R. Sabry

Modified extended tanh-function method for solving nonlinear partial differential equations

Abstract Based on an extended tanh-function method, a general method is suggested to obtain multiple travelling wave solutions for nonlinear partial differential equations (PDEs). The validity and reliability of the method is tested by its application to some nonlinear PDEs. The obtained results are compared with that of an extended tanh-function method and hyperbolic-function method. New exact solutions are found.

Fully nonlinear ion-acoustic solitary waves in a plasma with positive-negative ions and nonthermal electrons

Properties of fully nonlinear ion-acoustic solitary waves in a plasma with positive-negative ions and nonthermal electrons are investigated. For this purpose, the hydrodynamic equations for the pos ...

Nonlinear structures: Explosive, soliton, and shock in a quantum electron-positron-ion magnetoplasma

Theoretical and numerical studies are performed for the nonlinear structures (explosive, solitons, and shock) in quantum electron-positron-ion magnetoplasmas. For this purpose, the reductive perturbation method is employed to the quantum hydrodynamical equations and the Poisson equation, obtaining extended quantum Zakharov–Kuznetsov equation. The latter has been solved using the generalized expansion method to obtain a set of analytical solutions, which reflects the possibility of the propagation of various nonlinear structures. The relevance of the present investigation to the white dwarfs is highlighted.

Modified extended tanh-function method and its applications to nonlinear equations

New exact travelling wave solutions for the generalized shallow water wave equation, the improved Boussinesq equation and the coupled system for the approximate equations for water waves are found using a modified extended tanh-function method. The obtained results include rational, periodic, singular and solitary wave solutions.

Dust-acoustic solitary waves and double layers in a magnetized dusty plasma with nonthermal ions and dust charge variation

The effect of nonthermal ions and variable dust charge on small-amplitude nonlinear dust-acoustic (DA) waves is investigated. It is found that both compressive and rarefactive solitons exist and depend on the nonthermal parameter a. Using a reductive perturbation theory, a Zakharov–Kuznetsov (ZK) equation is derived. At critical value of a, ac, a modified ZK equation with third- and fourth-order nonlinearities, is obtained. Depending on a, the solution of the evolution equation reveals whether there is coexistence of both compressive and rarefactive solitary waves or double layers (DLs) with the possibility of their two kinds. In addition, for certain plasma parameters, the solitary wave disappears and a DL is expected. The variation of dust charge number, wave velocity, and soliton amplitude and its width against system parameters is investigated for the DA solitary waves. It is shown that the incorporation of both the adiabatic dust-charge variation and the nonthermal distributed ions modifies significa...

Nonlinear dust acoustic waves in a nonuniform magnetized complex plasma with nonthermal ions and dust charge variation

Propagations of nonlinear dust acoustic (DA) solitary waves and shock waves in a nonuniform magnetized dusty plasma are investigated. The incorporation of the combined effects of nonthermally distributed ions, nonadiabatic dust charge fluctuation, and the inhomogeneity caused by nonuniform equilibrium values of particle density, charging variable, and particle potential on the waves leads to a significant modification to the nature of nonlinear DA solitary waves. The nonlinear wave evolution is governed by a modified Zakhavov-Kusnetsov-Burgers (MZKB) equation, whose coefficients are space dependent. Using a generalized expansion method, new solutions for the MZKB equation are obtained. The form of solutions consists of two parts; one of them is the amplitude factor and the other is a superposition of bell-shaped and kink-type shock waves. New solutions are classified into three categories. A type of the solution is determined depending on the nonthermal parameter. Findings in this investigation should be useful for understanding the ion acceleration mechanisms close to the Moon and also enhancing our knowledge on pickup ions around unmagnetized bodies, such as comets, Mars, and Venus, including medium inhomogeneities with nonadiabatic dust charging processes.

Nonlinear Dynamics of Rotating Multi-Component Pair Plasmas and e-p-i Plasmas ∗)

The propagation of small amplitude stationary profile nonlinear electrostatic excitations in a pair plasma is investigated, mainly drawing inspiration from experiments on fullerene pair-ion plasmas. Two distinct pair ion species are considered of opposite polarity and same mass, in addition to a massive charged background species, which is assumed to be stationary, given the frequency scale of interest. In the pair-ion context, the third species is thought of as a background defect (e.g. charged dust) component. On the other hand, the model also applies formally to electron-positron-ion (e-p-i) plasmas, if one neglects electron-positron annihilation. A two-fluid plasma model is employed, incorporating both Lorentz and Coriolis forces, thus taking into account the interplay between the gyroscopic (Larmor) frequency ωc and the (intrinsic) plasma rotation frequency Ω0. By employing a multi-dimensional reductive perturbation technique, a Zakharov-Kuznetsov (ZK) type equation is derived for the evolution of the electric potential perturbation. Assuming an arbitrary direction of propagation, with respect to the magnetic field, we derive the exact form of nonlinear solutions, and study their characteristics. A parametric analysis is carried out, as regards the effect of the dusty plasma composition (background number density), species temperature(s) and the relative strength of rotation to Larmor frequencies. It is shown that the Larmor and mechanical rotation affect the pulse dynamics via a parallel-to-transverse mode coupling diffusion term, which in fact diverges at ωc →± 2Ω0. Pulses collapse at this limit, as nonlinearity fails to balance dispersion. The analysis is complemented by investigating critical plasma compositions, in fact near-symmetric (T− ≈ T+) “pure” (n− ≈ n+) pair plasmas, i.e. when the concentration of the 3rd background species is negligible, case in which the (quadratic) nonlinearity vanishes, so one needs to resort to higher order nonlinear theory. A modified ZK equation is derived and analyzed. Our results are of relevance in pair-ion (fullerene) experiments and also potentially in astrophysical environments, e.g. in pulsars.

Large amplitude ion-acoustic solitary waves and double layers in multicomponent plasma with positrons

A finite amplitude theory for ion-acoustic solitary waves and double layers in multicomponent plasma consisting of hot positrons, cold ions, and electrons with two-electron temperature distributions is presented. Conditions are obtained under which large amplitude stationary ion-acoustic solitary waves and double layers can exist. For the physical parameters of interest, the ion-acoustic solitary wave (double layers) profiles and the relationship between the maximum soliton (double layers) amplitude and the Mach number are found. Also, we have presented the region of existence of the large amplitude ion-acoustic waves by analyzing the structure of the pseudopotential. For the selected range of parameters, it is found that only positive solitary waves and double layers can exist. An analysis for the small amplitude limit through the Sagdeev pseudopotential analysis and the reductive perturbation theory shows the existence of positive and negative ion-acoustic solitary waves and double layers. The effects o...

New exact solutions for a generalized variable coefficients 2D KdV equation

Abstract Using homogeneous balance method an auto-Backlund transformation for a generalized variable coefficients 2D KdV equation is obtained. Then new exact solutions are found which include solitary one. Also, we have found certain new analytical soliton-typed solution in terms of the variable coefficients of the studied 2D KdV equation.

Exact travelling wave solutions for the generalized shallow water wave equation

Abstract Using homogeneous balance method an auto-Backlund transformation for the generalized shallow water wave equation is obtained. Then solitary wave solutions are found. Also, modified extended tanh-function method is applied and new exact travelling wave solutions are obtained. The obtained solutions include rational, periodical, singular and solitary wave solutions.