On the analytical approximations to the nonplanar damped Kawahara equation: Cnoidal and solitary waves and their energy

Abstract

In this work, the non-integrable nonplanar (cylindrical and spherical) damped Kawahara equation (ndKE) is solved and analyzed analytically. The ansatz method is implemented for analyzing the ndKE in order to derive some high-accurate and more stable analytical approximations. Based on this method, two-different and general formulas for the analytical approximations are derived. The obtained solutions are applied for studying the distinctive features for both cylindrical and spherical dissipative dressed solitons and cnoidal waves in a complex plasma having superthermal ions. Moreover, the accuracy of the obtained approximations is numerically examined by estimating the global maximum residual error. Also, a general formula for the nonplanar dissipative dressed solitons energy is derived in details. This formula can recover the energy of the nonplanar dissipative dressed solitons, the planar dressed solitons, the planar damped dressed solitons, and the nonplanar dresses solitons. Both the suggested method and obtained approximations can help a large sector of authors interested in studying the nonlinear and complicated phenomena in various fields of science such as the propagating of nonlinear phenomena in physics of plasmas, nonlinear optics, communications, oceans and seas.

A Comparative Study of the Fractional-Order Belousov–Zhabotinsky System

“…Finally, the proposed methods can be used to interpret and analyze many nonlinear phenomena that arises in plasma physics, such as soliton waves, rogue waves, shock waves, etc. [10][11][12][13][14][15][16][17][18][19]. Data Availability Statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.…”

“…Partial differential equations (PDEs) are mathematical equations that described physical processes involving multiple independent variables, such as time and space. They play a crucial role in modeling many phenomena in science and engineering, including fluid dynamics, electromagnetism, quantum mechanics, and plasma physics [10][11][12][13][14][15][16][17][18][19]. The important area of partial differential equations is the fractional system of PDEs, which extends the traditional integer-order calculus to non-integer orders [20][21][22].…”

On the oblique electrostatic waves in a dusty plasma with non-Maxwellian electrons for Saturn’s magnetosphere

“…The amplitude of the solitary structure has been shown to depend upon the dust concentration, temperature ratio, magnetic field, direction cosine, superthermality parameter q and the flatness parameter r. The present work may be extended to the laboratory and space plasma systems, where dusty plasmas and (r, q) distributed electrons have been predicted. Future work: In this plasma model, if the nonplanar geometrical effect, the collisional force between the plasma particles or the higher-order perturbation is/are considered, [39][40][41] we will get some non-integrable evolution equations which can be solved using some semi-analytical and numerical approaches. [42][43][44][45][46][47][48]…”

Nonplanar ion-acoustic solitary and cnoidal waves in a non-Maxwellian plasma: Study on nonplanar (modified) Kawahara equation

“…Therefore, some researchers have devoted significant efforts to analyzing and finding approximate analytical solutions for this family with some complex physical effects. For instance, El-Tantawy group used the ansatz method to derive for the first time many semi-analytical solutions for the following family of Kawahara-type equations: [22][23][24][25][26][27][28][29][30][31][32] (1) Damped Kawahara equation (KE) that arises as a result of taking the collisional force between the charged/or neutral particles into account [22][23][24]…”

“…It would be ensured that the family of fifth-order dispersion equations, say, KE and mKE, has been used for the interpretation and explanation of the propagation mechanism of the large amplitude localized and periodic waves that can be excited by external perturbations and noises in many different plasma systems. [22][23][24][25][26][27][28][29][30] Therefore, we will follow the same approach in the literature [22][23][24][25][26][27][28][29][30] to discuss the effect of C-T-distributed electrons on the formation and propagation of large amplitude localized waves (SWs) and periodic waves (CWs) in non-Maxwellian plasmas.…”