KdVB Equation Research Articles

Nonlinear propagation of dust-acoustic waves and its modulation instability

Investigation the nonlinear dust-acoustic waves in a complex plasma in Titan’s ionosphere sets to become a vital factor in understanding different wave profiles in such space plasma. As for dust on Titan, there is evidence that suggests the Moon has a dusty surface. The Cassini spacecraft, which explored Saturn and its moons, detected bright spots on Titan’s surface that could be the result of dust or ice particles reflecting sunlight. Based on the above finding, a convenient mathematical model is considered and an appropriate evolution equation (Korteweg–de Vries-Burgers KdVB equation) is derived. At low wavenumber, the KdVB equation is transformed to a complex Ginzburg-Landau (CGL) equation that describes the propagation of the wave packet in the system, which is an efficient tool for studying rogue waves in space plasmas. Different plasma parameters are examined on the propagating waves in Titan’s ionosphere.

KdV-type equations in projective Gevrey spaces

We prove well-posedness of the Cauchy problem for a class of third order quasilinear evolution equations with variable coefficients in projective Gevrey spaces. The class considered is connected with several equations in Mathematical Physics as the KdV and KdVB equation and some of their many generalizations.

Nonlinear behaviour of ion acoustic shock waves in a two-electron temperature nonthermal complex plasma

Abstract The nonlinear propagation of dust-ion-acoustic shock waves (DIASWs) in unmagnetized dusty plasma comprising inertial ions, non-Maxwellian electrons with two distinct temperatures, and negatively charged dust is investigated in this article using a different approach based on the Sagdeev pseudopotential theory. The reductive perturbation approach is used to produce the KdVB and mKdVB equations and a comparison of their analytical and numerical solutions is shown. The effects of various parameters consisting of macroscopic non-thermal, ion-kinematic viscosity, etc. that significantly alternate the qualitative properties of DIASW are discussed. Both oscillatory and monotonic natures of the dispersive-diffusive shock wave structures are described in the present study. It has also been concentrated on nonlinear dynamics in such a plasma environment. The findings of this study should aid in understanding the nonlinear dynamics of wave damping and interactions in space and laboratory dusty plasmas, where the most relevant plasma parameters are kinematic viscosity and macroscopic non-thermality.

An efficient approximate-analytical method to solve time-fractional KdV and KdVB equations

An efficient approximate-analytical method to solve time-fractional KdV and KdVB equations

Electron-Acoustic Solitary Waves in Fermi Plasma With Two-Temperature Electrons

Electron Acoustic waves in Fermi Plasma with two temperature electrons have various applications in space and laboratory-made plasma. In some dense plasma systems like the inside of compact stars, Fermi plasma is important. We have studied Fermi plasma system with three components, two temperature electrons, and ions. The hot electrons are mobile and produce restoring force to the system while cold electrons are immobile and produce inertia to the system. We have studied the dispersion behavior of electron acoustic waves in Fermi plasma with two temperature electrons and investigated its dependence with various plasma parameters. We have investigated KdV-B equation for the solitary profile of Fermi plasma with two temperature electrons and investigated its dependence with various plasma parameters.

A Numerical Approximation to the Unique Solution of KdVB Equation under the Prabhakar Function

A Numerical Approximation to the Unique Solution of KdVB Equation under the Prabhakar Function

On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burger’s equations using homotopy analysis transform method

In this paper we consider the homotopy analysis transform method (HATM) to solve the time fractional order Korteweg-de Vries (KdV) and Korteweg-de Vries-Burger’s (KdVB) equations. The HATM is a combination of the Laplace decomposition method (LDM) and the homotopy analysis method (HAM). The fractional derivatives are defined in the Caputo sense. This method gives the solution in the form of a rapidly convergent series with h-curves are used to determine the intervals of convergent. Averaged residual errors are used to find the optimal values of h. It is found that the optimal h accelerates the convergence of the HATM, with the rate of convergence depending on the parameters in the KdV and KdVB equations. The HATM solutions are compared with exact solutions and excellent agreement is found.

Solitary and shock wave in magnetized collisional pair-ion plasmas

The effect of ion–ion collision on the dynamics of nonlinear magnetosonic wave in a magnetized pair-ion plasma consisting of positive and negative ions has been studied. The external magnetic field is situated perpendicular to the wave propagation. Two fluid model is used to describe the dynamics of positive and negative ions. Lagrangian transformation technique is used to carry out the linear and nonlinear analysis. Linear analysis yields the dispersion relation of magnetosonic wave in pair-ion plasmas. In small amplitude limit, Korteweg-de Vries Burgers’ (KdVB) equation has defined the nonlinear propagations of magnetosonic wave. Ion–ion collisions are the source of dissipation in the system and also the origin of the Burgers’ term in KdVB equation. This Burgers’ term in KdVB equation is the source of the shock structures in pair-ion plasmas. Analytical and numerical analysis reveals that the wave exhibits the solitary wave in absence of collisions. In presence of collisions, the wave exhibits both oscillatory and monotonic shock structures depending on the balance between dispersion and dissipation of the system. For weak dissipation compare to dispersion, oscillatory shock structure is formed and in reverse case, nonlinear wave exhibits monotonic shock wave.

A complex solution to (3+1)-dimensional KdV and (3+1)-dimensional KdV-Burger equations in bubbly liquid

In this paper, we use secant hyperbolic ansatz with tanh-coth method, combined with Riccati equation to derive complex travelling wave solutions to the (3+1)-dimensional Korteweg-de-Vries (KdV) and (3+1)-dimensional KdV-Burgers equations. Both the complex solitary and periodic solutions for (3+1)-dimensional KdV equation were obtained. For the (3+1)-dimensional KdVB equation, the real part is the sum of the shock wave solution of a (3+1) dimensional Burgers equation and the solitary wave solution of a (3+1)-dimensional KdV equation, while the imaginary part is the product of a shock wave solution of (3+1)-dimensional Burgers equation with a solitary wave solutions of (3+1)-dimensional KdV equation.

Obliquely Propagating Electron Acoustic Shock Waves in Magnetized Plasma

Obliquely propagating electron acoustic shock waves in plasma with stationary ions, cold and superthermal hot electrons are investigated in magnetized plasma. Employing reductive perturbation method, Korteweg-de Vries-Burgers equation (KdVB) is derived in the small amplitude approximation limit. The analytical and numerical calculations of the KdVB equation show the variation of shock waves structure (amplitude, velocity, and width) with different plasma parameters. Particle density (α), superthermal parameter (κ), electron temperature ratio (𝜃), kinetic viscosity (η0), obliqueness (kz), and strength of magnetic field (ωc) significantly modify the properties of the shock waves structures. The present investigation is useful to understand dissipative structures observed in space or laboratory plasma where multielectrons population with superthermal electrons are prevalent.

Solitons and shocks in e-p-i magneto-rotating plasma using Cairns distribution

Solitons and shocks formation are studied in a magnetized rotating electron-ion-positron plasma using Cairns distribution. We derive an admitted solitary wave solution KdV equation and an admitted travelling wave solution KdVB equation. We apply HPM technique on derived KdV equation and tanh-method on derived KdVB equation. It is observed thatγ=Th/TP, the ratio of electron temperature to positron temperature, andα=n0P/n0h, the ratio of number density of positrons to electrons, affect both the soliton width and amplitude. It is also found thatγ=Te/TP, α=n0P/n0h, kinematic viscosity and angular frequency affects the structure of shocks. We have compared our results with publish papers and conclude our results are good. This work may be helpful in order to study the rotating flows of magnetized plasma.

The tanh solution of KdV-B equation in four-component quantum plasma by taking into account dust charge variation

Nonlinear propagation of quantum dust-ion acoustic shock and solitary waves in an unmagnetized and collisionless four-component plasma medium containing inertialess quantum electrons and positrons, and also classical ions and stationary negative dust grains, which are affected by dust charge variations and viscosity of ions, are investigated using a quantum hydrodynamic model by employing the tanh method. The Korteweg–de Vries-Burgers and the Korteweg–de Vries equations indicate the behavior of shocks and solitons in this plasma model, respectively. Consideration of the dust charge variation requires the evaluation of charging currents. These currents are determined with the orbit limited motion theory and using the corresponding velocity distribution function. In the following, the effects of chemical potential and the density of dust grains on the structure of oscillatory and monotonic shock waves and solitary waves are investigated. The current study extends our understanding of the characteristics of quantum nonlinear structures in superdense astrophysical systems, inertial confinement fusion plasmas, and ultrasmall micro- and nano- electronic devices.

Non homogeneous Dirichlet problem for the KdVB equation on a segment

Non homogeneous Dirichlet problem for the KdVB equation on a segment

Periodic traveling wave solution for non-homogeneous BBM and KdVB equations

In this paper the Green’s function method and results about fixed point are used to get existence results on periodic traveling wave solution for non-homogeneous problems of generalized versions of the BBM and KdVB equations. It is shown through the constructions of explicit Green’s functions that the periodic boundary value problems for the traveling wave solutions of the BBM and KdVB equations are equivalent to integral equations which generate compact operators in the space of periodic functions. These integral representations allowed us to prove that if the speed of the wave propagation is suitably chosen, then the BBM and KdVB equations will admit periodic traveling wave solution.

Comment on “Electrostatic compressive and rarefactive shocks and solitons in relativistic plasmas occurring in polar regions of pulsar”

The aim of this comment is to show the solution of the KdVB equation used by Shah et al. (Astrophys. Space Sci. 335:529–537, 2011, doi:10.1007/s10509-011-0766-y) is not correct. So, the numerical results that are predicted in this manuscript should not be helpful for further investigations in a plasma laboratory. For this reason, we have employed the Bernoulli’s equation method to obtain the correct form of analytical solution to this equation, which is appropriate for the study of electrostatic compressive and rarefactive shocks and solitons in relativistic plasmas occurring in polar regions of pulsar.

Exact Traveling Wave Solutions for the (1 + 1)-Dimensional Compound KdVB Equation via the Novel (G'/G)-Expansion Method

In this work, while applying a new and novel (G'/G)-expansion version technique, we identify four families of the traveling wave solutions to the (1 + 1)-dimensional compound KdVB equation. The exact solutions are derived, in terms of hyperbolic, trigonometric and rational functions, involving various parameters. When the parameters are tuned to special values, both solitary, and periodic wave models are distinguished. State of the art symbolic algebra graphical representations and dynamical interpretations of the obtained solutions physics are provided and discussed. This in turn ends up revealing salient solutions features and demonstrating the used method efficiency.

Exact travelling wave solutions of the (3+1)-dimensional mKdV-ZK equation and the (1+1)-dimensional compound KdVB equation using the new approach of generalized G ′ / G $\left (\boldsymbol { {G^{\prime }/G}} \right )$ -expansion method

In this paper, the new generalized ( $G^{\prime } /G)$ -expansion method is executed to find the travelling wave solutions of the (3+1)-dimensional mKdV-ZK equation and the (1+1)-dimensional compound KdVB equation. The efficiency of this method for finding exact and travelling wave solutions has been demonstrated. It is shown that the new approach of generalized ( ${G}^{\prime }/G)$ -expansion method is a straightforward and effective mathematical tool for solving nonlinear evolution equations in applied mathematics, mathematical physics and engineering. Moreover, this procedure reduces the large volume of calculations.

SOLUTION OF KDVB EQUATION VIA BLOCK PULSE FUNCTIONS METHOD Vol. 1(2) July 2013, No. 22, pp. 361-367 M. S. EL-AZAB, I. L. EL-KALLA AND S. A. EL MORSY

SOLUTION OF KDVB EQUATION VIA BLOCK PULSE FUNCTIONS METHOD Vol. 1(2) July 2013, No. 22, pp. 361-367 M. S. EL-AZAB, I. L. EL-KALLA AND S. A. EL MORSY

Half-optical-cycle damped solitons in quadratic nonlinear media

In this paper, Using a classical model of the radiation-matter interaction, we show that the propagation of (1+1) dimensional few-optical-cycle pulses in quadratic nonlinear media, taking moderate absorption into account, can be described by the Korteweg–de Vries-Burgers’ (KdVB) equation without using the slowly varying envelope approximation. To fulfill this purpose we use the reductive perturbation method and consider the long-wave approximation, assuming that the characteristic frequency of the pulse is much lower than the resonance frequency of the atoms. We also study both analytical and numerical solution of the KdVB equation describing damped few-optical-cycle soliton propagation.

Formation of electrostatic solitons, monotonic, and oscillatory shocks in pair-ion plasmas

The nonlinear electrostatic structures in homogeneous, unmagnetized pair-ion plasma are studied. The dissipation in the system is taken through kinematic viscosities of both pair-ion species. The one dimensional (Korteweg–de Vries–Burgers) KdVB equation is derived using reductive perturbation method. The analytical solution of KdVB is obtained using tanh method. It is found that solitons and monotonic shocks structures are formed in such type of plasmas depending on the value of dissipation in the system. Both compressive and rarefactive structures of solitons and monotonic shocks are obtained depending on the temperatures of negative and positive ions. The oscillatory shock structure in pair-ion plasmas is obtained and its necessary conditions for formation are discussed. The numerical illustrations of potential structures for different values of dissipation in the system are also shown, which may have some relevance in the future experiments of laboratory produced pair-ion plasmas.