Showing papers by "Vladimir E. Zakharov published in 1974"

On stochastization of one-dimensional chains of nonlinear oscillators

Abstract: 1. Fermi, Pasta, and Ulam performed in 1954 a series of numerical experiments aimed at ascertaining how randomization and the transition to a uniform en­ ergy distribution take place in dynamic systems with a large number of degrees of freedom[1,2]. The experi­ ment were performed on one-dimensional chains of nonlinear oscillators representing discrete models of a nonlinear string.The nonlinearity level and the num­ ber of oscillators were large enough (the chain con­ sisted of 64 oscillators in some experiments) for the experimenters to hope to discern rapid randomization of the chains and a transition to a uniform distribution of the energy over the degrees of freedom. They ob­ served instead a quasiperiodic energy exchange between several initially excited modes and were unable to ob­ serve a tendency to a stochastic transition of the energy to higher modes over a sufficiently large time (up to several hundred oscillation periods).

Nonlinear interaction of high-frequency and low-frequency waves

Abstract: The interaction of high-frequency waves with low-frequency (acoustic) waves is investigated. The analysis is carried out in the Hamiltonian formalism in the interest of generality. The instability problem is investigated for the high-frequency wave. The general results obtained in the article are applied to the stability analysis of electromagnetic waves in plasmas and dielectrics. Wave propagation in weakly dispersive media is considered. It is shown that the waves are unstable. The possibility of self-focusing of the waves is studied.

Dynamics of the Langmuir collapse

Abstract: The dynamics of the nonlinear stage of the modulation instability of long plasma waves (Langmuir waves) is analyzed. The dynamics of various initial spatial distributions of the plasma waves is studied by numerical simulation. It is shown that this instability results in a Langmuir collapse: the appearance of local singularities in the amplitude of plasma waves in regions of relatively low plasma density (cavitons). The electric field in a caviton is shown to have a nonspherical dipole structure. Asymptotic self-similar solutions for large values of the time are found. The influence of Landau damping on the dynamics of the collapse is taken into account. (AIP)

Breakdown of a monochromatic wave in a medium with an inertia-free nonlinearity

Abstract: The nonlinear instability mode of a monochromatic wave in a medium with an inertia-free non-linearity is analyzed theoretically and simulated numerically. It is shown that, if longitudinal and transverse instabilities occur simultaneously, the wave is splitinto three-dimensional clusters containing amplitude singularities. As a result, the monochromatic wave “breaks down,” which is accompanied by a considerable widening of its spectrum and angular divergence.