A. H. Khater

Solitary and freak waves in a dusty plasma with negative ions

It is shown that solitary and freak waves can propagate in a dusty plasma composed of positive and negative ions, as well as nonextensive electrons. The evolution of the solitary waves is described by the Korteweg-de Vries (KdV) equation. However, when the frequency of the carrier wave is much smaller than the ion plasma frequency then the KdV equation is also used to study the nonlinear evolution of modulationally unstable modified ion-acoustic wavepackets through the derivation of the nonlinear Schrodinger (NLS) equation. In order to show that the characteristics of the solitary and freak waves are influenced by the plasma parameters, the relevant numerical analysis of the appropriate nonlinear solutions is presented. The relevance of the present investigation to nonlinear waves in astrophysical plasma environments is discussed.

Two-dimensional force-free magnetic fields described by some nonlinear equations

A force-free magnetic field arises as a special case in the magnetostatic equation in plasmas when only the magnetic energy density is relevant while all other energy densities are negligible and so only the magnetic pressure is considered. In this article, we find the exact solutions of two-dimensional force-free magnetic fields described by Liouville, sine, double sine, sinh-Poisson, and power force-free magnetic equations. We use the generalized tanh method. In all those cases, the ratio of the current density and the magnetic field is not constant as it happens, e.g., in the solar atmosphere.

Travelling wave solutions to some important equations of mathematical physics

Exact travelling wave solutions of some nonlinear evolution equations of mathematical physics are obtained by using the mapping method and the extended F -expansion method. It is well known that different types of exact solutions of a given auxiliary ordinary differential equation produce new types of exact solutions to nonlinear evolution equations. Many new exact travelling wave solutions of Zakharov-Kuzentsov (ZK), modified Kadomtsev-Petviashvilli (KP) with square root nonlinearity and modified fifth-order Korteweg-de Vries (KdV) equations are constructed by using mapping method. We also apply the extended F -expansion method to the long-short-wave interaction system and the coupled modified KdV equations. The solutions obtained in this paper include single and combined Jacobi elliptic function solutions, rational solutions and hyperbolic function solutions. In the limiting case, the solitary wave solutions of such equations and systems are also studied.

Exact equilibria for nonlinear cylindrical ideal magnetohydrodynamic plasma with steady incompressible flow and arbitrary cross sectional shape

Translational symmetric ideal magnetohydrodynamic plasma with steady incompressible flow is considered. The domain then is of cylindrical shape with arbitrary cross section. Several exact solution classes for nonlinear cases are obtained. The obtained solutions are bounded at infinity, in which they are physically acceptable and have applications in astrophysics as well as solar magnetic fields. Some of the solutions are examined to describe low and high β-plasma equilibria in terms of elementary functions. They can be employed to describe plasma in the solar corona, the photosphere, and the upper corona.

Numerical solutions of integral and integro-differential equations using legendre polynomials

In this paper, a finite Legendre expansion is developed to solve singularly perturbed integral equations, first order integro-differential equations of Volterra type arising in fluid dynamics and Volterra delay integro-differential equations. The error analysis is derived. Numerical results and comparisons with other methods in literature are considered.

Exact solutions for axisymmetric nonlinear magnetohydrodynamic equilibria of aligned magnetic field and plasma flow with applications to astrophysics and plasma confinement devices

The steady state equations of magnetohydrodynamic (MHD) flows for an inviscid fluid of high electrical conductivity are considered for an axisymmetric case, in which the physical quantities are independent of the coordinate ϕ of a cylindrical coordinate system (r,ϕ,z). The magnetic field is taken to be aligned to the plasma velocity, i.e., the magnetic lines of force and the streamlines of the velocity field coincide. Two classes of exact solutions are obtained. The obtained solutions are smooth everywhere and satisfy all necessary physical conditions, in which they have applications in astrophysics as well as plasma confinement devices, e.g., tokamak and reversed field pinch.

Nonlinear periodic solutions for isothermal magnetostatic atmospheres

Magnetohydrodynamic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential A, known as the Grad–Shafranov equation. Specifying the arbitrary functions in the latter equation, one obtains three types of nonlinear elliptic equations (a Liouville equation, a sinh Poisson equation, and a generalization of those with a sum of exponentials). Analytical solutions are obtained using the tanh method; this is elaborated in the Appendix. The solutions are adequate to describe an isothermal atmosphere in a uniform gravitational field showing parallel filaments of diffuse, magnetized plasma suspended horizontally in equilibrium.

Solving integral equations with logarithmic kernels by Chebyshev polynomials

In this paper, a finite Chebyshev expansion is developed to solve Volterra integral equations with logarithmic singularities in their kernels. The error analysis is derived. Numerical results are given showing a marked improvement in comparison with the piecewise polynomial collocation method given in literature.

Legendre solutions of integral equations with logarithmic kernels

In this paper a finite Legendre expansion is developed to solve linear and nonlinear Volterra integral equations with logarithmic singularities in their kernels. The error analysis is derived. Numerical results and comparisons with other methods in the literature are considered.

Large Nonlinear periodic solutions for isothermal magnetostatic atmospheres

Magnetohydrodynamic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic partial differential equation for the magnetic potential A, known as the Grad-Shafranov equation. Specifying the arbitrary functions in the latter equation, one gets a nonlinear elliptic partial differential equation (the sinh Poisson equation). Analytical solutions of this equation are obtained for the case of an isothermal atmosphere in a uniform gravitational field. The solutions are obtained by using the tanh method, and are adequate for describing parallel filaments of diffuse, magnetized plasma suspended horizontally in equilibrium in a uniform gravitational field.