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In this article, we will present new and more general traveling‐wave solutions for the Dodd–Bullough–Mikhailov equation based on the extended Fan sub‐equation method. The two‐dimensional affine sphere's intrinsic geometry is determined by Dodd–Bullough in the three‐dimensional unimodular affine space. The most significant accomplishment of our method is that we are able to provide all forms of the previously obtained solutions in one action, using at least four different approaches. As a result, the general elliptic equation with five parameters can be connected to other sub‐equations with three parameters that already exist, such as the Riccati equation, the first sort of elliptic equation, the auxiliary ordinary equation, and the modified Riccati equation. The extended Fan sub‐equation method is utilized to construct traveling‐wave solutions, solitary wave solutions, dark soliton solutions, and bell soliton solutions for the nonlinear Dodd–Bullough–Mikhailov equation. Mathematica is utilized to depict the dynamics of various wave structures in 3D, 2D, and contour formats, utilizing a specific set of parameters. Our graphical comparative analysis suggests that the employed method is both reliable and powerful for obtaining exact solutions of nonlinear evolution equations. The approach employed in this study may also be applicable in enhancing the understanding of numerous other complex physical phenomena. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation is one of the most important physical model in optical fiber theory for comprehension of the fluctuations of optical bullet development. In this study, the exact bullet solutions for the (3+1)-dimensional Schrödinger equation which demonstrate the bullet behaviours in optical fibers can be accumulated through the Sardar sub-equation method and the unified method. The applied strategies may retrieve several kinds of optical bullet solutions within one frameworks as well as is quite simple and reliable. Mathematica are utilised for describing the dynamics of different wave structures as 3D, 2D, and contour visualisations for a given set of parameters. As a result, we are able to develop a variety of travelling wave structures namely the periodic, singular and V shaped soliton wave solutions. The stability analysis for the derived results is analysed efficiently while the modulation instability for the governing model has also been studied to demonstrate the reliability of the research. The approaches implemented here works perfectly and can be extended to deal with many advanced models in contemporary areas of science and engineering. The solutions attain by using these techniques are robust, unique and straight forward and has applications in different fields of physics, engineering and mathematical science. Specially physical applications of these obtain results are in the transmission of data in optical fibers. We also add the graphics for the better understanding of the attain solutions behaviour. [ABSTRACT FROM AUTHOR]

The Schrödinger–Hirota equation is one of the most important models of contemporary physics which is popular throughout the broad fields of fluid movement as well as in the study of thick-water crests, liquid science, refractive optical components and so on. In this paper, we utilize the Hirota bilinear technique and the unified technique to attain various soliton solutions of the governing model analytically. These approaches are robust, powerful and unique also have many applications in different fields of mathematical physics. The solutions attained from these techniques are highly valuable and useful in various fields of sciences specially in the transmissions of optical fibers, also they give different behaviors including V-shaped and periodic soliton solution behavior. Further, the approaches applied here are not applied in this model previously. Therefore, ours is a new work, which summarizes its novelty. The 3D, 2D and contour plots are included to grasp the understanding of solutions' behavior. These findings are valuable in electronic communications such as elliptical circuits and in investigation of solitude controlling. [ABSTRACT FROM AUTHOR]

The (3+1)-dimensional hyperbolic nonlinear Gardner–Kadomtsov–Petviashvili (GKP) equation is a mathematical model that describes the propagation of nonlinear waves in a four-dimensional space-time. The GKP equation is used as a mathematical tool for studying the dynamics of nonlinear waves in a variety of physical systems including fluid dynamics, nonlinear optics, plasma physics. The GKP equation is a nonlinear partial differential equation (PDE) with mixed spatial and temporal derivatives. It exhibits various interesting phenomena, such as soliton solutions, wave interactions and wave turbulence depending on the values of the conditions. The improved F-expansion technique and generalized Kudryashov technique are successfully employed to evaluate the soliton solutions of governing model. The GKP equation supports various types of soliton solutions, such as bright solitons and dark solitons, depending on the specific parameters. The study of solitons in the GKP equation provides insights into the stability, propagation, and interaction of nonlinear waves in higher-dimensional spaces. The nonlinear terms in the GKP equation are responsible for wave–wave interactions. When multiple waves co-exist in a medium, they can interact with each other, leading to complex phenomena. The nature of wave interactions in the GKP equation depends on the specific values of the coefficients. The GKP equation has many applications in various areas of physics and mathematics. It has been used to model nonlinear waves in several fields and Bose–Einstein condensates. [ABSTRACT FROM AUTHOR]

One of the most important physical models for describing the dynamics of optical soliton propagation in the theory of optical fibers is the nonlinear Schrödinger equation (NLSE). Since there are so many uses for ultrafast signal routing systems and brief light pulses in telecommunication, optical soliton propagation in nonlinear optical fibers is a subject of intense current interest. The conventional Khater approach and the extended direct algebraic method are used in this research to study a generalized unstable space–time fractional NLS model in mathematical physics. This leads to the discovery of solutions for the bright, dark, periodic, rational and elliptic functions. To illustrate the physical nature of the nonlinear model, the contour plots in 3D, 2D and 2D are produced by assigning the appropriate values to the arbitrary constants. Additionally, consideration is given to the stability analysis of the solutions to the governing equation. In the current era of communications network technology and nonlinear optics, the applied strategy appears to be a more potent and effective way for producing precise optical solutions to a number of various modern models of recent generations. [ABSTRACT FROM AUTHOR]

In this article, the fractional Schrödinger–Hirota equation which is a generalization of the standard Schrödinger equation which, in particular, explain how soliton transmission behaves on fiber optic systems in physics when data is transmitted over long distances with a wide bandwidth. A collection of comprehensive soliton structures are developed to study the behaviour of the governing model with the aid of some efficient explicit strategies namely the exp (- ψ (ζ)) -expansion method and the Sardar sub-equation method. By transforming the original equation into a system of ordinary differential equations, it becomes possible to obtain explicit solutions with a high degree of accuracy. These solutions incorporate dark soliton and trigonometric function solutions, dark singular solition plane wave, singular solition, opposite singular solition, smooth, bell shaped, w-shaped periodic, bright, anti kink, singular bell shaped solitons and traveling wave structures. [ABSTRACT FROM AUTHOR]

The Hamiltonian amplitude equation that describes the transmissions of wire optical technology along wave propagation in the physics linked disciplines can be solved by utilizing the unique analytical and semi-analytical approaches namely the polynomial expansion method and the Adomian decomposition technique. Various types of soliton solutions are attained that express different behaviours including singular and v-shaped soliton solutions behaviour. The developed solutions are highly important in dealings the wave behaviour in mathematical physics related fields and in the transmission of optical fibers. The approaches applied in this paper are not utilized in given model so we apply these strategies to attain the soliton solutions. Hence we conclude the the results attained are unique, robust and newly made that proves novelty of the work. Furthermore, we study the stability plus the modulation instability of the given model. The 3D, contours, and 2D graphics are visualised to demonstrate the dynamical structures of the established results more efficiently. [ABSTRACT FROM AUTHOR]

The nonlinear (4+1)-dimensional Fokas equation (FE) has been demonstrated to be the integrable extension of the Kadomtsev–Petviashvili (KP) and Davey–Stewartson (DS) equations. In nonlinear wave theory, the governing model is one of the fundamental structures that explains the surface waves and interior waves in straits or channels with different depths and widths. In this study, the generalized unified approach, the generalized projective ricatti equation technique, and the new F/G-expansion technique are applied to investigate the higher dimensional nonlinear model analytically. As a result, several solutions are successfully achieved, including dark soliton, periodic type solitons, w-shaped soliton, and single-bell shaped solitons. Along with an explanation of their behavior, we also display a few of the equation's exact solutions graphically. The results demonstrate the effectiveness and simplicity of the approaches mentioned in this article, demonstrating their applicability to a wide range of additional nonlinear evolution issues in numerous scientific and technical disciplines. [ABSTRACT FROM AUTHOR]

The problem studied in the quadratic-cubic fractional nonlinear Schrödinger model in superfast fibers is the propagation of optical pulses in fiber optic systems. Optical pulses in fiber optic systems are important because they enable the transmission of information over long distances at high speeds with minimal loss and interference. We utilized the unified method and the extended simple equation method. These methods have been instrumental in obtaining important results in the study of the quadratic-cubic fractional nonlinear Schrödinger model in superfast fibers. These methods have allowed researchers to gain a deeper understanding of the dynamics of optical pulses and have contributed to advancements in fiber optic communication systems. Superfast fibers have revolutionized data transmission by providing high-speed and reliable connectivity. They enable faster internet connections, support emerging technologies, and have implications for various industries. While there are challenges associated with their deployment and maintenance, the benefits of superfast fibers outweigh these limitations. One of the key novelties of the this model is its ability to accurately describe the propagation of ultrafast optical pulses in highly nonlinear fibers. The model takes into account both quadratic and cubic nonlinear effects, which are known to play crucial roles in the dynamics of intense light pulses. [ABSTRACT FROM AUTHOR]

The most important physical structure that is assumed to illustrate the geometry of optical soliton replication in optical fiber theory is the nonlinear Schrödinger equation (NLSE). Optical soliton generation in nonlinear optical fibers is a topic of great contemporary interest because of the numerous applications of ultrafast signal routing systems and short light pulses in communications. This analysis's main goal is to create a large number of soliton solutions for the dynamical model using a variety of contemporary analytical methods. This paper studies different soliton solutions to the perturbed NLSE with Kerr law nonlinearity using two sets of two distinct integration strategies: the mapping approach and the unified auxiliary equation method. The majority of solutions have been found as Jacobi elliptic functions with limiting ellipticity modulus values. Solitons like dark, bright, optical, lonely, and others are also retrieved. We were able to create various single‐type solutions with the help of these strategies. As a result, there is a variety of optical, bell‐shaped, single periodic, and multi‐periodic solutions. In order to validate the computations, the stability of the acquired findings must also be proven. The study provides a highly stunning and suitable strategy for combining numerous exciting wave demonstrations for more advanced models of the present era. Furthermore, we can assert that the outcomes reported here are unique and novel. [ABSTRACT FROM AUTHOR]

In this study, the space-time fractional generalised reaction duffing model is investigated analytically, which is a generalization for a collection of prominent fractional models describing various key phenomenon in science and engineering. The governing equation is converted to a nonlinear ODE by the compatible travelling wave transform. The investigation established that for analysing nonlinear evolution equations of fractional order, the recommended approach is more effective and realistic. The findings are given extensively in rational forms of trigonometric function series or clearly in powers of specific trigonometric functions. A collection of bright, dark, periodic, and optical solitons is discovered. Mathematica is used to flourish the presence of some obtained solutions in 3D graphs with different fractional orders. The results show that the recommended methods are more practical and effective ways to illustrate the dynamics of several complex wave structures in modern science and technology. [ABSTRACT FROM AUTHOR]

In this paper, the new generalized stochastic fractional potential-KdV model is investigated analytically which describes the nonlinear optical solitons and photons propagating in electric circuits and multicomponent plasmas. The governing model is transformed into a bilinear form using the Hirota bilinear method. A variety of novel waves structures are developed in the form of lump waves, collisions between lump waves and periodic waves, collisions between lump waves or single and double-kink soliton solutions, collisions between lump, periodic, single and double-kink soliton solutions. Furthermore, the ( G ′ / G 2 ) expansion method and the Adomian decomposition method are employed to construct a number of new configurations for the governing dynamical model. To validate the scientific computations, a comparison study and stability analysis are carried out efficiently. Finally, to understand the dynamical behaviour of the the nonlinear fractional model, the 3D, 2D, and contour plots have also been demonstrated with the appropriate set of parameters. Depending upon the efficiency and the validity of the applied techniques, it is established that these strategies can also be applicable to many contemporary models. [ABSTRACT FROM AUTHOR]

In this study, we solve by involving conformable fractional derivative on time-fractional perturbed nonlinear Schrödinger model which describe the behaviour of soliton transmission on fibers optical system in physics specially in transmission of data over long distances with large bandwidth. There are two modern techniques are utilized for solving the suggested model namely the Extended hyperbolic function technique and the polynomial expansion technique. These techniques give unique, robust, and powerful solutions that are useful in many research areas. We attain different types of solutions that give unique behaviour of V shaped, singular solution, and periodic soliton solutions. These solutions are play a vital role in the soliton theory along with data transmission in optical fibers. Further we also discuss the stability of obtain solutions as well as the modulation instability of the governing NLSE. To grasp and better understanding of the solution behavior we include the 3D, contour and 2D graphics. [ABSTRACT FROM AUTHOR]

The shallow water equation, which describes fluid flow below a pressure surface, is the structure that is most frequently used for displaying free surface flows. This particular kind of approach takes into consideration the impact of gravity on fluids through the inclusion of a source term proportional to the product of the water level and ground slopes. The given model is converted into a bilinear form using the Hirota bilinear method. Which refers to the development of lump waves, collisions between lump waves and periodic waves, collisions between lump waves and single- and double-kink soliton solutions, and collisions between lump, periodic, and single- and double-kink soliton solutions. Furthermore, the exp (- ϱ (ς)) approach are applied to obtain several forms of innovative combinations for the governing dynamical fractional model. In addition, it has been confirmed that the established results are stable, and it has been helpful to validate the calculations. Moreover, multiple intriguing exact solutions are utilized to illustrate the physical nature of 3D, contour, and 2D graphs. The solution yields a set of bright, dark, periodic, rational, and elliptic function solutions. [ABSTRACT FROM AUTHOR]

In this article, the (2+1)-dimensional nonlinear hyperbolic Schrödinger model is studied analytically which specifies the proliferation of optics signal in single-mode fiber optics by using the unified technique, the exp(- δ (ζ) ) technique and the polynomial expansion technique. These techniques can give solutions of different kind that has application in physical mathematical fields, optical wave structure, and many other fields that are related to wave transmission. As compared to other approaches these used methodology are effective, unique, and easy to applicable for solving the complex models in Physics. The waveform in electromagnetic domains topics are discussed using this model as a governing equation. For a given set of pertinent parameters, the dynamics of various wave structures are visualised in 3D, 2D, and contour using Mathematica. These solutions exhibit singular periodic, multi-periodic, optical, singular and multiple bell-shaped, and μ-shaped solitons solutions as their behaviour. Further we study the stability and modulation instability (MI), also some plots are drawn for better understanding of solution. The solutions derived from the utilized techniques describe that these approches are easy, potent and uncomplicated, and these can be applicable for solving many other NLSE in Mathematical physics, and also in different field of natural sciences. The solution obtain are newly made because according to my best knowledge these methodology are not applied to this model in previous. [ABSTRACT FROM AUTHOR]

A broad spectrum of opportunities for ultrafast information processing and light pulses in communications has sparked a lot of interest in the field of wave propagation in nonlinear fibers. In this work, some new analytical soliton solutions to the perturbed nonlinear Schrödinger equation are constructed. To illustrate several soliton solutions for various parameter values, 3D simulations were carried out. Dark, bright, optical, solitary, and other solitons are also retrieved. We were able to create multiple single-type solutions using these methods. The observed results are especially intriguing since they give researchers a better computational tool for developing numerous travelling wave solutions to nonlinear equations that have recently appeared in diverse disciplines of science and engineering. A variety of optical, bell-shaped, singular, periodic, and multiple periodic solutions are produced as a result. The stability of the results is also proven in order to validate the computations. The study provides a highly spectacular and acceptable way to mix many fascinating wave displays for more complex current period models. We can also claim that the outcome we're talking about is fresh and original. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE), one of the most fundamental physical models for understanding the fluctuations of optical soliton development, plays a significant role to demonstrate the dynamics of optical fibers. Therefore, the wave propagation in nonlinear dispersive medium is a subject of considerable interest due to the wide range of possibilities for ultrafast data processing and light pulses in communications. In this article, the coupled space-time fractional NLSE is investigated which is used to describe the non-relativistic quantum mechanical behaviour. A collection of comprehensive soliton structures are developed to study the dynamics of the governing model with the aid of some efficient analytical strategies. These solutions incorporate dark soliton and trigonometric function solutions, singular solition, dark singular solition plane wave, singular solition, opposite singular solition, smooth, bell shaped, periodic, bright, anti kink, singular bell and traveling wave with darkness. The presence of some attained solutions are flourish in 3D graphs with various fractional orders by using Mathematica. The results which we obtained reveal that the suggested approaches are more convenient and productive techniques to depict the dynamics of numerous complex wave structure in contemporary areas of science and technology. [ABSTRACT FROM AUTHOR]

In optical fiber theory, the nonlinear Schrödinger equation (NLSE) is one of the most important physical models for describing the transmission of optical soliton growth. Due to the large range of possibilities for superfast signal processing and light pulses in communications, the wave propagation in nonlinear fibers is currently a topic of substantial curiosity. In this study, the third order generalized NLSE is investigated analytically by using the extended hyperbolic function technique, the improved F-expansion method, and the Adomian decomposition method, which has great importance in applied physics especially in the study optical fibers. The obtained solutions are newly made also they have the form of optical solitons, singular bell shaped, multi-bell shaped, and singular periodic solitons wave behavior and have applications in the optical fibers transmission, physics and many other scientific fields. The stability along with the modulation instability (MI) of the governing model has also been examined to validate the results. The nonlinear model properties have been illustrated using 3D, 2D, and contour plots with the appropriate set of parameters, which is demonstrated to visualize the physical structures more productively. Finally, it is concluded that similar strategies can also be implemented to study many contemporary models. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviour of the coupled Higgs equation (CHE) is explored analytically, which explains the interaction between mesons and electrons to characterize smooth changes in states of a particle. In this paper, a collection of efficient analytical procedures is employed to develop a variety of intriguing travelling wave structures such as bright, dark, singular, optical, bell shaped, and periodic solitons. Mathematica is used to display the dynamics of various wave shapes in 3D, 2D, and contour structures for a given set of parameters. Furthermore, the consistency of the obtained results is evaluated by validating the stability of the established results. The computational complexities and results show the approaches' lucidity, efficacy, and simplicity, implying their applicability to diverse sets of dynamic and static nonlinear equations governing evolutionary phenomena in computational physics, as well as other practical domains and numerous areas of research. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE) is one of the most important physical model in optical fiber theory for explaining changes in optical soliton growth. Due to the large range of opportunities for ultrafast signal routing systems and brief pulses of light in communications, optical soliton transmission in nonlinear fibers is currently a topic of significant interest for researchers. In this article, we study the dynamics of the generalised unstable NLSE which specified the time evolution of perturbations in barely steady or unstable media. A set of efficient analytical approaches are implemented to develop various interesting travelling wave structures such as bright, dark, singular, optical, bell shaped and periodic solitons solutions. Additionally, the stability of the obtained results is validated. Finally, the modulation instability of the governing model has also been investigated to present change effects that occurs as a result of the irregular refractive indices in an optical fiber, leading to distinct behaviour patterns. The methods as reported are applicable in clarity, effectiveness, and simplicity, establishing their applicability to various sets of dynamic and static nonlinear equations concerning evolutionary phenomena in computational physics, as well as other practical domains and numerous areas of research. [ABSTRACT FROM AUTHOR]

In the theory of optical fibres, the nonlinear Schrödinger equation is one of the most significant physical models for understanding the dynamics of optical soliton proliferation. Optical soliton propagation in nonlinear optical fibres is a topic of considerable contemporary interest due to the wide range of applications for ultrafast signal routing systems and short light pulses in communications. Our basic objective of this investigation is to use different modern analytical techniques to develop several soliton solutions for the resonance model. As an outcome, a variety of periodic solitons and singular bell shaped solitons solutions are obtained. Furthemore, the findings of stability as well as its modulation instabilities are effectively analyzed. The adapted schemes seem very impressive and competent avenue to obtained several advanced moving pulse shapes of different coincident pattern in the recent decade. [ABSTRACT FROM AUTHOR]

The Allen–Cahn equation (ACE), which has applications in solid-state physics, imaging, plasma physics, material science and other fields, is one of the most important models of the modern era for describing the dynamics of oil pollution, reaction-diffusion mechanisms, and the mechanics of crystalline solids. By using the (1 G ′ ) -expansion method (GEM) and the Bernoulli sub-ODE schemes, some new traveling wave solutions for the governing model are created in this study (BSODE). The reduced integrable ordinary differential equation is produced using the traveling wave hypothesis. To better understand their behavior, the 3D, contour, and 2D graphs are displayed for a number of fascinating exact solutions. Additionally, we use numerical simulation to confirm the stability of the derived analytical solutions. It results the propagation of temporal soliton for long time of simulation. These results will be used to explain physical phenomenon in crystalline solids and others fields. [ABSTRACT FROM AUTHOR]

The generalized Korteweg–de Vries–Zakharov–Kuznetsov model is one of the dominant nonlinear complex structures to exhibit the influence of magnetic fields on weak ion‐acoustic waves in plasma made up of cool and hot electrons. In this study, the nonlinear higher dimensional model is subjected to the extended F$$ F $$‐expansion approach and the exp(−ϕ(ζ))$$ \exp \left(-\phi \left(\zeta \right)\right) $$‐expansion strategy in order to discover some novel traveling waves solutions and other exact solutions. There have been several solutions discovered, such as bright, dark, periodic, bell‐shaped, and single bell‐shaped wave structures. Furthermore, the stability analysis of the model is also reported along with the dynamical visualization of the charismatic behavior of various solitons with the aid of 3D, 2D, and contour plots. The proposed techniques can also be implemented to analyze a wide range of nonlinear evolution models emerging in diverse disciplines of science and technology including mathematical physics and plasma physics. [ABSTRACT FROM AUTHOR]

In this paper, we study the perturbed nonlinear Schrödinger equation (P-NLSE) representing the propagation of waves and may be regarded as a nonlinear complicated physical model. We use the modified extended Tan hyperbolic function (METhF) approach and the ( G ′ G 2 ) -expansion approach to construct some new traveling wave structures. Several solutions have been found including dark soliton, periodic-type solitons, bell-shaped solitons, single bell-shaped solitons. We also show a graphical representation of a number of exact solutions to the dynamical model together with a description of their behavior. The proposed techniques can also be extended to various nonlinear evolution models in mathematical physics and engineering. [ABSTRACT FROM AUTHOR]

The Schamel–Korteweg-de Vries (S-KdV) model is used to predict the influence of surface for deep water in the presence of solitary waves. The aim of the study is to study the governing model analytically by employing the extended modified auxiliary equation mapping approach and the extended FAN sub-equation method. The 3D, 2D and contour plots are drawn to demonstrate the physical nature of the nonlinear model for a set of parameters. As a result, dark solitons, light solitons, solitary waves, periodic solitary waves, rational functions, and elliptic function solutions are established. Furthermore, the the developed results are verified with the aid of latest computing tool such as Mathematica or Maple. The applied strategy appears to be a more powerful and efficient scheme for achieving exact solutions to a number of diversified contemporary models of recent eras. [ABSTRACT FROM AUTHOR]

In this study, some new soliton solutions to the (3+1)-dimensional Kadomtsev–Petviashvili equation describing the propagation of weakly dispersive waves in plasma, have been constructed analytically. The observed results are very encouraging and provide a powerful tool for analyzing the analytical solutions to nonlinear models emerging in the current generation of science and technology. The 3D, 2D and contours plots are drawn to demonstrate the physical nature of the nonlinear model by setting appropriate values to the arbitrary constants. The reported structures are bright, explicit, periodic, and combined wave solutions, which have been successfully demonstrated. Such approach can help to develop a better understanding for the dynamical behaviour of nonlinear structures and may also be applied to a variety of other complex models of recent era. [ABSTRACT FROM AUTHOR]

In this work, the Lakshmanan-Porsezian-Daniel model is investigated which is the generalization of the non-linear Schrödinger model, to describes the dynamical behavior of optical solitons. The extended modified auxiliary equation mapping method is employed to develop some new exact solitary wave solutions to the complex model with the ker law, the parabolic law and the anti-cubic law nonlinearities. As a result, dark solitons, light solitons, singular solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions are established. In the current era of communications network technology and nonlinear optics, the applied strategy appears to be a more powerful and efficient approach for achieving exact optical solutions to a number of diversified contemporary models. [ABSTRACT FROM AUTHOR]

In this article, the modified Khater method is utilized to extract new analytical solutions to the cubic-quintic nonlinear non-paraxial pulse propagation model in a planar waveguide with Kerr nonlinearity, fifth-order nonlinearity, and spatiality dispersion due to non-paraxial effects. As a result, a variety of some new exact solutions are observed. Moreover, the three-dimensional shapes are visualized with the aid of the latest scientific tools. [ABSTRACT FROM AUTHOR]

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified. [ABSTRACT FROM AUTHOR]

The focus of this article is to find some new exact solutions to the M-fractional paraxial nonlinear Schrödinger equation with Kerr media by employing the modified simple equation method and the auxiliary equation method. A set of novel travelling wave solutions are observed such as bright, dark, periodic and optical solitons. Moreover, the physical interpretation of nonlinear waves would also be demonstrated with the aid of scientific computing. [ABSTRACT FROM AUTHOR]

This paper studies the new closed form solutions to (2+1)-dimensional Bogoyavlenskii system that describes interaction of a Riemann wave propagation. The extended Fan sub-equation technique is used to investigate some new traveling wave solutions to the higher-dimensional coupled model. The obtained closed form solutions are named as shock, kink, shock and periodic soliton solutions. Clearly, the outcomes of the study confirm the strength of the current approach. Moreover, the obtained results are helpful for the understanding of non-linear wave propagation and are of great interest to present-day scientists and can be employed to deal with more complex models arising in diverse disciplines of contemporary science. [ABSTRACT FROM AUTHOR]

In this study, a state-of-the-art approach tilted as the extended modified auxiliary equation mapping method is employed to observe some novel exact solutions to the generalized (1 + 1)-dimensional unstable space–time fractional nonlinear Schrödinger model. A variety of solitons are discovered namely, bright, dark, singular, combo, optical, singular optical and combo solitons in an elegant scheme. The existence conditions are also presented by choosing appropriate parameters to visualize the physical interpretation of solitons in three-dimensions with the aid of the latest computational tools. [ABSTRACT FROM AUTHOR]

In this paper, some new traveling wave solutions to the resonant nonlinear Schrödinger's equation (R-NLSE) with time-dependent coefficients are constructed. The well-known auxiliary equation method is applied to develop numerous interesting classes of nonlinearities, namely the Kerr law and parabolic law. Such approach provides an extensive mathematical tool to develop a family of traveling wave solutions such as bright, dark, singular and optical solutions to the nonlinear evolution model. Moreover, with the aid of symbolic computation the three-dimensional plot and contour plot have been carried out to demonstrate the dynamical behavior of the nonlinear complex model. [ABSTRACT FROM AUTHOR]

In this paper, the auxiliary expansion equation method is applied to compute the analytical wave solutions for (3 + 1)-dimensional Boussinesq and Kadomtsev–Petviashvili (KP) equations. A simple transformation is carried out to reduce the set of nonlinear partial differential equations (NPDEs) into ODEs. These obtained results hold numerous traveling wave solutions that are of key importance in elucidating some physical circumstance. [ABSTRACT FROM AUTHOR]

In this paper, the new optical wave solutions of truncated M-fractional perturbed Kundu–Eckhaus model with full non-linearity are obtained by utilizing the exp a function technique and modified extended tanh expansion function technique. The equation was independently introduced by Wiktor Eckhaus and by Anjan Kundu to model the propagation of waves in dispersive media. The model Kundu–Eckhaus equation is a general form of integrable system that is Gauge-equivalent to the mixed nonlinear Schrödinger equation. The solutions are in the form of dark soliton, bright soliton, singular solitons and other form of solutions. The gained solutions are helpful for the further development of concerned model. The obtained results have also been presented graphically in both two-dimensional and three-dimensional formats to discuss the dynamical features as well as the parametric dependence of the constructed solutions. The obtained solutions may be used for the propagation of the ultra-short femtosecond pulses and the rogue wave in an optical fiber. The study offers a highly spectacular and acceptable techniques to combine various intriguing wave demonstrations for more sophisticated models of the modern day. [ABSTRACT FROM AUTHOR]

The mixed localized wave solutions and their interaction have fascinated the vast age of plasma physics, fluid mechanics, and ocean dynamics, thanks to advances in nonlinear shallow water waves theory. In this paper, we conduct an analytical investigation of the (3+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt model. For a deeper understanding of the integrability characteristics of the governing model, the Hirota's bilinear form has been used to exhibit a variety of wave patterns, such as the Rogue wave solution and interactions between lump wave and periodic wave solutions. Additionally, by using the exp (− ϕ (ξ)) expansion method, certain novel travelling wave shapes are created. In order to understand the dynamics of the governing model, many exciting graphical representations are demonstrated at the conclusion in the form of 3D, 2D, and contour plots. As a result, we created unique bell-shaped solutions, bright soliton solutions, and periodic wave solutions. The discovered results are highly positive and can also be used to assess the characteristics of different complex dynamical models that have emerged in more modern science and engineering. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation is one of the significant nonlinear complex models describing the optical solitons in dispersive media. In this study, the (2+1)-dimensional chiral nonlinear Schrödinger model has been investigated analytically which is of key importantance in the field of fluid sciences. A variety of exclusive travelling waveform solutions have been established for the complex dynamical model by employing a set of eminent analytical approaches namely the extended modified auxiliary equation mapping method, the improved F -expansion method, and the unified method, respectively. We obtained different forms of solitary types solutions with the help of these technique. The outcomes are a set of bell-shaped, single periodic, optical, and multi-periodic solutions. Finally, the stability of the developed results is also established to validate the computations. The study provides a very spectacular and appropriate way to put together several interesting wave demonstrations for more complex models of current era. [ABSTRACT FROM AUTHOR]

The paper studies the single and combined optical solitons in the physical model of telecommunication system that describes the dynamics of optical solitons with quadratic–cubic nonlinearity in the mitigate internet bottleneck. This fractional temporal evolution is considerd with the coefficients of self-steeping, group velocity, inter-model and nonlinear dispersions. The extended Fan sub-equation method is used to investigate the families of single optical solitons (bright, dark and singular) for the different values of parameters. [ABSTRACT FROM AUTHOR]

The article studies the dynamics of optical single and combined solitons in monomode optical fibers which are described by higher order nonlinear Schrödinger equation. As a result, the bright, dark and singular optical 1-solitons and dark-singular, combo singular optical solitons have been extracted. The beauty of this research work is, one can obtain single and combined optical solitons using five parameters in a single integration technique. Moreover, the graphical representations are also depicted. [ABSTRACT FROM AUTHOR]

In this article, we have discussed the analytical solution to the higher order nonlinear Schrödinger equation representing the propagation of short light pulses in the monomode optical fibers. A simple transformation is introduced to convert the nonlinear partial differential equation into an ordinary differential equation. A verity of traveling solutions are obtained by employing the auxiliary equation method. The technique can also be functional to other sorts of nonlinear evolution equations in contemporary areas of research. [ABSTRACT FROM AUTHOR]

In this article, the single (bright, dark, singular) and combined (bright-dark, dark-singular) optical solitons have been extracted using extended fan sub-equation method. The corresponding integrability criteria, also known as constraint conditions, naturally emerge during the analysis of this model which is considered with the coefficients of group velocity dispersion and second order spatiotemporal dispersion. [ABSTRACT FROM AUTHOR]

The purpose of this study is to employ the Sine–Cosine expansion approach to produce some new sort of soliton solutions for the cubic–quintic nonlinear Helmholtz problem. The nonlinear complex model compensates for backward scattering effects that are overlooked in the more popular nonlinear Schrödinger equation. As a result, a number of novel traveling wave structures have been discovered. We also investigate the stability of solitary wave solutions for the governing model. Furthermore, the modulation instability is discussed by employing the standard linear-stability analysis. The 3D, contour and 2D graphs are visualized for several fascinating exact solutions to comprehend their behaviour. [ABSTRACT FROM AUTHOR]

This current research is about some new exact wave solutions to the generalized ( 1 + 1 )-dimensional unstable nonlinear Schrödinger equation with novel truncated M-fractional derivative. The obtained solutions may be applied in the demonstration of this model in some better way. The novel derivative operator is applied to study the aforementioned model. These solutions are in the form of dark, bright and other solutions with certain conditions. The achieved solutions are also verified by using the MATHEMATICA software. Modified integration method, extended Jacobi's elliptic expansion functions method (EJEEFM) is utilized to secure the aforesaid solutions. These exact solitons suggest that this method are effective, straight forward and reliable as compare to other methods. [ABSTRACT FROM AUTHOR]

In this study, the (2+1)-dimensional generalized Hirota–Satsuma–Ito model is studied extensively to investigate wave dynamics of shallow water. A set of state of the art analytical approaches are being employed to acquire several fascinating wave structures. We have established a comparison between exact and numerical solutions to determine the absolute error. Moreover, the modulation instability as well as the stability of the solutions are adequately discussed. The 3D, contour and 2D plots are displayed for several interesting exact solutions to understand their behaviour. These strategies are demonstrated as more strong, effective, and efficient in developing various new wave structures which have numerous applications in several mathematical physics domains. [ABSTRACT FROM AUTHOR]

In this paper, the auxiliary equation method is successfully applied to compute analytical solutions for (3+1)-dimensional generalised Kadomtsev-Petviashvili and (2+1)-dimensional Gardner-Kadomtsev-Petviashvili equations, by introducing simple transformations. These results hold numerous travelling wave solutions that are of key importance which provide a powerful mathematical tool for solving nonlinear wave equations in recent era of applied science and engineering. The method can also be extended to other nonlinear evolution models arising in contemporary physics. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical solutions of the perturbed nonlinear Schrödinger equation through the modified Khater method. This method is considered one of the most recent accurate analytical schemes in nonlinear evolution equations where it obtained many distinct forms of solutions of the considered model. The investigated model in this paper is an icon in quantum fields where it describes the wave function or state function of a quantum-mechanical system. The physical characterization of some obtained solutions in our study is explained through sketching them in two- and three-dimensional contour plots. The novelty of our study is clear by showing the matching between our solutions and those that have been constructed in previously published papers. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE) is one of the most important physical models for explaining the dynamics of optical soliton proliferation in optical fibre theory. Due to an extensive variety of applications for ultrafast signal routing systems and short light pulses in communications, optical soliton propagation in nonlinear fibres is a topic of substantial present interest. To overcome the ill-posedness of the unstable NLSE, a new term L x t is introduced which leads to the Hamiltonian amplitude equation. In this work, we study exact wave solutions to the nonlinear complex model by utilizing a set of analytical approaches namely as the exp (− ξ (δ)) method, the improved F-expansion technique, and the extended modified auxiliary equation mapping method. Numerous solutions, including singular periodic, multi-periodic, single bell-shaped, and multi-bell-shaped wave frameworks were found. The proposed plan of action is swift, strong, as well as offers the requirements needed to verify the origin of these solutions. Additionally, a dynamics representation of the fascinating behaviour of several solidarity using 2D, 3D, and contour graphs is provided, also we discuss the stability analysis and modulation instability of the proposed model. [ABSTRACT FROM AUTHOR]