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This study focuses on the propagation behavior of traveling wave solution in microcrystalline materials using the polynomial complete discriminant system method. By establishing a complete discriminant system, we systematically analyze the formation and evolution process of traveling wave solution in microcrystalline materials. Specifically, we apply the cubic polynomial extension to the strain wave equation to obtain more accurate analytical solutions. Additionally, two-dimensional, three-dimensional, and contour plots are generated to visually illustrate the characteristics of the obtained solutions, facilitating a more intuitive understanding of their physical significance. These findings not only help reveal the propagation mechanism of traveling wave solution but also provide a theoretical foundation for the application of microcrystalline materials.

The higher dimensional Fokas equation is the integrable expansion of the Davey–Stewartson and Kadomtsev–Petviashvili equations. In wave theory, the Fokas model plays a crucial role in explaining the physical phenomena of waves both inside and outside of water. The (4+1)-dimensional fractional-order Fokas equation is the subject of this article. Two effective approaches are employed to obtain the solutions for the considered equation: the generalized auxiliary equation technique and the G′/(bG′+G+a) technique. Several novel soliton solutions are obtained, including periodic solitary waves, bright solitons, and dark solitons. Various parametric values are employed to produce these new soliton waves at certain fractional order levels α. Furthermore, the bilinear version of the equation helps to develop its two-wave, three-wave, and multi-wave, as well as lump and rogue wave solutions. The properties of the solutions to the underlying problem are most effectively analyzed through the use of graphical representations. These outcomes and techniques can be used to study various fractional-order problems that emerge in wave theory, such as those in physics, hydraulics, optical technology, quantum mechanics, and plasma particles.

In this paper, by using the bifurcation theory for dynamical system, we construct traveling wave solutions of a high-order nonlinear Schrödinger equation with a quintic nonlinearity. Firstly, based on wave variables, the equation is transformed into an ordinary differential equation. Then, under the parameter conditions, we obtain the Hamiltonian system and phase portraits. Finally, traveling wave solutions which contains solitary, periodic and kink wave solutions are constructed by integrating along the homoclinic or heteroclinic orbits. In addition, by choosing appropriate values to parameters, different types of structures of solutions can be displayed graphically. Moreover, the computational work and it's figures show that this technique is influential and efficient.

In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the conformable extended Korteweg–de-Vries (KdV) model are investigated. First, the conformal fractional order extended KdV model is transformed into ordinary differential equation through traveling wave transformation. Second, two-dimensional (2D) planar dynamical system is presented and its chaotic behavior is studied by using the planar dynamical system method. Moreover, some three-dimensional (3D), 2D phase portraits and the Lyapunov exponent diagram are drawn. Finally, many meaningful solutions are constructed by using the complete discriminant system method, which include rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. In order to facilitate readers to see the impact of fractional order changes more intuitively, Maple software is used to draw 2D graphics, 3D graphics, density plots, contour plots, and comparison charts of some obtained solutions.

The aim of current work is to establish novel traveling wave solutions of the nonlinear Atangana conformable Klein - Gordon equation using a new extended direct algebraic technique. The Klein - Gordon equation is the relativistic state of the Schrödinger equation with a second - order time derivative and zero spin. Complex wave variable transformation is used to convert Atangana conformable nonlinear differential equation into an ordinary differential equation. Using the proposed technique based on Maple software structure, various types of solutions, such as, generalized trigonometric, generalized hyperbolic, and exponential functions, are established. When special parameteric values are considered for this method, solitary wave solutions can be obtained through other methods, such as the (G′G)-expansion method, the modified Kudryashov method, the sub-equation method, and so forth. A physical explanation is provided for the solutions under consideration to enhance comprehension of the physical phenomena resulting from the obtained solutions, provided that the physical parameters are set appropriately using 3D, 2D, and contour simulations. The results demonstrated that the new extended direct algebraic method provides a more potent mathematical tool for solving numerous more nonlinear partial differential equations with the aid of symbolic computation.

In this article, the dynamic behavior and solitary wave solutions of the Akbota equation were studied based on the analysis method of planar dynamic system. This method can not only analyze the dynamic behavior of a given equation, but also construct its solitary wave solution. Through traveling wave transformation, the Akbota equation can easily be transformed into an ordinary differential equation, and then into a two-dimensional dynamical system. By analyzing the two-dimensional dynamic system and its periodic disturbance system, planar phase portraits, three-dimensional phase portraits, Poincaré sections, and sensitivity analysis diagrams were drawn. Additionally, Lyapunov exponent portrait of a dynamical system with periodic disturbances was drawn using mathematical software. According to the maximum Lyapunov exponent portrait, it can be deduced whether the system is chaotic or stable. Solitary wave solutions of the Akbota equation are presented. Moreover, a visualization diagram and contour graphs of the solitary wave solutions are presented.

This paper investigates the pure-cubic complex Ginzburg–Landau equation (PC-CGLE) with different nonlinearities such as Kerr law, power law and so on. We get the dynamic systems and show that solitons and periodic solutions exist through the complete discrimination system for the polynomial method (CDSPM). To verify these conclusions, we construct the traveling wave solution via the CDSPM, and some new solutions are also built. The soliton stability and modulation instability with two types of nonlinearities are discussed. Finally, by adding perturbed terms to the dynamic system, we obtain the largest Lyapunov exponents and the phase diagrams of the equation, which proves there are chaotic behaviors in PC-CGLE. The results such as Gaussian soliton solutions and chaotic behavior for PC-CGLE are initially discovered in the present paper.

In this paper, we considered a delayed predator-prey model with stage structure and Beddington-DeAngelis type functional response. First, we analyzed the stability of the constant equilibrium points of the model by the linear stability method. Furthermore, we considered the existence of traveling wave solutions connecting the zero equilibrium point and the unique positive equilibrium point. Second, we transformed the existence of traveling wave solutions into the existence of fixed points of an operator by constructing suitable upper and lower solutions, and combined with the Schauder fixed point theorem, we gave the existence of fixed points and obtained the existence of traveling wave solutions of the model.

We explored the (3+1)-dimensional negative-order Korteweg-de Vries-alogero-Bogoyavlenskii-Schiff (KdV-CBS) equation, which develops the classical Korteweg-de Vries (KdV) equation and extends the contents of nonlinear partial differential equations. A traveling wave transformation is employed to transform the partial differential equation into a system of ordinary differential equations linked with a cubic polynomial. Utilizing the complete discriminant system for polynomial method, the roots of the cubic polynomial were classified. Through this approach, a series of exact solutions for the KdV-CBS equation were derived, encompassing rational function solutions, Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. These solutions not only simplified and expedited the process of solving the equation but also provide concrete and insightful expressions for phenomena such as optical solitons. Presenting these obtained solutions through 3D, 2D, and contour plots offers researchers a deeper understanding of the properties of the model and allows them to better grasp the physical characteristics associated with the studied model. This research not only provides a new perspective for the in-depth exploration of theoretical aspects but also offers valuable guidance for the practical application and advancement of related technologies.

The behaviors of wave solutions of the fractional nonlinear space-time Sharma-Tasso-Olever equation and the fractional nonlinear space-time Estevez-Mansfield-Clarkson equation, representing a fluid dynamics equation and a shallow water equation, respectively, can be obtained by transforming the fractional nonlinear space-time partial differential equations into nonlinear ordinary differential equations with the Jumarie's Riemann-Liouville fractional derivative, and solving for a finite series form of solution in the Riccati sub-equation method. The newly discovered traveling wave solutions took the forms of generalized triangular functions and generalized hyperbolic functions, which ultimately led to the assessing physical wave behaviors. These behaviors are manifested in kink and periodic waves, and they are separately depicted by 2-D, 3-D and contour graphs. In addition, the results we received were more diverse than previous solutions.

By using the Riccati-Bernoulli (RB) subsidiary ordinary differential equation method, we proposed to solve kink-type envelope solitary solutions, periodical wave solutions and exact traveling wave solutions for the coupled Higgs field (CHF) equation. We get many solutions by applying the Bäcklund transformations of the CHF equation. The proposed method is simple and efficient. In fact, we can deal with some other classes of nonlinear partial differential equations (NLPDEs) in this manner.

The main object of this paper is to study the traveling wave solutions of the fractional coupled Konopelchenko–Dubrovsky model by using the complete discriminant system method of polynomials. Firstly, the fractional coupled Konopelchenko–Dubrovsky model is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the trigonometric function solutions, rational function solutions, solitary wave solutions and the elliptic function solutions of the fractional coupled Konopelchenko–Dubrovsky model are derived by means of the polynomial complete discriminant system method. Moreover, a two-dimensional phase portrait is drawn. Finally, a 3D-diagram and a 2D-diagram of the fractional coupled Konopelchenko–Dubrovsky model are plotted in Maple 2022 software.

The main object of this paper is to study the bifurcation, chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion. A traveling wave transformation is used to simplified the perturbed stochastic nonlinear Schrödinger equation into ordinary differential equation. The dynamic behavior of two-dimensional planar dynamical systems and their perturbed systems are studied, and bifurcation, phase portrait, and Poincaré section are presented. Furthermore, traveling wave solutions included Jacobian function solutions, trigonometric function solutions and hyperbolic function solutions are constructed.

This article investigates the traveling wave solution of the parabolic law nonlinear stochastic dispersive Schrödinger-Hirota equation with multiplicative white noise, and its conclusions can be applied to optical communication. Firstly, the equation is transformed into a nonlinear ordinary differential equation through traveling wave transformation, and its planar dynamic system is studied. The phase portraits of the equation in various cases are drawn by Maple software. Secondly, the traveling wave solutions of this equation are obtained by using the complete discriminant system theory of cubic polynomials. Finally, we draw two-dimensional graphics, three-dimensional graphics, density plots and contour plots for the module length of some solutions under specific parameters. In addition, the above research results can help readers have a more comprehensive and clear understanding of this equation and its application.

In this paper, the traveling wave solutions of the model of magneto-optic waveguide with anti-cubic nonlinearity are obtained by using the complete discrimination system for polynomial method, including singular solutions, solitary wave solutions,and double periodic solutions. And under specific parameter conditions, three types of optical wave patterns are obtained to visualize the model and demonstrate their accurate physical behavior. Then the dynamic properties of Schrödinger equation in magneto-optic waveguide with anti-cubic nonlinearity are analyzed, the existence of periodic and solitary solutions is proved based on the bifurcation method. Also the Hamiltonian properties and the classification of its equilibrium points are obtained. In final, we analyze the chaotic behavior of the model under some external perturbations.

This study presents a large family of the traveling wave solutions to the two fourth-order nonlinear partial differential equations utilizing the Riccati-Bernoulli sub-ODE method. In this method, utilizing a traveling wave transformation with the aid of the Riccati-Bernoulli equation, the fourth-order equation can be transformed into a set of algebraic equations. Solving the set of algebraic equations, we acquire the novel exact solutions of the integrable fourth-order equations presented in this research paper. The physical interpretation of the nonlinear models are also detailed through the exact solutions, which demonstrate the effectiveness of the presented method.The Bäcklund transformation can produce an infinite sequence of solutions of the given two fourth-order nonlinear partial differential equations. Finally, 3D graphs of some derived solutions in this paper are depicted through suitable parameter values.

In this paper, we approximate traveling wave solutions via artificial neural networks. Finding traveling wave solutions can be interpreted as a forward-inverse problem that solves a differential equation without knowing the exact speed. In general, we require additional restrictions to ensure the uniqueness of traveling wave solutions that satisfy boundary and initial conditions. This paper is based on the theoretical results that the bistable three-species competition system has a unique traveling wave solution on the premise of the monotonicity of the solution. Since the original monotonic neural networks are not smooth functions, they are not suitable for representing solutions of differential equations. We propose a method of approximating a monotone solution via a neural network representing a primitive function of another positive function. In the numerical integration, the operator learning-based neural network resolved the issue of differentiability by replacing the quadrature rule. We also provide theoretical results that a small training loss implies a convergence to a real solution. The set of functions neural networks can represent is dense in the solution space, so the results suggest the convergence of neural networks with appropriate training. We validate that the proposed method works successfully for the cases where the wave speed is identical to zero. Our monotonic neural network achieves a small error, suggesting that an accurate speed and solution can be estimated when the sign of wave speed is known.

In this paper, the traveling wave solution of the Fokas system which represents the irregular pulse propagation in monomode optical fibers is studied by using the complete discriminant system method of polynomials. Firstly, the Fokas system is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the Jacobian function solutions, the trigonometric function solutions, the hyperbolic function solutions and the rational function solution of Fokas system are obtained by using the complete discriminant system method of polynomials. Finally, in order to show the propagation of Fokas system in monomode optical fibers, three-dimensional diagram, two-dimensional diagram, contour plot and density plot of some solutions are drawn by using Maple software.

This article mainly studies the new traveling wave solutions of the stochastic modified Korteweg–de Vries equation with multiplicative noise. The traveling wave solutions in the form of hyperbolic function, trigonometric function, rational function and Jacobi elliptic function are obtained by transforming the modified Korteweg–de Vries equation into an ordinary differential equation through traveling wave transformation, and combining it with the complete discriminant system of polynomials. This article fully applies visual analysis technology and provides graphical representations of the obtained solutions in multiple dimensions such as phase maps, sensitivity maps, 3D maps and 2D maps, providing an intuitive and convenient channel for people to further understand the physical characteristics of the solutions for the stochastic modified Korteweg–de Vries equation. This article provides a rich graphical analysis of the solutions to the stochastic modified Korteweg–de Vries equation from multiple perspectives, and provides a fast and efficient solution method for the exact solution of the equation. The research method in this paper can also be used to study nonlinear partial differential equation in other fields.

This article investigates the traveling wave solutions of the stochastic Davey–Stewartson equations based on the theory of the complete discriminant system of quadratic polynomials. Firstly, the stochastic Davey–Stewartson equations are transformed into the nonlinear ordinary differential equations through traveling wave transformation. By using Maple software, some two-dimensional and three-dimensional phase portraits are drawn. Secondly, according to the classification of the roots of quadratic polynomial equation, new traveling wave solutions of the stochastic Davey–Stewartson equations are obtained by using the complete discriminant system, which include rational function solutions, triangle function solutions and the Jacobi elliptic function solutions. Finally, three-dimensional-surface plots and two-dimensional-shape plots of some obtained solutions are drawn by using Maple software.

In this paper, the stochastic space-fractional long-short-wave interaction system (SF-LSWIS) with multiplicative white noise is considered. The stochastic exact solutions, including triangular function solutions, hyperbolic function solutions, and Jacobi elliptic function solutions, are obtained via the complete discriminant system for polynomial method. For this purpose, the traveling wave transformation is applied to carry out the ordinary differential equation of this model, which is also converted to a dynamical system. Thereafter, qualitative behavior and bifurcation of the phase portraits of the dynamical system are investigated using the qualitative theory of dynamical systems. The chaotic behaviors of the system considering external periodic perturbation are studied. Furthermore, the 3D surface, graphs of contour plots, and level curves of some exact solutions are depicted by choosing proper parameter values. The influence and effect of fractional derivatives and noise strength on the solutions are also studied.

The paper mainly studies the traveling wave solutions, phase portrait and chaotic pattern of the generalized fractional stochastic Schrödinger equations forced by multiplicative Brownian motion. By using the polynomial complete discrimination method, the hyperbolic functions, rational functions and Jacobi elliptic function solutions are obtained. Moreover, the physics properties of the obtained solutions are shown through three-dimensional, two-dimensional and contour graphs. In order to help better analyze dynamic behavior, phase diagrams, chaos behavior and sensitivity are plotted by using Maple software.

To provide insights into the spreading speed and propagation dynamics of viruses within a host, in this paper, we investigate the traveling wave solutions and minimal wave speed for a degenerate viral infection dynamical model with a nonlocal dispersal operator and saturated incidence rate. It is found that the minimal wave speed c∗ is the threshold that determines the existence of traveling wave solutions. The existence of traveling fronts connecting a virus-free steady state and a positive steady state with wave speed c≥c∗ is established by using Schauder’s fixed-point theorem, limiting arguments, and the Lyapunov functional. The nonexistence of traveling fronts for c

We derive a reaction–diffusion model with time-delayed nonlocal effects to study an epidemic’s spatial spread numerically. The model describes infected individuals in the latent period using a structured model with diffusion. The epidemic model assumes that infectious individuals are subject to containment measures. To simulate the model in two-dimensional space, we use the continuous Runge–Kutta method of the fourth order and the discrete Runge–Kutta method of the third order with six stages. The numerical results admit the existence of traveling wave solutions for the proposed model. We use the COVID-19 epidemic to conduct numerical experiments and investigate the minimal speed of spread of the traveling wave front. The minimal spreading speeds of COVID-19 are found and discussed. Also, we assess the power of containment measures to contain the epidemic. The results depict a clear drop in the spreading speed of the traveling wave front after applying containment measures to at-risk populations.

The Bogoyavlenskii equation is used to describe some kinds of waves on the sea surface and discussed by many researchers. Recently, the $ G'/G^2 $ method and simplified $ \tan(\frac{\phi(\xi)}{2}) $ method are introduced to find novel solutions to differential equations. To the best of our knowledge, the Bogoyavlenskii equation has not been investigated by these two methods. In this article, we applied these two methods to the Bogoyavlenskii equation in order to obtain the novel exact traveling wave solutions. Consequently, we found that some new rational functions, trigonometric functions, and hyperbolic functions can be the traveling wave solutions of this equation. Some of these solutions we obtained have not been reported in the former literature. Through comparison, we see that the two methods are more effective than the previous methods for this equation. In order to make these solutions more obvious, we draw some 3D and 2D plots of them.

Inspired by the extensive applicability of solitons in nonlinear optics, advanced telecommunication industry, trans-continental, and trans-oceanic systems, coupled with the copious functionality of viscoelastic models in highway engineering pavement theory, civil engineering, and solid mechanics, unique closed-form solutions have been obtained for a highly nonlinear model built on a highly nonlinear partial differential equation, which describes the dynamics of the incompressible viscoelastic Kelvin–Voigt fluid viz: the modified (1＋1) dimension Oskolkov equation. Abundant solitary and traveling wave solutions to the prototypical Kelvin–Voigt incompressible model have been elucidated and put forward by means of the hyperbolic tangent method using the tangent function as an ansatz on the transformed nonlinear ordinary differential equation, thus, depicted breather, cusp, rouge, compacton, and kink waveforms via graphical illustrations for arbitrary intervals x∈a1,a2, t∈b1,b2, a1,a2,b1,b2∈Z.This present research yields new traveling wave profiles and structures with holomorphic terms, precise with minimized computational work, capturing all solitonic features of the Kelvin–Voigt model, contrary to the existing results with cumbersome and longwinded solution profiles, thus, the present research novelty established. The dynamical analyses, the mathematical, and solution framework presented here will be of substantial benefit to the geosciences, ocean, plasma, and, material sciences, civil engineering, nanotechnology, signaling, and signal processing to effectively tackle and convey theoretical explorations on cogent equations arising in these afore-mentioned fields. Notably, the impact of the diffusive parameter Ω

In this paper, the coupled Kundu–Mukherjee–Naskar equation is considered, which is usually used to describe the oceanic rogue waves, hole waves and Bragg grating fibers. Firstly, the traveling wave transformation is applied to convert the coupled Kundu–Mukherjee–Naskar equation into two-dimensional planar dynamic system. Secondly, bifurcation and two-dimensional phase portrait of the coupled Kundu–Mukherjee–Naskar equation are discussed by using the plane dynamics system. What is more, after adding periodic perturbation term, two-dimensional phase portrait, three-dimensional phase portrait, Poincaré section and sensitivity of its perturbed system are plotted by the Maple software. Finally, the optical soliton solutions of the coupled Kundu–Mukherjee–Naskar equation are obtained by using planar dynamical system method.

This study employed the planar dynamics system method to investigate the chaotic behavior and traveling wave solution of the fractional stochastic Zakharov system arising in plasma physics. Through the analysis of the phase portraits, the sensitivity of the dynamic system has been examined and various new exact solutions for traveling waves have been constructed. These obtained solutions are conducive to further understanding the propagation of plasma waves. Furthermore, an analysis of the impact of fractional derivative and random noise on the solutions is done. The chaotic behavior of dynamic systems is examined by time series, Poincaŕe sections, and 2D and 3D phase portraits. Finally, the comparative investigation reveals that the characteristics of the solutions are significantly impacted by both the fractional derivative and random noise. The research methodologies and results are efficacious and can be extended to the exploration of more complex phenomena.

In this paper, we investigate the traveling wave solutions to a cubic predator-prey diffusion model with stage structure for the prey. Firstly, using the upper and lower solutions method we prove the existence and non-existence of weak traveling wave solutions. Furthermore, we prove that the weak traveling wave solutions are actually traveling wave solutions under additional conditions by using Lyapunov function method and LaSalle's invariance principle.

This article describes the construction of optical solitons and single traveling wave solutions of Biswas-Arshed equation with the beta time derivative. By using the polynomial complete discriminant system method, a series of traveling wave solutions are constructed, including the rational function solutions, Jacobian elliptic function solutions, hyperbolic function solutions, trigonometric function solutions and inverse trigonometric function solutions. The conclusions of this paper comprise some new and different solutions that cannot be found in existing literature. Using the mathematic software Maple, the 3D and 2D graphs of the obtained traveling wave solutions were also developed. It is worth noting that these traveling wave solutions may motivate us to explore new phenomena which may be appear in optical fiber propagation theory.

The formation of grout sediment in the pores of loose rock increases the water resistance of the soil and strengthens the foundation. A one-dimensional model of filtration in a porous medium considers the particles transport by the flow of a carrier fluid and the deposition of particles on the framework of a porous medium. The purpose of the work is to study the concentrations of suspended and settled particles of a suspension over a long time. Exact and asymptotic methods are used to obtain a solution to the model. The exact solution is presented in an implicit integral form. A set of solutions in the form of traveling waves with an arbitrary initial condition and their asymptotics are constructed. For the exact solution, an explicit second-order asymptotic solution for a long time is obtained as an expansion in decreasing exponents. Comparison of the asymptotic solution with the traveling waves makes it possible to choose a single traveling wave corresponding to the exact solution. The closeness of the traveling wave to the exact solution of the filtration model is verified numerically. The traveling wave found determines the explicit asymptotics of the concentration of deposited particles for a long time.

The formation of grout sediment in the pores of loose rock increases the water resistance of the soil and strengthens the foundation. A one-dimensional model of filtration in a porous medium considers the particles transport by the flow of a carrier fluid and the deposition of particles on the framework of a porous medium. The purpose of the work is to study the concentrations of suspended and settled particles of a suspension over a long time. Exact and asymptotic methods are used to obtain a solution to the model. The exact solution is presented in an implicit integral form. A set of solutions in the form of traveling waves with an arbitrary initial condition and their asymptotics are constructed. For the exact solution, an explicit second-order asymptotic solution for a long time is obtained as an expansion in decreasing exponents. Comparison of the asymptotic solution with the traveling waves makes it possible to choose a single traveling wave corresponding to the exact solution. The closeness of the traveling wave to the exact solution of the filtration model is verified numerically. The traveling wave found determines the explicit asymptotics of the concentration of deposited particles for a long time.

The main aim of this article is to focus on the dynamics and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense. First, the considered model is converted to the nonlinear ordinary differential equations by using traveling wave transformation. Secondly, two-dimensional phase portrait of the nonlinear ordinary differential equation and its periodic perturbation system are drawn by using Maple software. Finally, the traveling wave solutions of the investigated equation are obtained via the planar dynamic system method. Moreover, three-dimensional graphs of some obtained solutions are drawn.

Analytical solutions are the most effective tools for understanding phenomena modeled with differential equations. On the other hand, analytical solvers may be limited in their ability to handle complex problems and we are forced to rely on approximate and numerical methods. Through the investigation of two types of fractional operators, this article aims to investigate both approaches to studying the interactive dynamics of an eco-epidemiological diffusion model. We consider two differential operators in our main model: local beta time and non-local Riesz space fractional derivative. Firstly, we propose novel classes of soliton wave solutions for the local time fractional version of the system using two efficient analytical methods. In the second part of the paper, we consider the case where the second-order space derivative operator in the problem is of the type of Riesz fractional operator that is non-local. A finite difference scheme is the main idea behind discretizing this fractional system. Throughout this article, we highlight the validity and efficiency of both analytical and numerical perspectives. The results can be examined by epidemiologists for a more comprehensive interpretation.

This research aims to look into alternative unique solitary wave solutions to the perturbed Chen–Lee–Liu (CLL) equation to explain the kinetic and physical aspects of an optical fiber pulse. The perturbed CLL equation is one of the most well-known icon models derived from the well-known Schrödinger equation. Two different analytical methodologies are used to develop novel solitary wave solutions. These responses are then thoroughly evaluated using a well-known numerical approach to establish their underlying veracity. Various graphics are used to show the pulse wave analysis process and outcomes in an optical cable. We show how scientifically unique the study is by explaining the comparison figure that was made from our data and has been used in many recent research studies

In this article, (2+1)-dimensional Ito equation that models waves motion on shallow water surfaces is analyzed for exact analytic solutions. Two reliable techniques involving the simplest equation and modified simplest equation algorithms are utilized to find exact solutions of the considered equation involving bright solitons, singular periodic solitons, and singular bright solitons. These solutions are also described graphically while taking suitable values of free parameters. The applied algorithms are effective and convenient in handling the solution process for Ito equation that appears in many phenomena.

In Zhang et al. (2007) and Zhang (2021) we constructed all single-peak traveling wave solutions of the Camassa–Holm equation including some explicit solutions. In general it is a challenge to construct exact multi-peak traveling wave solutions. As an example a periodic traveling wave (or wavetrain), a special type of spatiotemporal oscillation that is a periodic function of both space and time, plays a fundamental role in many mathematical equations such as shallow water wave equations. In this paper we will construct some new exact periodic traveling wave solutions of the Camassa–Holm equation.

This paper considers the generalized Davey-Stewartson equation that is used to investigate the dynamics of wave propagation in water of finite depth under the effects of gravity force and surface tension. The model is considered in the presence of full nonlinearity. The main objective of this paper is to extract soliton solutions of the generalized Davey-Stewartson equation. Three state-of-the-art integration schemes, namely exp(-Φ(ξ))-expansion method, the first integral method and the Sine-Gordon expansion method have been employed for obtaining the desired soliton solutions. The proposed methods successfully attain different structures of explicit solutions such as bright, dark, singular, rational and periodic solitary wave solutions. All the newly found solutions are discussed with their existence criteria. The 2D and 3D portraits are also shown for some of the reported solutions.

In this article, we transform the (1 + 1)-dimensional non-linear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and the (2 + 1)-dimensional cubic Klein Gordon (cKG) equation, which are the non-linear partial differential equations, into the non-linear ordinary differential equations by using the traveling wave transformation and solve these solutions with the simple equation method (SEM) with the Bernoulli equation. Two classes of exact explicit solutions-hyperbolic and trigonometric solutions of the associated NLEEs are characterized with some free parameters; we obtain the kink waves and periodic waves.

In the paper, the nonlinear variable-coefficient fifth-order Schrödinger (NLVS) equation is researched. The NLVS equation is an integrable equation, which can be described the spreading of ultrashort pulses in an inhomogeneous optical fiber. Firstly, by using the modified traveling wave transformation, the NLVS equation is changed into an ordinary equation. Secondly, by the Jacobian elliptic function expansion method for the ordinary equation, we obtain the exact solutions for the ordinary equation, and then, we obtain the exact solutions to the NLVS equation. These solutions mainly include three types: Jacobi elliptic function solutions, hyperbolic function solutions, and triangular function solutions. Finally, according to the special parameters, we show the figures of the exact solutions.

In this paper, we investigate the conformable time fractional Burgers type equations. First, we construct the explicit solutions of Riccati equation by means of modified tanh function method and modified extended exp-function method respectively. In addition, based on the formulas obtained above, the traveling wave solutions of conformable time fractional Burgers equation and (2+1)-dimensional generalized conformable time fractional Burgers equations are established applying functional separation variables method. Furthermore, the three-dimensional diagrams of the obtained exact solutions are presented for the purpose of visualization.

The exact traveling wave solutions to coupled KdV equations with variable coefficients are obtained via the use of quadratic Jacobi’s elliptic function expansion. The presented coupled KdV equations have a more general form than those studied in the literature. Nine couples of quadratic Jacobi’s elliptic function solutions are found. Each couple of traveling wave solutions is symmetric in mathematical form. In the limit cases m→1, these periodic solutions degenerate as the corresponding soliton solutions. After the simple parameter substitution, the trigonometric function solutions are also obtained.

This paper is devoted to studying the existence and nonexistence of traveling wave solution for a nonlocal dispersal delayed predator-prey system with the Beddington-DeAngelis functional response and harvesting. By constructing the suitable upper-lower solutions and applying Schauder's fixed point theorem, we show that there exists a positive constant $ c^* $ such that the system possesses a traveling wave solution for any given $ c > c^* $. Moreover, the asymptotic behavior of traveling wave solution at infinity is obtained by the contracting rectangles method. The existence of traveling wave solution for $ c = c^* $ is established by means of Corduneanu's theorem. The nonexistence of traveling wave solution in the case of $ c < c^* $ is also discussed.

Abstract In this paper we investigate the existence of traveling wave for a one-dimensional reaction diffusion system. We show that this system has a unique translation traveling wave solution.

The non-linear space-time fractional Estevez-Mansfield-Clarkson (EMC) equation and the non-linear space-time fractional Ablowitz-Kaup-Newell-Segur (AKNS) equation showed the motion of waves in the shallow water equation and the optical fiber equation, respectively. The process used to solve these equations is to transform the non-linear fractional partial differential equations (PDEs) into the non-linear ordinary differential equations by using the Jumarie's Riemann-Liouville derivative and setting the solution in the finite series combined with the simple equation (SE) method with the Bernoulli equation. The new traveling wave solutions were the exponential functions resulting in the physical wave effects are produced in the form of kink waves and represented by the two-dimensional graph, three-dimensional graph, and contour graph. In addition, the comparison of the solutions revealed that the new solutions have a more convenient and easier format.

In this article, the dynamics of modified nonlinear waves for the generalized breaking soliton (gBS) equation have been explored by using analytical and numerical approaches. Firstly, the Tanh method is used to derive the solitonic solutions of the considered equation and their graphical illustrations for different values of parameters have been discussed. By using the proposed strategy, a variety of solutions can be found. Furthermore, the Galilean transformation is used to investigate the bifurcation behavior of the model. All possible phase portraits have been drawn and examined in detail. More specifically, it is shown that under defined parametric restrictions, the model has a straightforward specific value. At each fixed point, the existence of periodic points and bifurcation analysis is also confirmed. To clarify theoretical results, detailed numerical simulations are offered. Then by introducing perturbed term, the periodic and quasi-periodic behaviors of the discussed equation for various initial values are reported. Different periodic solutions can also be found in this parameter zone. Furthermore, it is reported that discussed equation is multi-stable for different values of parameters. Finally, sensitivity analysis for different initial value problems has been elaborated.

Mathematical modeling for cancerous disease has attracted increasing attention from the researchers around the world. Being an effective tool, it helps to describe the processes that happen to the tumour as the diverse treatment scenarios. In this paper, a density-dependent reaction-diffusion equation is applied to the most aggressive type of brain cancer, Glioblastoma multiforme. The model contains the terms responsible for the growth, migration and proliferation of the malignant tumour. The traveling wave solution used is justified by stability analysis. Numerical simulation of the model is provided and the results are compared with the experimental data obtained from the reference papers.

This article concerns traveling wave solutions of the fully parabolic Keller-Segel chemotaxis system with logistic source, $$\displaylines{ u_t=\Delta u -\chi\nabla\cdot(u\nabla v)+u(a-bu),\quad x\in\mathbb{R}^N,\cr \tau v_t=\Delta v-\lambda v +\mu u,\quad x\in\mathbb{R}^N, }$$ where $\chi, \mu,\lambda,a,b$ are positive numbers, and $\tau\ge 0$. Among others, it is proved that if $b>2\chi\mu$ and $\tau \geq \frac{1}{2}(1-\frac{\lambda}{a})_{+}$, then for every $c\ge 2\sqrt{a}$, this system has a traveling wave solution $(u,v)(t,x)=(U^{\tau,c}(x\cdot\xi-ct),V^{\tau,c}(x\cdot\xi-ct))$ (for all $\xi\in\mathbb{R}^N $) connecting the two constant steady states $(0,0)$ and $(\frac{a}{b},\frac{\mu}{\lambda}\frac{a}{b})$, and there is no such solutions with speed $c$ less than $2\sqrt{a}$, which improves the results established in [30] and shows that this system has a minimal wave speed $c_0^*=2\sqrt a$, which is independent of the chemotaxis.

In this paper, we apply the two variable (φ'/φ, 1/φ)-expansion method to seek exact traveling wave solutions (solitary wave solutions, periodic function solutions, rational function solution) for time-fractional Kuramoto-Sivashinsky (K-S) equation, (3+1)-dimensional time-fractional KdV-Zakharov-Kuznetsov (KdV-ZK) equation and time-fractional Sharma-Tasso-Olver (FSTO) equation. The solutions are obtained in the form of hyperbolic, trigonometric and rational functions containing parameters. The results show that the two variable (φ'/φ, 1/φ)-expansion method is simple, effctivet, straightforward and is the generalization of the (G'/G)-expansion method.