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Nonlinear evolution equations play a crucial role in modelling various physical phenomena, such as fluid flow dynamics and plasma waves. Among them, the generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt (WBKBK) system, along with its special cases, has garnered significant attention in the study of soliton theory due to its intricate mathematical structure and broad range of applications. This study explores the properties and behaviour of soliton solutions in the WBKBK system. The study presents six new classes of wavelike solutions of the WBKBK system and investigate the unique characteristics exhibited by dark and anti-dark solitons, as well as the phenomenon of forward and backward propagating solitons in the system. These findings contribute to a deeper understanding of the fundamental dynamics of the WBKBK system and offer valuable insights into soliton solutions of the system.

New optical soliton solutions for the coupled conformable Fokas–Lenells equation with spatio-temporal dispersion are obtained via Atangana’s derivative operator using an integration method. The used method in this article is sine–Gordon expansion method. The obtained solutions include dark, bright, singular as well as periodic waves solitons which are graphically discussed through 2D- and 3D-plots.

Discrete networks, nonlinear networks, high speed computer data buses, computer network connections, and cable signal distribution are analyzed using the modified Zakharov-Kuznetsov (mZK) equation as a model governing the propagation in the electrical transmission lines. In this study, we establish viable extensive traveling wave solutions with the familiar singular kink type, kink type, bell-type solution, singular bell-type solution, t-type solution, and some advanced soliton solutions through the modified simple equation (MSE) and the sine-Gordon expansion methods. The established solutions are highly stable, resilient and capable to travel long distance. The solutions obtained can determine the voltage and current flow that can be used to design the transmission lines. The density plot and three-dimensional diagrams of the solutions obtained illustrate the voltage and current distributions in the transmission line. The traveling wave solutions are found in closed-form which simplify their utilization. The implemented methods are compatible for the extraction of traveling wave solutions.

Stable soliton solutions for the nonlinear Klein–Gordon equation in condensed matter physics, particle physics, nonlinear optics, solid state physics and the gas dynamics equation ensuing in shock fronts have been established by putting use of the sine-Gordon expansion procedure. We establish the broad-ranging familiar stable wave solutions related with some free parameters, for particular values of which some of the subsisting solutions in the literature are re-established and several fresh solutions are formulated. The contour and 3D shapes are depicted to describe the physical phenomena of the established solutions. Scores of solitary wave solutions are obtained such as, compacton, smooth soliton, anti-bell type solitary wave, bell-type solitary wave, kink and some other new types of solitary wave solutions. The obtained results might play important role in optics, quantum mechanics, mathematical physics and engineering. It is noticeable to inspect that the sine-Gordon expansion procedure is an efficient, competent and straightforward mathematical tool to ascertain exact solitary wave solutions.

Nonlinear evolution equations play a crucial role in modelling various physical phenomena, such as fluid flow dynamics and plasma waves. Among them, the generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt (WBKBK) system, along with its special cases, has garnered significant attention in the study of soliton theory due to its intricate mathematical structure and broad range of applications. This study explores the properties and behaviour of soliton solutions in the WBKBK system. The study presents six new classes of wavelike solutions of the WBKBK system and investigate the unique characteristics exhibited by dark and anti-dark solitons, as well as the phenomenon of forward and backward propagating solitons in the system. These findings contribute to a deeper understanding of the fundamental dynamics of the WBKBK system and offer valuable insights into soliton solutions of the system. • The generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system is studied. • The sine-Gordon expansion method is used to obtain solitary waveform solutions to the system. • The system is characterized by forward and backward propagating kink, antikink and bell-shaped solitary waves. [ABSTRACT FROM AUTHOR]

New optical soliton solutions for the coupled conformable Fokas–Lenells equation with spatio-temporal dispersion are obtained via Atangana's derivative operator using an integration method. The used method in this article is sine–Gordon expansion method. The obtained solutions include dark, bright, singular as well as periodic waves solitons which are graphically discussed through 2D- and 3D-plots. • The coupled conformable Fokas–Lenells equation with spatio-temporal dispersion is investigated. • The Sine–Gordon expansion method is applied to that equation. • A graphical representation of the obtained soliton solutions is presented. [ABSTRACT FROM AUTHOR]

• Abundant solutions of the electrical transmission lines in mZK equation are found. • Performance was done using the strategy of two different techniques. • Physical explanations are discussed for the obtained solutions. Discrete networks, nonlinear networks, high speed computer data buses, computer network connections, and cable signal distribution are analyzed using the modified Zakharov-Kuznetsov (mZK) equation as a model governing the propagation in the electrical transmission lines. In this study, we establish viable extensive traveling wave solutions with the familiar singular kink type, kink type, bell-type solution, singular bell-type solution, t-type solution, and some advanced soliton solutions through the modified simple equation (MSE) and the sine-Gordon expansion methods. The established solutions are highly stable, resilient and capable to travel long distance. The solutions obtained can determine the voltage and current flow that can be used to design the transmission lines. The density plot and three-dimensional diagrams of the solutions obtained illustrate the voltage and current distributions in the transmission line. The traveling wave solutions are found in closed-form which simplify their utilization. The implemented methods are compatible for the extraction of traveling wave solutions. [ABSTRACT FROM AUTHOR]

• The broad-ranging general stable soliton solutions to some nonlinear equations have been established associated with parameters and integral constants in physics and gas dynamics. • The characteristics of gas flow ensuing shock fronts and nonlinear optics have been analysed. • The 3D and contour plots of the obtained solutions are depicted to illustrate profile of the phenomena. Stable soliton solutions for the nonlinear Klein–Gordon equation in condensed matter physics, particle physics, nonlinear optics, solid state physics and the gas dynamics equation ensuing in shock fronts have been established by putting use of the sine-Gordon expansion procedure. We establish the broad-ranging familiar stable wave solutions related with some free parameters, for particular values of which some of the subsisting solutions in the literature are re-established and several fresh solutions are formulated. The contour and 3D shapes are depicted to describe the physical phenomena of the established solutions. Scores of solitary wave solutions are obtained such as, compacton, smooth soliton, anti-bell type solitary wave, bell-type solitary wave, kink and some other new types of solitary wave solutions. The obtained results might play important role in optics, quantum mechanics, mathematical physics and engineering. It is noticeable to inspect that the sine-Gordon expansion procedure is an efficient, competent and straightforward mathematical tool to ascertain exact solitary wave solutions. [ABSTRACT FROM AUTHOR]