Hybrid solitary waves for the generalized Kuramoto-Sivashinsky equation

Effects of dust charges on the nonlinear ion acoustic solitary waves in a fully relativistic dusty plasma for both cases of negative and positive dusts are numerically studied based on the pseudopotential method. In the presence of dusty particles, it is found that various types of nonlinear acoustic waves exist in forms which can be viewed as sequential combinations of three kinds of elementary solitary waves: bump, dip, and kink-type solitary waves. The number and the sequence of the constituent elementary solitary waves in a given nonlinear waves depend more sensitively on dust particle density than any other parameters. For negatively charged dust particles of low density, the nonlinear wave is in the shape of bumpy solitary wave. For a somewhat higher density, the wave changes into a form which can be viewed as a combination of bump and dip-type solitary waves. As the density is increased further, a more complex nonlinear wave composed of bump, kink, and dip-type solitary waves emerges. For a much higher density of dust particle, the nonlinear wave can have a shape that can be considered as a combination of bump and kink-type solitary waves. For the case of positively charged dust particles, two kinds of nonlinear waves can exist: bump-type solitary wave and a combination of bump and kink solitary waves. For both cases of negative and positive dust particles, it is found that single dip-type solitary wave does not exist. It is also found that as dust particle density increases, the signature of the elementary waves becomes less prominent.

Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms

ABSTRACTThis paper addresses the (2+1)-dimensional cubic-quintic nonlinear Schrödinger equation (CQNLS) that serves as the model to study the light propagation through nonlinear optical media and non-Kerr crystals. A dark–bright optical solitary wave solution of this equation is retrieved by adopting the complex envelope function ansatz. This type of solitary wave describes the properties of bright and dark optical solitary waves in the same expression. The integration naturally lead to a constraint condition placed on the solitary wave parameters which must hold for the solitary waves to exist. Additionally, the modulation instability (MI) analysis of the model is studied based on the standard linear stability analysis and the MI gain spectrum is got. Numerical simulation and physical interpretations of the obtained results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the CQNLS.

Propagation of three-dimensional nonlinear ion-acoustic solitary waves and shocks in a homogeneous magnetised electron–positron–ion plasma is analysed. Modified extended mapping method is introduced to find ion-acoustic solitary wave solutions of the three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. As a result, solitary wave solutions (which represent electrostatic field potential), electric fields, magnetic fields and quantum statistical pressures are obtained with the aid of Mathematica. These new exact solitary wave solutions are obtained in different forms such as periodic, kink and antikink, dark soliton, bright soliton, bright and dark solitary wave etc. The results are expressed in the forms of hyperbolic, trigonometric, exponential and rational functions. The electrostatic field potential and electric and magnetic fields are shown graphically. These results demonstrate the efficiency and precision of the method that can be applied to many other mathematical and physical problems.

Nonlinear steady states of hyperelastic membrane tubes conveying a viscous non-Newtonian fluid

Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of the dispersive shock can be determined. In this paper an approximate method is presented for calculating the amplitude of the lead solitary waves of a dispersive shock for general nonlinear wave equations, even if these equations are not hyperbolic in the dispersionless limit. The approximate method is validated using known dispersive shock solutions and then applied to calculate approximate dispersive shock solutions for equations governing nonlinear optical media, such as nematic liquid crystals, thermal glasses and colloids. These approximate solutions are compared with numerical results and excellent comparisons are obtained.

In this work, we investigated the modulated, nonlinear, ion-acoustic wave (IAW) propagating in an unmagnetized, collisionless, homogeneous plasma where a charged density source of varying velocity due to the debris particles is present. The space-time dynamics of the one dimensional nonlinear ion-acoustic wave is modelled by a forced Nonlinear Schrodinger equation (fNLSE) where the forcing function is generated because of the space debris objects. We obtained : the exact 1 solitary wave, 2 solitary wave followed by N solitary wave solutions and the 1st order rogue wave, the 2nd order rogue wave and then the Nth order rogue wave solutions of the fNLSE for some specific forms of pinned debris functions. The combined system of the nonlinear ion-acoustic wave and the debris function propagates with acceleration. These debris induced, accelerated, exact, modulated nonlinear wave solutions are new in this field of plasma physics as per our study. The condition of modulation instability of such forced system is also discussed with plot. Such exact, nonlinear wave solutions with acceleration may be useful in modelling the experimental data of real astrophysical plasma system.

In this study, the nonlinear Klein–Gordon equation the relativistic equivalents of the nonlinear Schrödinger equations is presented which characterizes the connection between relativistic energy-momentum in a quantized manner. A variety of exact solutions are developed by employing the extended Fan sub-equation approach and the Sardar sub-equation method. The study includes three dimensional surface plots of some derived solutions like dark and bright solitary waves, kink solitary waves, anti-kink solitary waves, periodic solitary waves and hyperbolic functions, with several solutions being novel. Furthermore, the stability analysis of the observed solutions is also established to validate the scientific computations. The 3D, 2D and contour visualizations of the wave dynamics are demonstrated using Mathematica for a suitable choice of parameters. The applied strategies are recognized as a significant mathematical tool for obtaining analytical solutions to partial differential equations in mathematical physics.

Generating stable solitary waves with a piston-type wavemaker

Piezoelectric elements have been commonly used because of their wide applications in sensors, transducers, and some micro intelligent structures. However, in the fields of aviation, aerospace, and automation, some relevant equipment works in a harsh environment and is susceptible to the temperature change, thereby leading its performances to be greatly affected. Therefore, the problem of nonlinear wave relating to piezoelectric circular rods in different temperature fields is studied by modeling and numerical analysis. Firstly, based on the theory of finite deformation, we take infinite piezoelectric circular rod as a research object and consider the effects of transverse inertia and equivalent Poisson's ratio under the thermoelectric coupling action. Using the Hamilton principle and introducing the Euler equation, the longitudinal wave equation of piezoelectric circular rod is obtained. Secondly, Jacobi elliptic cosine function and Jacobi elliptic sine function expansion method are used to solve the wave equation of the piezoelectric circular rod, and the solitary wave solution and the exact periodic solution of the wave equation are obtained. It is found that the periodic solution can be reduced into a solitary wave solution under certain conditions, and it is proved theoretically that there may be solitary wave stably propagating in a piezoelectric circular rod. Finally, the dispersion curves of different wave velocity ratios and the curves about influences of temperature field on the waveform, amplitude and wave number of the piezoelectric rod are obtained by Matlab. The numerical results show that the wave velocity decreases with the increase of temperature when the wave velocity ratio is constant. Given the temperature is constant, it can be found that with the increase of the ratio, the amplitude of solitary wave gradually increases while the wavelength gradually decreases. In addition, the images obtained show that although temperature change can cause the characteristics of solitary waves to change, the solitary waves are always symmetrical bell shaped waves in the propagation process, reflecting the stability characteristics under the combined action of nonlinear and dispersion effects. Therefore, the variation of temperature field can influence and control some propagation characteristics of solitary waves. Moreover, the wave theory has been widely used in the nondestructive testing of structures and the improving of information transmission quality due to its special stability.

This article focuses on the exact periodic solutions of nonlinear wave equations using the well-known Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. The periodic solutions are found using this method which contains both solitary wave and shock wave solutions. In this paper, the new results are computed using the closed-form solution including solitary or shock wave solutions which are obtained using Jacobi elliptic function method. The corresponding solitary or shock wave solutions are compared with the actual results. The results are visualised and the periodic behaviour of the solution is described in detail. The shock waves are found to break with time, whereas, solitary waves are found to be improved continuously with time.

The propagation characteristics of (2 + 1) dimensional nonlinear dust acoustic solitary wave in an unmagnetized hot dusty plasma composed of dust particles, electrons and nonthermal ions are studied in the paper. Firstly, the Kadomtsev-Petviashvili (KP) equation, which is used to describe the (2 + 1) dimensional nonlinear dust acoustic solitary wave, is derived by the reduced perturbation method, and the phase diagram and the Sagdeev potential equation of the system are obtained by using the traveling wave solution method. Then, the effects of different parameters in the hot dusty plasma system on the nonlinear coefficient, dispersion coefficient of the KP equation, system phase diagrams, Sagdeev potential function and the solitary wave solution in isothermal and adiabatic states are discussed by using numerical simulation and analysis method of mathematical software. Finally, the results show that the mass of dust particles, temperature, number density and distribution state of electrons and nonthermal ions have important effects on the amplitude, width and waveform of the nonlinear dust acoustic solitary wave under isothermal and adiabatic conditions.

This paper considers a progressive solitary wave of permanent form in an ideal fluid of constant depth and explores Davies' approximation [Proc. R. Soc. Lond. A, 208 (1951), pp. 475--486] with high-order corrections to Levi-Civita's surface condition for the logarithmic hodograph variable. Using a complex plane that was originally introduced by Packham [Proc. R. Soc. Lond. A, 213 (1952), pp. 234--249], it is shown that a singularity at infinity can be regularized. Therefore, the solutions in Packham's complex plane under high-order Davies' approximation maintain two critical properties of a solitary wave, the correct exponential decay in the outskirt of wave and the harmonic property of a solution, that are often violated in classical long wave approximations. After introducing an accurate numerical method to compute solitary wave solutions in Packham's complex plane, we compare high-order Davies' approximate solutions with fully nonlinear solutions as well as long wave approximate solutions. The results demonstrate that high-order Davies' approximation produces rapidly converging series solutions even for relatively large amplitude waves and that Davies' approximate solutions compare much better with the fully nonlinear solutions than the long wave approximate solutions.

Generalized (2 + 1)-dimensional mKdV-Burgers equation and its solution by modified hyperbolic function expansion method

Head-on collision and overtaking collision between a KdV solitary wave and an envelope solitary wave are first studied in present paper by using Particle-in-cell (PIC) method in a dusty plasma. There are phase shifts of the KdV solitary wave in both head-on collision and the overtaking collision, while no phase shift is found for the envelop solitary wave in any cases. The remarkable difference between head-on collision and the overtaking collision is that the phase shift of KdV solitary wave increases as amplitude of KdV solitary wave increases in head-on collision, while it decreases as amplitude of the KdV solitary wave increases in the overtaking collision. It is found that the maximum amplitude during the collision process is less than sum of two amplitudes of both solitary waves, but is larger than either of the amplitude.