On the propagation of cnoidal wave and overtaking collision of slow shear Alfvén solitons in low β magnetized plasmas

Abstract

The overtaking collisional phenomenon of slow shear Alfvén solitons are studied in a low beta (β = kinetic pressure/magnetic pressure) collisionless, magnetized plasma consisting of electron and ion fluids. By employing a reductive perturbation technique, the Korteweg–de Vries (KdV) equation is deduced for investigating the nonlinear slow shear Alfvén wave. Before embarking on the study of the overtaking collisions, the stability analysis of the KdV equation is studied using the bifurcation theory. Also, a nonlinear periodic solution of the KdV equation is derived for the first time in the Weierstrass elliptic function formula. Moreover, the condition for converting the Weierstrass elliptic function expression to soliton is discussed. Furthermore, it is found that only density dip (rarefactive) solitons are formed in the super-Alfvénic regime. The next step includes the use of the Hirota bilinear method, which results in two and three shear Alfvén soliton solutions and their subsequent phase shifts. The influence of the plasma parameters on the amplitude as well as width of the slow shear Alfvén wave solitons are examined analytically and numerically. We also find out the profiles of overtaking interaction of slow shear Alfvén dip solitons having different amplitudes and speeds numerically. This study is important for understanding the phenomena of nonlinear slow shear Alfvén wave structures both in space and in laboratory plasmas.

A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero- bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena

“…In future work, we can apply the proposed method to study and analyze many evolution equations that are used to describe the characteristics of many nonlinear phenomena that arise in different plasma systems. For example, it is possible to study the effect of the time and space fractional parameters on the properties of solitary and shock waves that arise within the various plasma systems, which are described by the Korteweg-de Vries (KdV)-type equations with third-order dispersion (e.g., KdV, modified KdV (mKdV), Extended/Gardner KdV equation, KdV-Burgers equation, and so on) [41][42][43][44][45][46][47][48][49][50][51][52] and higher-order dispersion (e.g., Kawahara-type equation with some physical effects) and many other related equations [53][54][55][56][57][58][59][60][61][62][63][64], whether in their integral or nonintegral form. Also, this method can be applied to investigating modulated nonlinear structures such as modulated envelope solitons, modulated cnoidal waves, rogue waves and breathers that are described by the standard form of undamped planar nonlinear Schrödinger equation (NLSE) [65][66][67][68][69][70][71][72][73][74] and the damped or nonplanar NLSE [75][76][77][78][79]…”

On the approximations to fractional nonlinear damped Burger’s-type equations that arise in fluids and plasmas using Aboodh residual power series and Aboodh transform iteration methods

“…The investigation shows that this improves the effectiveness of ARPSM in evaluating problem 1 and other strong nonlinear and more complicated fractional evolution equations.The approximation (61) is analyzed graphically against the fractional parameter p and for different values of η as evident in Figures 4, 5. It is shown that the amplitude of the wave, which is described by approximation (61), increases with increasing the fractional parameter p. To make sure that the approximation ( 61) is highly accurate, we calculated its absolute error compared to the exact solution (52), which can be seen in Figure 6; Table 2. Furthermore, the numerical results indicate that the derived approximations are consistently stable across the study domain.…”

“…Here, we considered the absolute error of the approximation (88) as compared to the exact solution (83) for the integer case, i.e., p = 1. Frontiers in Physics frontiersin.org are derived from the fluid equations to some plasma models, such as KdV-type equations with third-order dispersion [52][53][54], Burger's-type equations [55-57], Kawahara-type equations with fifth-order dispersion [58-60], nonlinear Schrödinger-type equations [61,62], and many other evolution equations. Therefore, the characteristics of the many nonlinear phenomena that can be generated and propagated in various plasma systems can be accurately described and examined by studying the effect of the fractional parameters on the behavior of these phenomena, Frontiers in Physics frontiersin.org such as solitons, dissipative solitons, shocks, dissipative shocks, rogue waves, dissipative rogue waves, periodic waves, dissipative periodic waves, etc., which are among the most famous phenomena that spread in multicomponent plasmas.…”

On the analytical soliton approximations to fractional forced Korteweg–de Vries equation arising in fluids and plasmas using two novel techniques

“…One of the PDEs that describes a broad spectrum of physical nonlinear phenomena as well as engineering materials is the well-known Korteweg-de Vries (KdV) equation [21][22][23][24][25][26]. This equation and its family, whether it contains third-order dispersion [27][28][29] or fifth-order dispersion [30][31][32] or other physical effects such as collision and viscosity forces, have effectively elucidated numerous nonlinear structures that emerge and propagate in diverse physical and engineering systems, including fluid physics and plasma physics. It has been widely used, especially in interpreting soliton and cnoidal waves propagating in various plasma systems [33][34][35][36].…”