Schamel Korteweg De Vries Research Articles

A new study on fractional Schamel Korteweg–De Vries equation and modified Liouville equation

Traveling wave solutions of fractional partial differential equations have great importance in the literature. The diversity of solutions plays an important role in understanding the physical structure of the model it represents. For this reason, two important differential equations with the fractional order, which have a significant role in applied sciences and can model real-life problems most accurately, have been solved by the generalized (G′G)-expansion method in this study. This method is a generalization of the classical (G′G)-expansion method. With this developed method, the non-linear fractional Schmael Korteweg–De Vries equation and fractional modified Liouville differential equations are discussed to find their exact solutions. In this way, new exact solutions of these equations that were not previously included in the literature have been found. The presented method has been applied to these two equations for the first time, and various new traveling wave solutions have been obtained. Thus, the study goes beyond other studies. To understand the physical behavior of these new exact solutions, three-dimensional graphs have been drawn according to different parameter values.

Optimal system and dynamics of optical soliton solutions for the Schamel KdV equation

In this research, we investigate the integrability properties of the Schamel–Korteweg–de Vries (S-KdV) equation, which is important for understanding the effect of electron trapping in the nonlinear interaction of ion-acoustic waves. Using the optimal system, we come over reduced ordinary differential equations (ODEs). To deal with reduced ODEs for this problem, Lie symmetry analysis is combined with the modified auxiliary equation (MAE) procedure and the generalized Jacobi elliptic function expansion (JEF) method. The analytical solutions reported here are novel and have a wide range of applications in mathematical physics.

Exact Traveling Wave Solutions of the Schamel-KdV Equation with Two Different Methods

The Schamel-Korteweg-de Vries (S-KdV) equation including a square root nonlinearity is very important pattern for the research of ion-acoustic waves in plasma and dusty plasma. As known, it is significant to discover the traveling wave solutions of such equations. Therefore, in this paper, some new traveling wave solutions of the S-KdV equation, which arises in plasma physics in the study of ion acoustic solitons when electron trapping is present and also it governs the electrostatic potential for a certain electron distribution in velocity space, are constructed. For this purpose, the Bernoulli Sub-ODE and modified auxiliary equation methods are used. It has been shown that the suggested methods are effective and give different types of function solutions as: hyperbolic, trigonometric, power, exponential, and rational functions. The applied computational strategies are direct, efficient, concise and can be implemented in more complex phenomena with the assistant of symbolic computations. The results found in the paper are of great interest and may also be used to discover the wave sorts and specialities in several plasma systems.

On some soliton structures to the Schamel–Korteweg-de Vries model via two analytical approaches

The Schamel–Korteweg-de Vries (S-KdV) model is used to predict the influence of surface for deep water in the presence of solitary waves. The aim of the study is to study the governing model analytically by employing the extended modified auxiliary equation mapping approach and the extended FAN sub-equation method. The 3D, 2D and contour plots are drawn to demonstrate the physical nature of the nonlinear model for a set of parameters. As a result, dark solitons, light solitons, solitary waves, periodic solitary waves, rational functions, and elliptic function solutions are established. Furthermore, the the developed results are verified with the aid of latest computing tool such as Mathematica or Maple. The applied strategy appears to be a more powerful and efficient scheme for achieving exact solutions to a number of diversified contemporary models of recent eras.

Explicit, periodic and dispersive soliton solutions to the Schamel-KdV equation with constant coefficients

In plasma physics, the Schamel–Korteweg-de Vries (S-KdV) model determines the electrostatic potential for a given electron distribution in velocity space and is used to explore ion acoustic solitons during the trapping of electrons. The key model reduces to the generalised KdV equation for α=0, and the Schamel equation when β=0. In this article, some new traveling wave solutions to the Schamel-KdV equation are constructed with the help of the classical Khater and the modified Khater methods. The observed results are very encouraging, which provide more powerful mathematical tool for analyzing exact traveling wave solution to the nonlinear equations arising in the recent era of applied sciences and engineering.

Two-component plasma and electron trapping’s influence on the potential of a solitary electrostatic wave with the dust-ion-acoustic speed

In this paper, the ion-acoustic waves’ dynamic behavior in plasma and dusty plasma is investigated to explain the electron trapping’s effect on the potential of a solitary electrostatic wave with the dust-ion-acoustic speed in two-components plasma. This study investigates the soliton wave solutions of the Schamel–Korteweg–de Vries (S–KdV) equation that Hans Schamel derived in 1973. In a nonlinear dispersive medium, the S–KdV equation is used to describe a coherent localized wave structure propagates are demonstrated. One-dimensional, unmagnetized plasma with positive ions, negative ions, trapped electrons, and stationary dust grains with both positive and negative charges is studied for the nonlinear propagation of dust-ion-acoustic (DIA) waves. The Bernoulli sub-equation function (BSEF) method is applied to the S–KdV equation for constructing some novel soliton wave solutions that explain the effect of the plasma parameters on the DIA solitary waves. The obtained solitary wave solutions are represented through some different graphs by Mathematica 12 software. Additionally, the axisymmetric pulse propagation is demonstrated in some stream plots. All the obtained solutions are checked by putting them back into the original model.

Dynamics of exact closed-form solutions to the Schamel Burgers and Schamel equations with constant coefficients using a novel analytical approach

In this paper, we investigate two different constant-coefficient nonlinear evolution equations, namely the Schamel Burgers equation and the Schamel equation. These models also have a great deal of potential for studying ion-acoustic waves in plasma physics and fluid dynamics. The primary goal of this paper is to establish closed-form solutions and dynamics of analytical solutions to the Schamel Burgers and the Schamel equations, which are special examples of the well-known Schamel–Korteweg-de Vries (S-KdV) equation. We derive completely novel solutions to the considered models using a variety of computation programmes and a newly proposed extended generalized [Formula: see text] expansion approach. The newly formed solutions, which include hyperbolic and trigonometric functions as well as rational function solutions, have been produced. The annihilation of three-dimensional shock waves, periodic waves, single soliton, singular soliton, and combo soliton, multisoliton as well as their three-dimensional and contour plots are used to show the dynamical representations of the acquired solutions. These results demonstrate that the proposed extended technique is efficient, reliable and simple.

Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel–Korteweg–de Vries equation* *Project supported by the National Natural Science Foundation of China (Grant No. 11461022).

The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics. The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses. The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given. All possible exact explicit parametric representations of the waves are also presented. Along with the details of the analyses, the analytical results are numerically simulated lastly.

Nonlinear Schamel–Korteweg deVries equation for a modified Noguchi nonlinear electric transmission network: Analytical circuit modeling

• Anew capacitance-voltage characteristics containing a square root nonlinearity is built and used for circuit modeling of a modified Noguchi nonlinear transmission network. • The perturbation method in the continuum limit is used to show that the weakly modulated waves propagating in the our model can be governed by the nonlinear Schamel-Korteweg-de Vries equation. • New soliton and periodic solutions of the Schamel-Korteweg-de Vries equation are derived and used to investigate the dependency of the characteristics of parameters of bright soliton and periodic signals on the network parameters. A modified Noguchi nonlinear electric transmission network that consists of a large number of identical sections is theoretically studied in the present work. A new capacitance-voltage (C-V) characteristics of the diodes containing a square root nonlinearity is introduced and used to show that in the continuum limit, the voltage for the transmission line is described by the Schamel Korteweg de Vries (S-KdV) equation. The dependency of the characteristics of modulated wave parameters on our nonlinear electric transmission network are shown in the work. The results found in this paper are of great interest and can be exploited in other areas of physics such as in plasma physics .

Nonlinear ion acoustic solitary waves with dynamical behaviours in the relativistic plasmas

This work investigates the basic features of Nonlinear Ion Acoustic Solitary waves (NIASWs) and their dynamical behaviours in an unmagnetized relativistic collisionless plasma system via the Schamel Korteweg-de Vries (SKdV) equation. Such plasma is composed by the generalized distributed electrons, Boltzmann distributed positrons and relativistic warm ions. The influences of plasma parameters on NIASWs and their dynamical behaviours are investigated by comparing 26−term expansion of relativistic Lorentz factor (RLF) with both of weakly (2−term expansion of RLF) and highly (3−term expansion of RLF) regimes. It is found that the 26−term expansion of RLF are significantly changed NIASWs instead of both weakly and highly relativistic regimes. Therefore, the theoretical results would be very useful for understanding the nature (amplitude, width, polarity, etc.) of wave dynamics not only in astrophysical and space environments but also in further laboratory studies, where the proposed plasma assumptions are existed.

Painlevé Analysis and Abundant Meromorphic Solutions of a Class of Nonlinear Algebraic Differential Equations

In this paper, a class of nonlinear algebraic differential equations (NADEs) is studied. The Painlevé analysis of the NADEs is considered. Abundant meromorphic solutions of the NADEs are obtained by means of the complex method. Then, meromorphic exact solutions of the Schamel‐Korteweg‐de Vries (S‐KdV) equation and (2 + 1)‐dimensional sine‐Gordon equation are derived via the applications of the NADEs.

New travelling solitary wave solutions for an evolution equation by three schemes

In this study, we establish new travelling wave Solitons solutions by using three methods. These methods are Tan-Cot method, Extended Tanh-Coth method, and Modified Simple equation Method. These methods are used to obtain new solitary wave trigonometric and hyperbolic functions solutions for the generalized Schamel-Korteweg-de Vries (S-KdV) equation. These methods have been successfully applied to construct new solitary solutions as illustrated in figures. The three methods are efficient and reliable for solving great many nonlinear partial differential equations in physics.

Exact Solitary Wave Solutions for the Generalized Schamel Korteweg–De Vries Equation

The generalized Schamel-Korteweg-de Vries (S-KdV) equation containing root of degree nonlinearity is a very attractive model for the study of ion-acoustic waves in plasma and dusty plasma. In this work, we obtain the soliton-like solutions, the kink solutions, and the plural solutions of the generalized S-KdV equation by using the sine-cosine method. These solutions may be of important significance for the explanation of some practical physical problems. It is shown that these two methods provide a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.

On the exact solutions, lie symmetry analysis, and conservation laws of Schamel–Korteweg–de Vries equation

In this work, we study the integrability aspects of the Schamel–Korteweg–de Vries equation that play an important role in studying the effect of electron trapping on the nonlinear interaction of ion‐acoustic waves by including a quasi‐potential. Lie symmetry analysis together with the simplest equation method and Kudryashov method is used to obtain exact traveling wave solutions for this equation. In addition, conservation laws are constructed using two different techniques, namely, the multiplier method and the new conservation theorem. Using the conservation laws and symmetries of the underlying equation, double reduction and exact solution were also constructed. Copyright © 2017 John Wiley & Sons, Ltd.

Travelling wave solutions of the Schamel–Korteweg–de Vries and the Schamel equations

Abstract In this paper, the extended (G′/G)-expansion method has been suggested for constructing travelling wave solutions of the Schamel–Korteweg–de Vries (s-KdV) and the Schamel equations with aid of computer systems like Maple or Mathematica. The hyperbolic function solutions and the trigonometric function solutions with free parameters of these equations have been obtained. Moreover, it has been shown that the suggested method is elementary, effective and has been used to solve nonlinear evolution equations in applied mathematics, engineering and mathematical physics.

Parametric study of a Schamel equation for low-frequency dust acoustic waves in dusty electronegative plasmas

In this paper, our attention is first concentrated on obliquely propagating properties of low-frequency (ω ≪ ωcd) “fast” and “slow” dust acoustic waves, in the linear regime, in dusty electronegative plasmas with Maxwellian electrons, kappa distributed positive ions, negative ions (following the combination of kappa-Schamel distribution), and negatively charged dust particles. So, an explicit expression for dispersion relation is derived by linearizing a set of dust-fluid equations. The results show that wave frequency ω in long and short-wavelengths limit is conspicuously affected by physical parameters, namely, positive to negative temperature ion ratio (βp), trapping parameter of negative ions (μ), magnitude of the magnetic field B0 (via ωcd), superthermal index (κn,κp), and positive ion to dust density ratio (δp). The signature of the penultimate parameter (i.e., κn) on wave frequency reveals that the frequency gap between the modes reduces (escalates) for k<kcr (k>kcr), where kcr is critical wave number. Alternatively, for weakly nonlinear analysis, reductive perturbation theory has been used to construct 1D and 3D Schamel Korteweg-de Vries (S-KdV) equations, whose nonlinearity coefficient prescribes only compressive soliton for all parameter values of interest. The survey manifests that deviation of ions from Maxwellian behavior leads intrinsic properties of solitary waves to be evolved in opposite trend. Additionally, at lower proportion of trapped negative ions, solitary wave amplitude mitigates, whilst the trapping parameter has no effect on both spatial width and the linear wave. The results are discussed in the context of the Earth's mesosphere of dusty electronegative plasma.

All traveling wave exact solutions of two nonlinear physical models

In this paper, we employ the complex method to obtain all meromorphic exact solutions of complex Schamel–Korteweg-de Vries (S–KdV) equation and modified Zakharov–Kuznetsov (mZK) at first, then find all exact solutions of Eqs. (mZK) and (S-Kdv). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that the complex method is simpler than other methods, and there exist some rational solutions wr,2(z),w2r,2(z),w2r,3(z) and simply periodic solutions ws,1(z),w2s,6(z) which are not only new but also not degenerated successively by the elliptic function solutions. In Section 4, we give some computer simulations to illustrate our main results.

Dust ion acoustic soliton in pair-ion plasmas with non-isothermal electrons

Dust ion acoustic (DIA) solitons in an unmagnetized pair-ion (PI) plasmas with adiabatic pair-ions, non-isothermal electrons, and negatively charged background dust are investigated, using both small and arbitrary amplitude techniques. An energy integral equation involving the Sagdeev potential is derived, and basic properties of the large amplitude solitary structures are investigated. The effects of dust concentration, resonant electrons, and ion temperatures on the profiles of the Sagdeev potential and corresponding solitary waves are studied. The related Schamel-Korteweg-de Vries (S-KdV) equation with mixed-nonlinearity is derived by expanding the Sagdeev potential. Asymptotic solutions for different orders of nonlinearity are discussed for DIA solitary waves. The present work is applicable to understand the wave phenomena and associated nonlinear electrostatic perturbations in the doped pair ion plasmas, not completely filtered e.g., pair ion-electron plasmas, enriched with an extra massive charged component (e.g., dust defects), which may be academic for the moment but might be of interest for forthcoming experiments in laboratory (space) plasmas.

Exact Travelling Wave Solutions of the Schamel–Korteweg–de Vries Equation

The Schamel–Korteweg–de Vries (S-KdV) equation containing a square root nonlinearity is a very attractive model for the study of ion-acoustic waves in plasma and dusty plasma. In this work, we obtain exact travelling wave solutions of the S-KdV equation by employing the exp function method. In general, the exact travelling wave solutions will be helpful in the theoretical and numerical study of the nonlinear evolution equations. The work emphasizes the power of the method in providing distinct solutions of different physical problems.

New Exact Solutions of Two Nonlinear Physical Models

Abundant new exact solutions of the Schamel–Korteweg-de Vries (S-KdV) equation and modified Zakharov–Kuznetsov equation arising in plasma and dust plasma are presented by using the extended mapping method and the availability of symbolic computation. These solutions include the Jacobi elliptic function solutions, hyperbolic function solutions, rational solutions, and periodic wave solutions. In the limiting cases, the solitary wave solutions are obtained and some known solutions are also recovered.