Solitons, chaos, and energy transfer in the Zakharov equations

Abstract

In the present paper we investigate the process of energy transfer in the Zakharov equations. Energy is initially injected into modes with small wave vectors. When the modulational instability threshold is exceeded, some additional modes with small wave vectors are excited and solitons are formed if one lies in a quasiintegrable regime and if the number of excited modes is large enough. These solitons are formed as a direct result of the modulational instability and in fact saturate the instability. However, use of a low-dimensional formalism based on collective variables shows that if the largest length scale of the linearly excited modes is much longer than the most unstable, these solitons may be greatly influenced as they interact with ion-acoustic waves. In those cases, full simulation of the space-time problem indicates that energy is progressively transferred to modes with very small length scales. Since we work with one spatial dimension, collapse is absent and energy transfer is due to the stochastic dynamics. ͓S1063-651X͑98͒02303-4͔ PACS number͑s͒: 05.45.ϩb, 52.35.Ra

Spatiotemporal chaos and the dynamics of coupled Langmuir and ion-acoustic waves in plasmas

“…From the analysis of the previous sections, we find that the three-wave model can be a good approximation [16] for the interaction of coupled LWs and IAWs in the plane wave region k c /2 < k < k c , where the system exhibits stable oscillations. It can relatively be accurate in the region 0.3 k < k c /2 where the system exhibits temporal chaos for a given value of the pump electric field E 0 .…”

“…In this way, Galerkin expansions and truncations to a few normal modes are commonly used to describe the basic features of the full dynamics by a low-dimensional model. However, we will see that the specific details of such model are highly dependent on the range for the basic wave number of modulation k. So, considering the dynamics of few coupled waves, we expand the envelope E(x, t) and the density ν(x, t) as [16] …”

“…It can be shown that the adiabatic limit of Eqs. (17)- (20) gives an integrable system, and hence nonchaotic [13,15,16]. Next, the local stability of the system about the fixed points (x 10 , x 20 , x 30 , 0) given by sin x 20 = 0, cot…”

“…However, the mechanism leading to such STC is still not clear. A few works in this direction can be found in the literature [12,16,19].…”

Breakdown of modulational approximations in nonlinear wave interaction

“…What usually happens in all those cases is that due to generic nonlinear interactions, the amplitude of a high-frequency carrier develops slow modulations in space and time. If the modulations are indeed much slower than the high-frequencies involved, one can obtain simplified equations describing the dynamics of the slowly varying amplitudes solely, the amplitude equations [3][4][5][6][7]. In the present analysis we consider systems that become integrable in this modulational limit, a feature often displayed.…”