On multiple soliton similariton‐pair solutions,conservation laws via multiplier and stability analysis for the Whitham–Broer–Kaup equations in weakly dispersive media

In this article, the generalized unified method (GUM) is used for finding multiwave solutions of the coupled Whitham‐Broer‐Kaup (WBK) equation with variable coefficients. Which describes the propagation of of shallow water waves. Here, we study the effects of the indirect nonlinear interaction of one‐, two‐ and three‐solitonic similaritons on the behavior of propagation of waves, in quasi‐periodic distributed system. This study can unable us to control the dynamics of type soliton (soliton, anti‐soliton) similaritons waves in dispersive waveguides. To give more physical insight to the obtained solutions, they are shown graphically. Their different structures are depicted by taking appropriate arbitrary functions. Further, with the suitable parameters, the indirect nonlinear interaction between two and three‐soliton waves are shown weal, in the sense that their amplitude does not blow up. Moreover, because of the importance of conservation laws Cls and stability analysis SA in the investigation of integrability, internal properties, existence, and uniqueness of a differential equation, we compute the Cls via multiplier technique and stability analysis via the concept of linear stability analysis for the WBK equations using the constant coefficients.     

Theoretic study of dust acoustic waves in a dusty plasma with dust charge variation

The nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation is investigated by using the formally variable separation approach. New solutions for the governing equation of this system have been obtained for dust acoustic waves in a dust plasma firsthand. We derive exact mathematical expressions and numerical simulation studies for the general case of the nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation.     

Effects of the dust charge variation and non-thermal ions on multi-dimensional dust acoustic solitary structures in magnetized dusty plasmas

The combined effects of both adiabatic dust charge variation and non-thermally (fast) distributed ions on dust acoustic solitary structures are studied in a magnetized dusty plasmas consisting of the negatively and variably charged hot dust fluid, Boltzmann distributed electrons and non-thermally distributed ions. By using the reductive perturbation method, we derive the Korteweg-de Vries (KdV) equation governing the dust acoustic solitary waves. It is shown that the dust charge variation and the presence of non-thermally distributed ions would modify the nature of dust acoustic solitary structures significantly and may excite both dust acoustic solitary holes (soliton with a density dip) and positive solitons (soliton with a density hump).     

Interactions of solitary waves under the conditions of the (3 + 1)-dimensional Kadomtsev-Petviashvilli equation

Using an extended mapping method with a linear variable separation process, a new family of the exact solutions of the (3 + 1)-dimensional Kadomtsev-Petviashvilli (KP) equation was derived. By applying the solitary wave solutions, this paper studied some newly localized excitations and the interactions of various solitary waves under the conditions of the (3 + 1)-dimensional KP equation.     

The formally variable separation approach for the dust-acoustic solitary waves with dust charge variation

The formally variable separation approach is used for handling the dust-acoustic solitary waves in a dusty plasma, including consideration of dust charge variation. New analytical solutions of nonlinear waves are formally derived for the governing equation of the system. We have triumphantly derived the exact analytical expressions and some approximate expressions of the nonlinear dust-acoustic waves in a dusty plasma under some special cases. The work introduces entirely new solutions and emphasizes the power of the newly developed method that can be used in problems with identical nonlinearities.     

Multidimensional dust acoustic solitary waves in inhomogeneous magnetized dusty plasmas with nonthermal ions

The linear dispersion relation and a modified variable coefficients Korteweg–de Vries (MKdV) equation governing the three-dimensional dust acoustic solitary waves are obtained in inhomogeneous dusty plasmas comprised of negatively charged dust grains of equal radii, Boltzmann distributed electrons and nonthermally distributed ions. The numerical results show that the inhomogeneity, the nonthermal ions, the external magnetic field and the collision have strong influence on the frequency and the nonlinear properties of dust acoustic solitary waves and both dust acoustic solitary holes (soliton with a density dip) and positive solitons (soliton with a density hump) are excited.     

A hybrid type of soliton equations with self-consistent sources: KP and Toda cases

Source generation procedure is applied to construct a hybrid type of soliton equations with self-consistent sources (SESCSs). The examples include the KP equation with self-consistent sources (KPESCS) and two-dimensional TodaESCS. One typical feature for this hybrid type of SESCSs is that soliton solutions of these new systems contain arbitrary functions of a linear combination of two independent variables, which is different from the normal SESCSs where soliton solutions only contain arbitrary functions of one independent variable. What's more, the obtained two hybrid SESCSs can be reduced to two different simpler SESCSs respectively.     

（2＋1）维KP方程的三类精确解

研究了（2＋1）维KP方程的孤子解问题．应用Riccati方程映射法,得到了（2＋1）维KP方程的新的显式精确解的结构．根据得到的精确解结构,构造出了该方程的三类精确解．     

Dust-acoustic solitary waves in a two-temperature electrons with charge fluctuations and nonisothermal ions

In this paper, a modified Korteweg–de Vries (mKdV) equation and Korteweg–de Vries (KdV) equation at critical ion density are derived for dusty plasmas consisting of hot dust fluid, nonisothermal ions and two-temperature electrons. The charge fluctuation dynamics of the dust grains has also been considered. It has been shown that the presence of a second component of electrons modifies the nature of dust acoustic (DA) solitary structures. The effects of two-temperature electrons, obliqueness and external magnetic field on the properties of DA solitary waves are discussed. Numerical investigations show that there exists only rarefactive solitary waves.     

On soliton solutions for the Fitzhugh–Nagumo equation with time-dependent coefficients

We consider a generalized Fitzhugh–Nagumo equation exhibiting time-varying coefficients and linear dispersion term. By means of specific solitary wave ansatz and the tanh method, a new variety of soliton solutions are derived. The physical parameters in the soliton solutions are obtained as function of the time-dependent model coefficients. The conditions of existence and uniqueness of solitons are presented. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous media that is described by the variable coefficients Fitzhugh–Nagumo equation. Clearly, adaptive methods are straightforward and concise and their applications for the Fitzhugh–Nagumo equation with t-dependent coefficients enable one to construct soliton-like solutions.     

Jacobi Elliptic Function Solutions and Traveling Wave Solutions of the (2 + 1)-Dimensional Gardner-KP Equation

The fully integrable KP equation is one of the models that describes the evolution of nonlinear waves, the expansion of the well-known KdV equation, where the impacts of surface tension and viscosity are negligible. This paper uses the Modified Extended Direct Algebraic (MEDA) method to build fresh exact, periodic, trigonometric, hyperbolic, rational, triangular and soliton alternatives for the (2 + 1)-dimensional Gardner KP equation. These solutions that we discover in this article will help us understand the phenomena of the (2 + 1)-dimensional Gardner KP equation. Comparing the study in this paper and existing work, we find more exact solutions with soliton and periodic structures and the rational function solution in this paper is more general than the rational solution in existing literature. Most of the Jacobi elliptic function solutions and the mixed Jacobi elliptic function solutions to the (2 + 1)-dimensional Gardner KP equation discovered in this paper, to the best of our highest understanding are not seen in any existing paper until now.     

New periodic wave and soliton solutions for a Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation

The periodic wave solutions for Kadomtsev–Petviashvili (KP) like equation coupled to a Schrödinger equation are obtained by using of Jacobi elliptic function method, in the limit cases, the multiple soliton solutions are also obtained. The properties of some periodic and soliton solution for this system are shown by some figures.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

The Kadomtsev–Petviashvili (KP), MKP,and coupled KP equations for two-ion-temperature dusty plasmas

It is well known that the Korteweg–de Vires (KdV) equation can describe small but finite amplitude dust acoustic waves in a dusty plasmas. In this paper, we use the reductive perturbation method and derive a Kadomtsev–Petviashvili (KP) equation, a modified KP (MKP) equation and a coupled KP equation for unmagnetized, collisionless, cold, and two-ion-temperature dusty plasmas with N different species of dust grains. We find that if a solitary wave exist in this system, the smaller grains have larger velocities and propagate longer distances than that of larger particles. The comparisons are given between the dusty plasma composed by different dust particles and the mono-sized dusty plasma.     

Quasiperiodic Wave Solutions of Supersymmetric KdV Equation in Superspace

A direct and unifying scheme for explicitly constructing quasiperiodic wave solutions (multiperiodic wave solutions) of supersymmetric KdV equation in a superspace is proposed. The scheme is based on the concept of super Hirota forms and on the use of super Riemann theta functions. In contrast to ordinary KdV equation with purely bosonic field, some new phenomena on super quasiperiodic waves occur in the supersymmetric KdV equation with the fermionic field. For instance, it is shown that the supersymmetric KdV equation does not possess an N ‐periodic wave solution for N≥ 2 for arbitrary parameters. It is further observed that there is an influencing band occurred among the quasiperiodic waves under the presence of the Grassmann variable. The quasiperiodic waves are symmetric about the band but collapse along with the band. In addition, the relations between the quasiperiodic wave solutions and soliton solutions are rigorously established. It is shown that quasiperiodic wave solution convergence to the soliton solutions under certain conditions and small amplitude limit.     

Envelop solitons in dusty plasmas for warm dust

A nonlinear Schrödinger equation is obtained for the warm dusty plasmas. The modulational instability of envelop soliton is investigated in this paper. Both the temperature of the dust grains and the charge variations of dust grains affect the instability regions of the dusty plasmas. It also affect the amplitude and the width of the envelop soliton.     

变形色散水波方程的Auto-Backlund变换、多孤子解和有理分式解

利用未知数变换并借助Ｍａｔｈｅｍａｔｉｃａ软件,给出了变形色散水波方程的Ａｕｔｏ－Ｂａｅｃｋｌｕｎｄ变换以及它与热传导方程和线性方程之间的Ｄａｒｂｏｕｘ变换。进而用此变换,获得了变形色散水波方程的多孤子解,有理分式解及其他精确解。这种思路也适用于其它的非线性方程。     

Higher order solution of the dust ion acoustic solitons in a warm dusty plasma with vortex-like electron distribution

The ratios of dust to free electron and free to trapped electron temperatures are examined in warm dusty plasmas with vortex-like electron distribution through the derivation of a modified Korteweg–de Vries (MKdV) equation using a reductive perturbation theory. As the wave amplitude increases, the width and velocity of the soliton deviate from the prediction of the MKdV equation, i.e., the breakdown of the MKdV approximation. To describe the soliton of larger amplitude, the MKdV equation with the fifth-order dispersion term is employed and its higher-order solutions are obtained.     

Analytical studies of the steady solution for optical soliton equation with higher-order dispersion and nonlinear effects

The exact analytical solution of the optical soliton equation with higher-order dispersion and nonlinear effects has been obtained by the method of separating variables. The new type of optical solitary wave solution, which is quite different from the bright and dark soliton solutions, has been found under two special cases. The stability of the solitary wave solutions for the optical soliton equation is discussed. Some new conclusion of the stability are obtained, for the solitary wave solutions of the nonlinear wave equations, by using the Liapunov direct method.     

Analytical study of nonlinear dust acoustic waves in two-dimensional dust plasma with vortex-like ion distribution

The nonlinear dust acoustic waves in two-dimensional dust plasma with vortex-like ion distribution are analytically investigated by using the formally variable separation approach. New analytical solutions for the governing equation of this system have been obtained for dust acoustic waves in a dust plasma for the first time. We derive exact analytical expressions for the general case of the nonlinear dust acoustic waves in two-dimensional dust plasma with vortex-like ion distribution.