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The brake mechanism is one of the inevitable and safety-critical components in an automobile. A proper rotor design and superior heat dissipating material provide better performance during the braking mechanism. In this study, modified ventilated disk brake rotor has been developed with curved vents, holes, and slots and analyzes the stress and temperature distribution. Finite element models of the rotor are shaped with SolidWorks and simulated using ANSYS. Structural and thermal characteristics are compared with a reference disk brake rotor. It is found that the proposed rotors outperformed the conventional one in terms of stress generation, temperature distribution, and factor of safety. Furthermore, computation has been done to find out the best-suitedmaterial for one of the proposed designs. The result provides a physical insight of the structural and thermal characteristics of the geometrically modified rotor that can be implemented in the automotive industry.

This work deals with the ion-acoustic shock waves (IASWs) around the critical values in an unmagnetized pair–ions plasma with ( $$\alpha , q$$ )-distributed electrons by formulating the correct stationary solution of Burgers-type equations with higher-order corrections. By considering higher-order correction of the reductive perturbation technique, the modified Burgers(mB)-, and mixed modified Burgers(mmB)-type equations are derived. With the changes of viscosity coefficients of positive and negative ions, the electrostatic IASWs and normalized electric fields are investigated around the critical values. The current studies might be very useful to understand the behavior of shocks around the critical values in the F- and D-regions of Earth’s ionosphere, and the later experimental verification in plasma laboratory.

The present work investigates the single and overtaking collision of multi-shock wave excitations having space fractional evolution in a thin viscoelastic tube filled with incompressible inviscid fluid. The previously proposed model equations are considered to study such physical scenarios. The spatial fractional Burgers equation is formulated by implementing the reductive perturbation method form the considered model equations. The new analytical solutions for single and overtaking collision of multi-shocks are constructed by implementing the rational exponential functions directly. With the changes of physical parameters, the behaviors of single and overtaking collision of multi-shocks are displayed graphically and described physically. It is found that the overtaking collisions of multi–shocks are produced with the presence of beta nonlocal operator. The single and interactions of multi-shocks are also significantly changed with the change of physical parameters. The obtained results are very useful in describing the nature of overtaking collision of multi-shocks in various environments, particularly in large blood vessels and further laboratory studies.

This work is reported the analytical solutions for describing the nonlinear directional couplers with metamaterials by including spatial–temporal fractional beta derivative evolution. The auxiliary ordinary differential equation method and the generalized Riccati method with the properties of beta derivative are implemented to secure such solutions. The solutions are obtained in the new forms by involving of some useful mathematical functions. The constraint conditions among the traveling wave parameters are evaluated. Some of the obtained solutions are presented graphically to illustrate the effectiveness of beta derivative parameter and mathematical techniques. It is investigated that the amplitudes of soliton are increased with the increase of fractional beta derivative parameter. The obtained results would be very useful to examine and understand the physical issues in nonlinear optics, especially in twin-core couplers with optical metamaterials.

This paper explores the head-on collision between two-counter propagating positron acoustic solitons and double layers (DLs) in an unmagnetized collisionless plasma having mobile cold positrons fluid, immobile positive ions and ([Formula: see text])-distributed hot positrons, and hot electrons. By employing the extended Poincaré-Lighthill-Kuo method, the coupled Korteweg-de Vries (KdV), modified KdV (mKdV) and Gardner equations are derived to archive this goal. The effect of dimensionless parameters on the propagation characteristics of interacting KdV solitons (KdVSs), mKdV solitons (mKdVSs), Gardner solitons (GSs) and DLs are examined in detail by considering the limiting cases of ([Formula: see text])-distribution. It is noted that the interaction of GSs and DLs are reported for the first time. The outcomes might be comprehended and beneficial not only in space and astrophysical environments but also in laboratory studies.

This chapter deals with the ion-acoustic (IA) solitons, rogue waves (RWs), and double layers (DLs) in multi-component plasmas consisting of positively charged ion fluid and \((\alpha , q)\)-distributed electrons as well as positrons. The nonlinear evolution equations, namely, Korteweg–de Vries equation (KdVE), modified KdV equation (mKdVE), nonlinear Schrodinger equation (NLSE) and standard Gardner equation (SGE) along with their solutions are derived using the reductive perturbation method. It is examined that KdV, and SG solitons, RWs and DLs are supported both of positive and negative electrostatic potential structures, but mKdV supports only negative electrostatic potential structure by depending on the related plasma parameters. The outcomes presented may be very helpful in understanding the behavior of nonlinear coherent structures in many astrophysical and space environments and the future experimental studies in laboratory.

This article examines the essential phenomena of nonlinear ion-acoustic waves (IAWs) in the unmagnetized relativistic collisionless plasmas by considering generalized distributed electrons, Boltzmann distributed positrons, and relativistic ions having kinematic viscosity. The nonlinear electrostatic IA shock and solitary wave dynamics are conducted by new nonlinear evolution equations, such as Korteweg-de Vries (KdV)–Burgers-type Schamel equation and the well-known KdV-type Schamel equation, respectively, due to the involvement of superthermal electrons, having trapping efficiency and kinematic viscosity of ions. It is first time reported that the nonlinear IAWs are supported by only positive polarity shock structures and solitons for superthermal electrons, having trapping efficiency with and without the influence of kinematic viscosity of ions, respectively, in the relativistic plasmas.

Turbulent flows play a major role in many fields of science and industry. Noticeable attention is seen on turbulent flows of suspending fibers because of the sensitivity of the electrical, thermal, and mechanical properties of the connecting fiber composites to the spatial configuration and orientation of fibers. The involvement of fibers in the turbulent flow greatly affects the turbulent energy. It is more influenced when the turbulent flow occurs in a rotating system. The effect of fibers on the turbulent energy in the rotating frame must therefore be investigated. For turbulent energy with fiber suspension, a mathematical model can be built in a rotating system that is very important to enhance the quality of industrial goods. This paper, therefore, develops a mathematical model for turbulent energy motion in a rotating frame with a fiber suspension. The model was formulated using the averaging procedure. The momentum equation for incompressible and viscous fluid turbulent flow was considered to develop the model. The turbulent energy motion of the fiber suspensions was presented in the rotating frame in second-order correlation tensors, W i , j , S i , j , L i , j , F i , j , G i , j , D i , j , Q i , j , and H i , j , where all the tensors are the function of time, distance, and space coordinates.

The plane waves with their dynamical behaviors of (2+1)-dimensional nonlinear Schrodinger equation (NLSE) having beta derivative spatial-temporal evolution are investigated. In order to study such phenomena, NLSE is converted to a nonlinear ordinary differential equation with a planar dynamical system (PDS) by considering the variable wave transform and the properties of the beta derivative. By employing two distinct solution methods, namely, the auxiliary ordinary differential equation method and the extended simplest equation method, some more new general form of analytical solutions of NLSE are constructed. The effect of beta derivative parameter and obliqueness on several types of wave structures along with the phase portrait diagrams are reported. It is found that the PDS is not supported by any type of orbits for Θ = 45°. It is also confirmed from the obtained solutions that no plane waves are generated for Θ = 45°. The presented studies on bifurcation analysis and analytical solutions for (2+1)-dimensional NLSE would be very useful to understand the physical scenarios for Heisenberg models of ferromagnetic spin chain with magnetic interactions.

This work intends to focus on the dynamical behaviors for the discrete nonlinear electrical transmission line(NETL) with the presence of Caputo-Fabrizio fractional derivative. With the changes of some parameters for the local solution and the nonlocal solution of NETL, the results are analyzed based on the response of bifurcation diagrams. It is immersed that the stability state in the local derivative has converted to a durable, stable spiral node in Caputo-Fabrizio fractional derivatives, which is a notable outcome of this exertion.

This work deals with the interaction of complex short wave envelope and real long wave described by the coupled Schrodinger–Boussinesq equation with variable-coefficients and beta space fractional evolution. The above interacting wave phenomena are studied by evaluating new non-autonomous analytical solutions of the considered coupled equation. The solutions of the equations are obtained by modifying the auxiliary ordinary differential equation method and taking the conformable beta derivative properties into account. The behaviors of the resonance nonautonomous structures are discussed with physical insight for diverse consideration of the chance functions of time-dependent in the solutions.

Turbulent energy plays a vital role in science and industries. Fiber suspension in turbulent flows has received significant attention since the electrical, thermal, and mechanical characteristics of the relating fiber composites are tactful to the spatial configuration and orientation of fibers. Turbulent energy can be affected by the fibers passes through the turbulent flow. It is further influenced by the occurrence of dust or other particles. The impact of the fibers along with such particles needs to be studied. To analyze the impact, it is very important to model dusty fluid turbulent fiber motion which can be substantially utilized in science and industries. Therefore, this study aims to construct a model for dusty fluid turbulent energy of fiber suspensions. The energy equation for turbulent flow was taken into account to develop the model. The turbulent fiber motion for dusty fluid was derived in correlation tensors, where the tensors are the function of distance, time, and space coordinates.

In this work, the effects of plasma parameters on overtaking collisions of ion acoustic multi-shocks are studied in an unmagnetized collisionless plasma with positive and negative ions, and (&alpha, q)-distributed electrons. To investigate such phenomena, the reductive perturbation technique is implemented to derive the Burgers equation. The N-shock wave solution is determined for this equation by directly implementing the exponential function. The result reveals that both the amplitudes and thicknesses of overtaking collisions of N-shock wave compressive and rarefactive electrostatic potential are significantly modified with the influences of viscosity coefficients of positive and negative ions. In addition, the density ratios also play an essential role to the formation of overtaking collisions of N-shocks. It is observed from all theoretical and parametric investigations that the outcomes may be very useful in understanding the dynamical behavior of overtaking collisions of multi-shocks in various environments, especially the D- and F-regions of the Earth&rsquo, s ionosphere and the future experimental investigations in Q-machine laboratory plasmas.

The present work discusses the basic features of bifurcation properties with chaotic motion of oblique plane wave in the discrete nonlinear electrical transmission lines having conformable derivative evolution. The planar dynamical systems are obtained to understand such physical issues for this model. In addition, the analytical plane wave solutions of nonlinear mathematical leading equation are determined by using two different mathematical methods. Both methods are supported only by the shock plane wave solutions. The conditions for other types of oscillatory plane wave solutions are also determined. The effect of obliqueness on the bifurcation properties, chaotic motion and shock plane waves structures are focused graphically with discussions by considering both of locality and non-locality in the system. It is found that the parameter of non-local operator is only affected on the nonlinear shock wave phenomena, whereas all basic features of discrete nonlinear electrical transmission line are changed with the changes of obliqueness.

This work studies fractional temporal evolution of oblique resonant optical solitons in (3+1)-dimensions with Kerr- and parabolic-law nonlinearities. The generalized exp(−Φ(ξ))-expansion method along with the Khalil's conformable fractional derivatives is implemented to locate several forms of oblique resonant solitons. It is observed that obliqueness significantly modified resonant wave dynamics. The obtained results are very useful for understanding the dynamics of obliquely propagating resonant optical solitons and optical bullets.

This work investigates the basic features of Nonlinear Ion Acoustic Solitary waves (NIASWs) and their dynamical behaviours in an unmagnetized relativistic collisionless plasma system via the Schamel Korteweg-de Vries (SKdV) equation. Such plasma is composed by the generalized distributed electrons, Boltzmann distributed positrons and relativistic warm ions. The influences of plasma parameters on NIASWs and their dynamical behaviours are investigated by comparing 26−term expansion of relativistic Lorentz factor (RLF) with both of weakly (2−term expansion of RLF) and highly (3−term expansion of RLF) regimes. It is found that the 26−term expansion of RLF are significantly changed NIASWs instead of both weakly and highly relativistic regimes. Therefore, the theoretical results would be very useful for understanding the nature (amplitude, width, polarity, etc.) of wave dynamics not only in astrophysical and space environments but also in further laboratory studies, where the proposed plasma assumptions are existed.

This work investigates the interactions among ion acoustic (IA) single- and multi-soliton and their corresponding phase shifts in an unmagnetized plasma composed of degenerate electrons, positrons, and positive ions. Two-sided Korteweg-de Vries (KdV) equations are derived by employing the extended Poincare-Lighthill-Kuo (PLK) method for the stretched coordinates. The single- and multi-soliton solutions of the KdV equations are constructed by using the Hirota’s method. The phase shifts are determined for two-, four-, six-, and eight-IA scattering solitons. The effect of positron concentration on electrostatic IA resonances due to the interactions among solitons and their corresponding phase shifts are investigated.

The oblique plane waves with their dynamical behaviors for a (2+1)-dimensional nonlinear Schrödinger equation (NLSE) having beta derivative spatial-temporal evolution are investigated. In order to study such phenomena, NLSE is converted to a nonlinear ordinary differential equation with a planar dynamical system by considering the variable wave transform with obliqueness and the properties of the beta derivative. Some more new general forms of analytical solutions, like bright, dark, singular, and pure periodic solutions of NLSE are constructed by employing the auxiliary ordinary differential equation method and the extended simplest equation method. The effect of obliqueness and beta derivative parameter on several types of wave structures along with the phase portrait diagrams are reported by considering some special values of parameters for the existence of attained solutions. It is found that the planar dynamical system is not supported by any type of orbit for

The paper deals with the obliquely propagating wave solutions of fractional nonlinear evolution equations (NLEEs) arising in science and engineering. The conformable time fractional (2 + 1)-dimensional extended Zakharov-Kuzetsov equation (EZKE), coupled space-time fractional (2 + 1)-dimensional dispersive long wave equation (DLWE) and space-time fractional (2 + 1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation are considered to investigate such physical phenomena. The modified Kudryashov method along with the properties of conformable and modified Riemann-Liouville derivatives is employed to construct the oblique wave solutions of the considered equations. The obtained results may be useful for better understanding the nature of internal oblique propagating wave dynamics in ocean engineering. Keywords: Fractional nonlinear evolution equations, Conformable derivative, Modified kudryashov method, Oblique wave solutions, MSCClassification codes: 35E99, 35N05, 35Q40

This work investigates the obliquely propagating wave solutions of conformable time fractional (2 + 1)-dimensional extended Zakharov–Kuzetsov equation (eZKE) for understanding the behavior of physical issues in science and engineering, especially in magnetized plasmas. The generalized exp(− Φ(ξ))-expansion method along with the conformable fractional derivatives is employed to obtain various types of exact solutions to eZKE. The traveling wave solutions of eZKE are represented in the forms of hyperbolic, trigonometric and rational functions with physical as well as some additional free parameters. It is found that the obliqueness and physical parameters are significantly modified the wave dynamics taking the appropriate values of free and physical parameters.

We wish to point out a mistake in the paper [Indian J Phys 90, 603 (2016)], which partially changes a few results presented.

The Burgers equation is obtained to study the characteristics of nonlinear propagation of ionacoustic shock, singular kink, and periodic waves in weakly relativistic plasmas containing relativistic thermal ions, nonextensive distributed electrons, Boltzmann distributed positrons, and kinematic viscosity of ions using the well-known reductive perturbation technique. This equation is solved by employing the (G'/G)-expansion method taking unperturbed positron-to-electron concentration ratio, electron-to-positron temperature ratio, strength of electrons nonextensivity, ion kinematic viscosity, and weakly relativistic streaming factor. The influences of plasma parameters on nonlinear propagation of ion-acoustic shock, periodic, and singular kink waves are displayed graphically and the relevant physical explanations are described. It is found that these parameters extensively modify the shock structures excitation. The obtained results may be useful in understanding the features of small but finite amplitude localized relativistic ion-acoustic shock waves in an unmagnetized plasma system for some astrophysical compact objects and space plasmas.

This study deal with the oblique plane wave solutions with dynamical behaviours for (2 + 1)-dimensional resonant nonlinear Schrodinger equations having Bhom’s quantum potential with distinct law of nonlinearities (Kerr and parabolic law) and fractional temporal evolution. The considered equations are converted to solvable form by assuming conformable Khalil’s fractional derivatives. The bifurcation behaviors and chaotic motion for the existence of traveling waves are investigated by forming the planar dynamical system from the considered equations. The novel auxiliary ordinary differential equation method is therefore used to divulge several forms of plane wave solutions of these equations. It is investigated that the widths of resonant wave dynamics are significantly modified with the influence of obliqueness. The evaluated results may be very useful for better examining the resonant optical solitons in nonlinear dynamics because of obliqueness are existed in various nonlinear systems, specifically in optical bullets, Madelung fluids, etc. Keywords: Oblique resonant soliton, Resonant nonlinear Schrodinger equation, Dynamical behavior, The novel auxiliary ordinary differential equation method

This work investigates the obliquely propagating resonance wave phenomena described by coupled model equations with fractional temporal evolution arising in many branches of physics. The (2 + 1) -dimensional coupled nonlinear Schrodinger equation (NLSE), long wave-short wave resonance interaction (LSRI) equation and Maccari system (MS) having fractional time evolution are considered to study such physical phenomena. The modified Kudryashov method (mKM) with the properties of Khalil's conformable fractional derivatives is implemented to divulge scattered oblique wave solutions of considered equations. The effects of obliqueness and fractional parameter on obtained results are demonstrated graphically along with the physical descriptions. It is found that the resonance wave phenomena are changed with the increase of obliqueness as well as fractional parameter.

In this article, the oblique resonance wave phenomena are investigated by considering nonlinear coupled evolution equations with fractional time evolution. In order to investigate such physical phenomena arising in many branches of physics, the time fractional coupled (2 + 1)-dimensional nonlinear Schrodinger and long-short wave resonance interaction evolution equations are considered. The analytical solutions of considered equations are achieved by implementing the proposed auxiliary ordinary differential equation method along with the properties of Khali’s fractional derivatives. The obtained outcomes may be useful for better understanding the basic properties of internal oblique propagating wave dynamics in many branches of science and engineering.

In this article, the exp ( - Φ ( ξ ) ) -expansion method is modified for (3+1)-dimensional space–time coordinate system and successfully implemented to construct the new exact traveling wave solutions of the (3+1)-dimensional coupled Klein–Gordon–Zakharov equation. The solutions of this equation are expressed in terms of hyperbolic, trigonometric, exponential and rational functions. The results illustrate its effectiveness for solving nonlinear coupled partial differential equations arises in mathematical physics and engineering. The annihilation phenomena of the wave propagation in the x – y plane are also investigated. Furthermore, the three-dimensional surface plots due to the obtained solutions are also given to make the dynamics of the equation visible.

In this paper, the novel (G′/G)-expansion method is applied to construct exact travelling wave solutions of the cubic nonlinear Schrodinger equation. This technique is straightforward and simple to use, and gives more new general solutions than the other existing methods. Various types of solitary and periodic wave solutions of this equation are derived. The obtained results may be helpful to describe the wave propagation in soliton physics, such as soliton propagation in optical fibers, modulus instability in plasma physics, etc. and provided us the firm mathematical foundation in soliton physics or any varied instances. Furthermore, three-dimensional modules plot of the solutions are also given to visualize the dynamics of the equation.

In this article, the oblique resonant traveling waves and dynamical behaviors of (2+1)-dimensional Nonlinear Schrodinger equation along with dual-power law nonlinearity, and fractal conformable temporal evolution are reported. The considered equation is converted to an ordinary differential equation by taking the traveling variable wave transform and properties of Khalil's conformable derivative into account. The modified Kudryashov method is implemented to divulge the oblique resonant traveling wave of such an equation. It is found that the obliqueness is only affected on width, but not on amplitude and phase patriots of resonant nonlinear propagating wave dynamics. The research outcomes are very helpful for analyzing the obliquely propagating nonlinear resonant wave phenomena and their dynamical behaviors in several nonlinear systems having Madelung fluids and optical bullets.

This article investigates the dust ion acoustic multi-shock wave excitations in weakly relativistic multi-component plasma by assuming nonthermal, nonextensive electrons and positrons, relativistic ion fluid having kinetic viscosity and immobile dust. Burgers equation is derived to investigate such excitations by applying the reductive perturbation method. The exponential functions are directly implemented to determine the novel multi-shock wave solution of Burgers equation. The dust ion acoustic (DIA) multi-shock wave excitations are investigated systematically to reveal the effects of parameters, namely, viscosity coefficient of ions, positron to electron density ratio, immobile dust to electron density ratio, ion to electron temperature ratio, electron to positron temperature ratio, and relativistic streaming factor of ions in the presence of nonthermal, nonextensive, and concurrently acting nonthermal and nonextensive electrons as well as positrons. It is found that the amplitudes and widths of not only single, but also multi-shock wave compressive and rarefactive electrostatic potential structures are changed with the influence of all plasma parameters. The obtained results may be useful to analyze the nature of DIA multi-shock wave phenomena in various astrophysical as well as space environments (particularly, in pulsar relativistic winds with supernova ejecta) and future studies in plasma laboratory.

This work investigates the wave-wave interactions by considering two-sided time fractional Korteweg-de Vries equations (TFKdVEs). The generalized $\exp(-\Phi(\xi))$ -expansion method (GEEM) along with Khalil's fractional derivatives is employed to divulge several types of scattered wave solutions of TFKdVEs. It is found that the fractional parameters significantly modified the interaction of nonlinear wave dynamics. The results obtained may be useful for clarifications of the interaction between two waves not only in non-conservative fluid-filled elastic tubes but also in shallow water and plasma physics.

The head-on collision between positron acoustic solitary waves (PASWs) as well as the production of rogue waves (RWs) in homogeneous and PASWs in inhomogeneous unmagnetized plasma systems are investigated deriving the nonlinear evolution equations. The plasmas are composed of immobile positive ions, mobile cold and hot positrons, and hot electrons, where the hot positrons and hot electrons are assumed to follow the Kappa distributions. The evolution equations are derived using the appropriate coordinate transformation and the reductive perturbation technique. The effects of concentrations, kappa parameters of hot electrons and positrons, and temperature ratios on the characteristics of PASWs and RWs are examined. It is found that the kappa parameters and temperature ratios significantly modify phase shifts after head-on collisions and RWs in homogeneous as well as PASWs in inhomogeneous plasmas. The amplitudes of the PASWs in inhomogeneous plasmas are diminished with increasing kappa parameters, concentration and temperature ratios. Further, the amplitudes of RWs are reduced with increasing charged particles concentration, while it enhances with increasing kappa- and temperature parameters. Besides, the compressive and rarefactive solitons are produced at critical densities from KdV equation for hot and cold positrons, while the compressive solitons are only produced from mKdV equation for both in homogeneous and inhomogeneous plasmas.

The novel (G′/G)-expansion method is one of the powerful methods accredited at the present time for establishing exact traveling wave solutions to nonlinear evolution equations (NLEEs). In this article, the method has been implemented to find the traveling wave solutions to the positive Gardner-KP equation. The efficiency of this method for finding exact and traveling wave solutions has been demonstrated. The obtained solutions have been compared with the solution obtained by other methods. The solutions have also been demonstrated by figures. It has been shown that the method is straightforward and an effective tool for solving NLEES that occur in applied mathematics, mathematical physics, and engineering.

In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling wave solutions are established in the form of trigonometric, hyperbolic, exponential and rational functions with some free parameters. It is shown that this method is standard, effective and easily applicable mathematical tool for solving nonlinear partial differential equations arises in the field of mathematical physics and engineering.

The theoretical and numerical studies have been investigated on the nonlinear propagation of electrostatic ion-acoustic waves (IAWs) in an un-magnetized Thomas–Fermi plasma system consisting of electron, positrons, and positive ions for both of ultra-relativistic and non-relativistic degenerate electrons. Korteweg-de Vries (K-dV) equation is derived from the model equations by using the well-known reductive perturbation method. This equation is solved by employing the generalized Riccati equation mapping method. The hyperbolic functions type solutions to the K-dV equation are only considered for describing the effect of plasma parameters on the propagation of electrostatic IAWs for both of ultra-relativistic and non-relativistic degenerate electrons. The obtained results may be helpful in proper understanding the features of small but finite amplitude localized IAWs in degenerate plasmas and provide the mathematical foundation in plasma physics.

The theoretical and numerical studies have been investigated on nonlinear propagation of weakly relativistic ion acoustic solitary waves in an unmagnetized plasma system consisting of nonextensive electrons, positrons and relativistic thermal ions. To study the characteristics of nonlinear propagation of the three-component plasma system, the reductive perturbation technique has been applied to derive the Korteweg–de Vries equation, which divulges the soliton-like solitary wave solution. The ansatz method is employed to carry out the integration of this equation. The effects of nonextensive electrons, positrons and relativistic thermal ions on phase velocity, amplitude and width of soliton and electrostatic nonlinear propagation of weakly relativistic ion acoustic solitary waves have been discussed taking different plasma parameters into consideration. The obtained results can be useful in understanding the features of small amplitude localized relativistic ion acoustic solitary waves in an unmagnetized three-component plasma system for hard thermal photon production with relativistic heavy ions collision in quark–gluon plasma as well as for astrophysical plasmas.

The exp(?(?))-expansion method is applied to find exact traveling wave solutions to the (2+1)-dimensional Boussinesq equation which is an important equation in mathematical physics. The traveling wave solutions are expressed in terms of the exponential functions, the hyperbolic functions, the trigonometric functions and the rational functions. The procedure is simple, direct and constructive without the help of a computer algebra system. The applied method will be used in further works to establish more new solutions for other kinds of nonlinear evolution equations arising in mathematical physics and engineering.

In this article, the novel (G′/G)-expansion method is used to construct exact travelling wave solutions of the coupled nonlinear evolution equation. This technique is uncomplicated and simple to use, and gives more new general solutions than the other existing methods. Also, it is shown that the novel (G′/G)-expansion method, with the help of symbolic computation, provides a straightforward and vital mathematical tool for solving nonlinear evolution equations. For illustrating its effectiveness, we apply the novel (G′/G)-expansion method for finding the exact solutions of the (2 + 1)-dimensional coupled integrable nonlinear Maccari system.