The transverse evolution of ions in front of two juxtaposed electrodes whose electric potentials are independently applied is calculated. The particle motion is described with nonlinear fluid equations and a self-consistent electric field.

We develop a new asymptotic method of resolution of the two-dimensional equilibrium equation of collisionless plasmas described by the Maxwell–Vlasov equations. This method differs from the classical one proposed by K. Schindler [Earth’s Magnetospheric Processes (ed. B. M. McCormac). Norwood, MA: Reidel, 1972, pp. 200–209.] since we consider free-boundary plasmas. Our method is a generalization of the usual multiscale asymptotic developments. The first-approximation asymptotic solutions are found from the elimination of increasing and singular terms in the next approximation. We apply the method to the mathematical description of nonlinear structures that may form in neutral sheets. Particular solutions describing localized plasmoids (O-point configuration) as well as X-point magnetic configurations are obtained. We also find more general solutions describing a finite number of ‘magnetic islands’ (multiplasmoid solutions) separated by X points.

This paper presents an investigation of the time-dependent setting-up of nonlinearity in a collisionless plasma in the presence of a short-duration laser pulse. Using the paraxial-ray approximation, transient self-focusing of the incident laser pulse and Brillouin back-scattering have also been studied, including pump depletion. The effect of this transient nature on reflectivity has been discussed.

It is known that the wavefront of a shock wave in a shock tube becomes unstable at particular Mach numbers and unperturbed gas pressure. This instability leads to the distortion of the wave front, i.e. particular Mach numbers define a threshold beyond which the wavefront is no longer plane (see Annou and Ferhat 1997, and references therein). In a recent paper (Annou and Ferhat 1997, subsequently referred to as AF), the authors proposed a microscopic model based on an ionization mechanism described by a set of two reaction–diffusion equations in an effort to interpret the wavefront distortion. Using an analytical approach, they argued that their equations have a bifurcation point beyond which the wavefront instability occurs and that the solution thus obtained describes the distortion of the wavefront. In this brief comment, we present analytical and numerical arguments supporting the fact that the proposed model, although it has an instability point, is unable to explain the front behavior.

We respond here to the comment by Houili et al. (2001) on our paper on instabilities of intense shock waves in monatomic gases (Annou and Ferhat 1997).

It has been reported that a strong shock-wave front in a shock tube, ceases to be plane and becomes distorted for particular experimental conditions (Schreffler and Christian 1954). Two different approaches have been developed to explain the instability (Annou 1995; Annou and Ferhat 1997). By means of the macroscopic approach, we can reproduce the experimental data. Nonetheless, many parameters suspected to be at the root of the instability have not been taken into account. Consequently, a microscopic approach has been proposed that copes with the atomic reactions taking place due to the shock wave. An irreducible system of atomic reactions leading to ionization of the gas is proposed. This is an attempt to develop a mathematical tool that has to be improved to account for the front distortion. A system of two nonlinear differential equations is obtained. As a matter of fact, we have been greatly inspired by the Brusselator model. Our achievement has been to link the experimental data to a reaction-diffusion equation system, analogous to the Brusselator. Hence, the analytical treatment of the stability, along with finding the solution beyond the instability, is based on the work of Nicolis and Prigogine (1977). It should be noted, as well, that the paper by Annou and Ferhat (1997) is based essentially on a thesis submitted (1994) for the award of the Magister degree (Annou 1995).One of the authors of the paper (R.A.) had asked Mr Houili to investigate the system numerically, and provided him with the necessary literature. Unfortunately, the comment was submitted for publication without consulting either of the authors of the original paper.