Application of higher-order KdV——mKdV model with higher-degree nonlinear terms to gravity waves in atmosphere

Higher-order Korteweg-de Vries (KdV)-modified KdV (mKdV) equations with a higher-degree of nonlinear terms are derived from a simple incompressible non-hydrostatic Boussinesq equation set in atmosphere and are used to investigate gravity waves in atmosphere. By taking advantage of the auxiliary nonlinear ordinary differential equation, periodic wave and solitary wave solutions of the fifth-order KdV--mKdV models with higher-degree nonlinear terms are obtained under some constraint conditions. The analysis shows that the propagation and the periodic structures of gravity waves depend on the properties of the slope of line of constant phase and atmospheric stability. The Jacobi elliptic function wave and solitary wave solutions with slowly varying amplitude are transformed into triangular waves with the abruptly varying amplitude and breaking gravity waves under the effect of atmospheric instability.     

Ion-acoustic waves in plasma of warm ions and isothermal electrons using time-fractional KdV equation

The ion-acoustic solitary wave in collisionless unmagnetized plasma consisting of warm ions-fluid and isothermal electrons is studied using the time fractional KdV equation. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation for small but finite amplitude ion-acoustic wave in warm plasma. The Lagrangian of the time fractional KdV equation is used in a similar form to the Lagrangian of the regular KdV equation with fractional derivative for the time differentiation. The variation of the functional of this Lagrangian leads to the Euler-Lagrange equation that gives the time fractional KdV equation. The variational-iteration method is used to solve the derived time fractional KdV equation. The calculations of the solution are carried out for different values of the time fractional order. These calculations show that the time fractional can be used to modulate the electrostatic potential wave instead of adding a higher order dissipation term to the KdV equation. The results of the present investigation may be applicable to some plasma environments,such as the ionosphere plasma.     

The role of Cairns–Gurevich distributed electrons on obliquely propagating ion-acoustic waves

We used a new distribution of electrons in a two-component magnetized plasma to study the non-linear ion-acoustic solitary structures. The distribution called “Cairns–Gurevich distribution” describes simultaneously the evolution of the energetic electrons and those trapped in the plasma potential well. A modified KdV equation describing the non-linear comportment of the ion-acoustic wave (IAW) was found by using the standard reductive perturbation technique and the appropriate independent variables. The behaviour of the soliton by changing the plasma parameters has been investigated, and we demonstrated that by decreasing the non-thermality parameter, the soliton solution amplitude is enhanced. In addition, we have discussed the growth rate of the solitary waves by calculating the instability criterion. Through discussion, we have conferred how different plasma parameters, such as the trapping, non-thermality, Mach number, obliqueness via the angle of propagation, and magnetic field via the ion-cyclotron frequency, can affect the solitary wave structures. This kind of theoretical studies can be relevant to understand the non-linear propagation of IA solitary waves plasmas of electrons and particles in laser-plasma interaction, pulsar magnetosphere, the auroral zone, and the upper ionosphere, where plasma with trapped and energetic electrons are often present.     

Dust ion-acoustic solitary waves in a hot adiabatic magnetized dusty plasma

The basic features of obliquely propagating dust ion-acoustic (DIA) solitary waves in a hot adiabatic magnetized dusty plasma (containing adiabatic inertia-less electrons, adiabatic inertial ions, and negatively charged static dust) have been investigated. The reductive perturbation method has been employed to derive the Korteweg-de Vries (KdV) equation which admits a small amplitude solitary wave solution. The combined effects of plasma particle (electron and ion) adiabaticity, ion-dust collision, and external magnetic field (obliqueness), which are found to significantly modify the basic features of the small but finite-amplitude DIA solitary waves are explicitly examined. The implications of our results in space and laboratory dusty plasmas are briefly discussed.     

Video solitons in a dispersive transmission line with the nonlinear capacitance of a p-n junction

Possible types of low-frequency electromagnetic solitary waves in a dispersive LC transmission line with a quadratic or cubic capacitive nonlinearity are investigated. The fourth-order nonlinear wave equation with ohmic losses is derived from the differential-difference equations of the discrete line in the continuum approximation. For a zero-loss line, this equation can be reduced to the nonlinear equation for a transmission line, the double dispersion equation, the Boussinesq equations, the Korteweg-de Vries (KdV) equation, and the modified KdV equation. Solitary waves in a transmission line with dispersion and dissipation are considered.     

Obliquely propagating ion-acoustic solitary and shock waves in magnetized quantum degenerate multi-ions plasma in the presence of trapped electrons

The properties of obliquely propagating ion-acoustic waves have been investigated in multi-ions magnetized plasma comprising of inertial, positively and negatively charged ion fluids, trapped electrons, and negatively charged stationary heavy ions. The propagation of the waves is oblique to the ambient magnetic field which is along the z-direction. Only fast type of modes exists in the linear regime. The reductive perturbation method was adopted to derive the Korteweg– de Vries (KdV) and Burger equations, as well as the solitary and shock wave solutions of the evolved equations, have been used to analyze the properties of the small but finite amplitude waves. The effects of the constituent plasma parameters, namely, the trapping effect of electrons, the electron degenerate temperature and the viscosity coefficient on the dynamics of the small amplitude solitary and shock waves have been examined. The influence of the magnetic field and the obliquity parameter on the propagation characteristics of ion-acoustic waves are discussed.     

Bäcklund transformation,infinite number of conservation laws and fission properties of an integro-differential model for ocean internal solitary waves

This paper presents an analytical investigation of the propagation of internal solitary waves in the ocean of finite depth. Using the multi-scale analysis and reduced perturbation methods, the integrodifferential equation is derived, which is called the intermediate long wave(ILW) equation and can describe the amplitude of internal solitary waves. It can reduce to the Benjamin–Ono equation in the deep-water limit, and to the Kd V equation in the shallow-water limit. Little attention has been paid to the features of integro-differential equations, especially for their conservation laws. Here,based on Hirota bilinear method, B?cklund transformations in bilinear form of ILW equation are derived and infinite number of conservation laws are given. Finally, we analyze the fission phenomenon of internal solitary waves theoretically and verify it through numerical simulation. All of these have potential value for the further research on ocean internal solitary waves.     

Nonlinear propagation of weakly relativistic ion-acoustic waves in electron–positron–ion plasma

This work presents theoretical and numerical discussion on the dynamics of ion-acoustic solitary wave for weakly relativistic regime in unmagnetized plasma comprising non-extensive electrons, Boltzmann positrons and relativistic ions. In order to analyse the nonlinear propagation phenomena, the Korteweg–de Vries (KdV) equation is derived using the well-known reductive perturbation method. The integration of the derived equation is carried out using the ansatz method and the generalized Riccati equation mapping method. The influence of plasma parameters on the amplitude and width of the soliton and the electrostatic nonlinear propagation of weakly relativistic ion-acoustic solitary waves are described. The obtained results of the nonlinear low-frequency waves in such plasmas may be helpful to understand various phenomena in astrophysical compact object and space physics.     

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利用数值方法研究了双温离子、磁场、非均匀性和波的斜向传播对三维非线性尘埃声孤波振幅和宽度的影响。运用约化摄动法得到描述三维非线性尘埃声孤波的非标准变系数Korteweg-de Vries(KdV)方程。然后把非标准变系数KdV方程变为标准变系数KdV方程,并且得到了此标准变系数KdV方程的近似解析解。研究结果表明,此系统中存在着两种形式的孤波,即压缩型孤波和稀疏型孤波,外部磁场对三维非线性尘埃声孤波的宽度有影响,而对其振幅没有影响。此外,波的相速度与波的斜向传播和非均匀性有着非常紧密的联系。     

Quantum ion-acoustic solitary waves in weak relativistic plasma

Small amplitude quantum ion-acoustic solitary waves are studied in an unmagnetized two- species relativistic quantum plasma system, comprised of electrons and ions. The one-dimensional quantum hydrodynamic model (QHD) is used to obtain a deformed Korteweg–de Vries (dKdV) equation by reductive perturbation method. A linear dispersion relation is also obtained taking into account the relativistic effect. The properties of quantum ion-acoustic solitary waves, obtained from the deformed KdV equation, are studied taking into account the quantum mechanical effects in the weak relativistic limit. It is found that relativistic effects significantly modify the properties of quantum ion-acoustic waves. Also the effect of the quantum parameter H on the nature of solitary wave solutions is studied in some detail.     

Low frequency electrostatic nonlinear structures in an inhomogeneous magnetized non-Maxwellian electron–positron–ion plasma

The properties of low frequency (coupled acoustic and drift wave) nonlinear structures including solitary waves and double layers in an inhomogeneous magnetized electron–positron–ion (EPI) nonthermal plasma with density and temperature inhomogeneities are studied in a simplified way. The nonlinear differential equation derived here for the study of double layers in the inhomogeneous EPI plasma resembles with the modified KdV equation in the stationary frame. But the method used for the derivation of nonlinear differential equation is simple and consistent to give both the stationary solitary waves and double layers. Further, the illustrations show that superthermality κ, drift velocity and temperature inhomogeneity have significant effects on the amplitude, width, and existence range of the structures.     

Model Equations and Their Solitary Wave Solutions in Cylindrical Coordinate System

The method of multiple-scales is used to investigate the evolution of a weak nonlinear internal waves between two-layer fluids in cylindrical coordinate system. Two reduced model wave equations, which we call a modified cylindrical KdV equation for axially symmetric case and a modified cylindrical KP equation for non-axially symmetric case, are derived and their solitary wave solutions are also obtained by relating them i to the modified KdV equation by means of an appropriate variable transformation.     

Solitary Potential in a Space Plasma Containing Dynamical Heavy Ions and Bi-Kappa Distributed Electrons of Two Distinct Temperatures

The heavy ion-acoustic solitary waves(HIASWs) in a magnetized, collisionless, space plasma system(containing dynamical heavy ions and bi-kappa distributed electrons of two distinct temperatures) have been theoretically investigated. The Korteweg-de Vries(K-dV), modified K-dV(MK-dV), and higher-order MK-dV(HMK-dV) equations are derived by employing the reductive perturbation method. The basic features of HIASWs(viz. speed, polarity,amplitude, width, etc.) are found to be significantly modified by the effects of number density and temperature of different plasma species, and external magnetic field(obliqueness). The K-dV and HM-Kd V equations give rise to both compressive and rarefactive solitary structures, whereas the MK-dV equation supports only the compressive solitary structures. The implication of our results in some space and laboratory plasma situations are briefly discussed.     

Head-on collision of ion-acoustic solitary waves in an unmagnetized electron-positron-ion plasma

The head-on collision between two ion-acoustic solitary waves in an unmagnetized electron-positron-ion plasma has been investigated. By using the extended Poincaré-Lighthill-Kuo perturbation method, we obtain the KdV equation and the analytical phase shift after the head-on collision of two solitary waves in this three-component plasma. The effects of the ratio of electron temperature to positron temperature, and the ratio of the number density of positrons to that of electrons on the phase shift are studied. It is found that these parameters can significantly influence the phase shifts of the solitons. Moreover, the compressive solitary wave can propagate in this system.     

非热离子对非均匀碰撞热尘埃等离子体中三维孤波的影响

从理论上研究了非热离子、外部磁场、碰撞对非均匀热尘埃等离子体中三维非线性尘埃声孤波的影响。运用约化摄动法得到描述三维非线性尘埃声孤波的非标准的变系数Korteweg-de Vries(KdV)方程。然后把非标准KdV方程变为标准的变系数KdV方程,并且得到了标准的变系数KdV方程的近似解析解。由此解析解可以看出,非热离子的数目、碰撞、非均匀性、波的斜向传播、尘埃颗粒和非热离子的温度对三维非线性尘埃声孤波的振幅和宽度有很大的影响。外部磁场对三维非线性尘埃声孤波的宽度有影响,而对其振幅没有影响。此外,波的相速度与非热离子、波的斜向传播、尘埃颗粒的温度和非均匀性有关。     

Particle-in-Cell Simulation of the Reflection of a Korteweg-de Vries Solitary Wave and an Envelope Solitary Wave at a Solid Boundary

Reflections of a Korteweg-de Vries(KdV) solitary wave and an envelope solitary wave are studied by using the particle-in-cell simulation method.Defining the phase shift of the reflected solitary wave,we notice that there is a phase shift of the reflected KdV solitary wave,while there is no phase shift for an envelope solitary wave.It is also noted that the reflection of a KdV solitary wave at a solid boundary is equivalent to the head-on collision between two identical amplitude solitary waves.     

Consistent Riccati expansion solvability,symmetries, and analytic solutions of a forced variable-coefficient extended Korteveg-de Vries equation in fluid dynamics of internal solitary waves

We study a forced variable-coefficient extended Korteweg-de Vries (KdV) equation in fluid dynamics with respect to internal solitary wave. Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevé expansion. When the variable coefficients are time-periodic, the wave function evolves periodically over time. Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations. One-parameter group transformations and one-parameter subgroup invariant solutions are presented. Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method. The consistent Riccati expansion (CRE) solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE. Interaction phenomenon between cnoidal waves and solitary waves can be observed. Besides, the interaction waveform changes with the parameters. When the variable parameters are functions of time, the interaction waveform will be not regular and smooth.     

Electrostatic Solitary Waves in Pair-ion Plasmas with Trapped Electrons

Electrostatic solitons in an unmagnetized pair-ion plasma comprising adiabatic fluid positive and negative ions and non-isothermal electrons are investigated using both arbitrary and small amplitude techniques. An energy integral equation involving the Sagdeev potential is derived, and the basic properties of large amplitude solitary structures are investigated. Various features of solitons differ in different existence domains. The effects of ion adiabaticity, particle concentration, and resonant electrons on the profiles of Sagdeev potential and corresponding solitary waves are investigated. The generalized Korteweg-de Vries equation with mixed-nonlinearity is derived by expanding the Sagdeev potential. Asymptotic solutions for different orders of nonlinearity are discussed for solitary waves. The present work is applicable to understanding the wave phenomena and associated nonlinear electrostatic perturbations in pair/bi-ion plasmas which may occur in space and laboratory plasmas.     

Electrostatic solitary waves associated with relativistic nonextensive electrons in magnetized fusion plasma

Properties of nonlinear electrostatic solitary waves in a magnetized multicomponent system of plasma containing of warm fluid ions, weakly relativistic warm fluid electrons and q-nonextensive distributed electrons using reductive perturbation method, have been surveyed. For this purpose, a KdV soliton type solution has been employed. The dependence of solitary wave structure, solitary wave maximum amplitude, and phase velocity of soliton on the plasma parameters is defined numerically.     

Solitons at the critical density of negative ions in multicomponentplasmas

The derivation of solitary waves in generalized multicomponent plasmas shows that the negative ion introduces a critical density at which the characteristics of the solitons are studied. The soliton's behavior derived using the Korteweg-deVries (KdV) equation at the critical density shows that the nonlinearity of the wave vanishes. Thus the mode of study is augmented through a modified KdV equation. Using a higher-order equation involving quadratic and cubic nonlinearities, the transition of the KdV equation to a modified KdV equation along with the conservation of the Sagdeev potential, which is not affected by the negative ions, is studied in detail. The results are compared with experimental observations, especially those made by Y. Nakamura et al. (J. Plasma Phys., vol.33, p.237-48, 1985)