled oscillators.

Abstract: The Fermi-Pasta-Ulam model has been studied following the time evolution of the space Fourier spectrum through the numerical integration of the equations of motion for a system of 128 non-linearly coupled oscillators. One-mode and multimode excitations have been considered as initial conditions; in the former case, an approximate analytic technique has been applied to describe the "short-time" behavior of the system, which fits well the experiment. The main result in both cases is the presence of different stationary states towards which the system is evolving: a $\frac{1}{{k}^{2}}$ spectrum (corresponding to the equipartition of energy) or an exponential spectrum can be reached, depending on the value of some parameter, which takes into account the relative weight of the nonlinear to the linear term of the equations of motion.

64 citations

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TL;DR: In this article, the behavior at low energy of two-dimensional Lennard-Jones systems with square or triangular cells and a number of degrees of freedom up to 128 was studied.

Abstract: We study by computer simulation the behavior at low energy of two-dimensional Lennard-Jones systems, with square or triangular cells and a number of degrees of freedom $N$ up to 128. These systems exhibit a transition from ordered to stochastic motions, passing through a region of intermediate behavior. We thus find two stochasticity borders, which separate in the phase space the ordered, intermediate, and stochastic regions. The corresponding energy thresholds have been determined as functions of the frequency $\ensuremath{\omega}$ of the initially excited normal modes; they generally increase with $\ensuremath{\omega}$ and appear to be independent of $N$. Their values agree with those found by other authors for one-dimensional LJ systems. We computed also the maximal Lyapunov characteristic exponent ${\ensuremath{\chi}}^{*}$ of our systems, which is a typical measure of stochasticity; this analysis shows that even in the ordered region certain stochastic features may persist. At higher energies, ${\ensuremath{\chi}}^{*}$ increases linearly with the energy per degree of freedom $e$. The law ${\ensuremath{\chi}}^{*}(e)$ has been determined in the thermodynamic limit by extrapolation. The values found for the stochasticity thresholds fall in a physically significant energy range. The behavior of the thresholds as a function of $\ensuremath{\omega}$ and $N$ is compatible with the hypothesis on the existence of a classical zero-point energy, advanced by Cercignani, Galgani, and Scotti.

40 citations

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