Reaction-Diffusion Equation with Stationary Wave Perturbation in Weakly Ionized Plasmas
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GENERAL AND APPLIED PHYSICS
Reaction-Diffusion Equation with Stationary Wave Perturbation in Weakly Ionized Plasmas S. T. da Silva1 · R. L. Viana1 Received: 14 July 2020 / Published online: 1 October 2020 © Sociedade Brasileira de F´ısica 2020
Abstract The influence of external harmonic forcing in nonlinear chaotic systems is a subject of active investigation, chiefly in lowdimensional dynamical systems, but fewer results are available for high-dimensional systems like plasmas. In this paper, we consider a theoretical model for a weakly ionized plasma, in which the following effects are taken into account: (i) ambipolar diffusion; (ii) the inflow of plasma particles through ionization processes; (iii) the outflow of plasma particles due to recombination. The spatiotemporal patterns of the resulting nonlinear system, as revealed by numerical integration of the reaction-diffusion partial differential equations, can be partially or totally suppressed through the action of acoustic waves forming stationary patterns in the plasma container. These suppression effects are quantitatively investigated by a number of numerical diagnostics like the average Lyapunov exponents and spatial correlation function. Suppression of spatiotemporal chaos can be achieved through adequate choices of amplitude and frequency of the applied waves. Keywords Nonlinear reaction-diffusion equations · Weakly ionized plasmas · Fisher-KPP equation · Spatiotemporal chaos
1 Introduction Chaos and turbulence are often seen as a great challenge in many situations of plasma physics interest [1, 2]. In fusion plasma physics, the existence of chaotic behavior often leads to turbulence [3, 4]. The combination of nonlinearity with other instability phenomena is a severe problem in the magnetic confinement of plasmas in fusion devices [2]. Generally, one has to distinguish between temporal low-dimensional chaos, related to oscillations, and spatiotemporal chaos, related to waves, also classified as weakly developed turbulence [5]. In the framework of nonlinear dynamical systems, several strategies have been developed to obtain active control over complex temporal or spatiotemporal behavior [5, 6]. Recent applications of control theory have been developed to control chaotic behavior [7, 8]. For example, small perturbations have been used to control chaotic oscillations of the current and plasma potential in plasma diodes [9]. R. L. Viana
[email protected] 1
Departament of Physics, Federal University of Paran´a, Curitiba, Paran´a, 81531-990, Brazil
Successful approaches in controlling chaos in lowdimensional systems have motivated the search of techniques for taming fully developed spatiotemporal chaos. As an example, continuous global control can be used to stabilize plasma turbulence due to weakly developed ionization waves [9, 10]. A system that presents a rich dynamics with spatiotemporal chaos is the reaction-diffusion equation in weakly ionized plasmas [11]. Depending on the parameter values, such systems can display spatiotemporal chaos. Particu
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