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In this paper, an improved nonlinear Riccati equation method is established and then used to study nonlinear evolution equations with variable coefficients. The idea behind this current study is to use the nonlinear Riccati equation with constant coefficients and convert it to second-order linear ordinary differential equations using appropriate transformations. The solutions of these second-order linear ordinary differential equations are then used to propose the solutions to the nonlinear Riccati equation. This improved method consists of the well-known (G ′ / G) -expansion and (1 / G ′) -expansion methods as a special case. It also consists of a newly proposed (in this current study) (G / G ′) -expansion method. As a result, this improved method is employed to investigate the traveling wave solutions of (2 + 1) -dimensional models with variable coefficients, such as the extended Kadomtsev–Petviashvili equation, nonlinear Schrödinger equation, and Davey–Stewartson equations. Solitary waves are obtained by taking the parameters involving these derived traveling wave solutions as special values. Furthermore, the constraint conditions for the existence of solutions to these variable coefficient equations are also revealed. This improved method has a wider range of applications for addressing nonlinear wave equations. [ABSTRACT FROM AUTHOR]

In this work, the traveling wave solutions for a modified Leslie-Gower predator–prey model under continuous and discrete diffusion cases are investigated respectively. First, the existence of weak traveling wave solutions is shown by Schauder's fixed point theorem with the help of positive upper and lower solutions, which can remove the singularity of the system with b = 0 at u = 0 . Then we use two methods, a squeeze method and a Lyapunov function method, to prove that, under additional conditions, the weak traveling wave solutions are actually traveling wave solutions. Finally, by showing the nonexistence of traveling wave solutions, the minimal wave speed is obtained. [ABSTRACT FROM AUTHOR]

This paper explores the oceanic wave characteristics exhibited by a modified integrable generalized (2+1)-dimensional nonlinear Schrödinger system of equations through variable coefficients. Two newly modified methods, specifically the improved nonlinear Ricatti equation method and the improved sub-equation method, have been proposed to investigate the aforementioned nonlinear system. Through the utilization of these methods, we successfully obtain traveling and solitary waves solutions for this nonlinear system. We emphasize several constraint conditions that serve to guarantee the existence of these solutions. More comprehensive information about the physical dynamical representation of some of the solutions presented is illustrated through graphical depictions. The Mathematica software package is employed to produce both three-dimensional and their corresponding contour plots, thereby improving the visualization and comprehension of the solutions. This paper illustrates that the two proposed approaches provide straightforward and efficient means of acquiring various types of soliton, rational, trigonometric, hyperbolic and exponential solutions. Moreover, they present a more potent mathematical tool for addressing a variety of other nonlinear partial differential equations that hold significance in the field of applied science and engineering. [ABSTRACT FROM AUTHOR]

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps. [ABSTRACT FROM AUTHOR]

The researchers have developed numerous analytical and numerical techniques for solving fractional partial differential equations most of which provide approximate solutions. Exact solutions, however, are vitally important in a convenient conception of the qualitative properties of the concerned phenomena and processes. In this paper, the pseudo-parabolic-type equations with conformable fractional derivatives are reduced to conformable fractional nonlinear ordinary differential equations by implementing a simple wave transformation. An important benefit of the proposed transformation is that it yields analytical solutions of the conformable pseudo-parabolic type equations by applying the exponential rational function strategy. The sensitivity behaviour of the model has been mentioned thoroughly. [ABSTRACT FROM AUTHOR]

This paper discusses the newly discovered acoustic waves that arise from the equilibrium between the inertia of dust particles and plasma pressure. The propagation of these waves is demonstrated to be non-linear, with supersonic solitons exhibiting either a positive or negative electrostatic potential, and linear, with normal modes in a dusty plasma. Implementing the unified method and the generalized projective Ricatti equation technique, we generate a variety of optical wave structures for the governing framework. Using 3 D and 2 D dimensional illustrations, the fundamental behavior of the derived solutions is visually analyzed by choosing appropriate values for arbitrary parameters that satisfy the specified constraints. The employed methodologies are acknowledged to be a powerful and effective mechanism for creating solitary wave solutions for non-linear evolution models. [ABSTRACT FROM AUTHOR]

In this paper, new exact traveling wave solutions are obtained by Hirota–Ramani equation. The many exact complex solutions of several types of nonlinear partial differential equations (NPDEs) are presented using the modified extended direct algebraic approach and new extended direct algebraic method, which is among the most effective mathematical techniques for finding a precise solution to NPDEs and put into a framework of algebraic computation. By selecting different bright and solitary soliton forms and by creating various analytical optical soliton solutions for the investigated equation, we hope to demonstrate how the analyzed model's parameter impacts soliton behavior. It is possible to obtain new, complex solutions for nonlinear equations like the (1 + 1) -dimensional Hirota–Ramani equation. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in propagation dynamics of a nonlocal dispersal Zika transmission model with general incidence. When the threshold R$$ \mathcal{R} $$ is greater than one, we prove that there is a wave speed c∗>0$$ {c}^{\ast }>0 $$ such that the model has a traveling wave solution with speed c>c∗$$ c>{c}^{\ast } $$, and there is no traveling wave solution with speed less than c∗$$ {c}^{\ast } $$. When the threshold R$$ \mathcal{R} $$ is less than or equal to one, we show that there is no nontrivial traveling wave solution. The approaches we use here are Schauder's fixed point theorem with an explicit construction of a pair of upper and lower solutions, the contradictory approach, and the two‐sided Laplace transform. [ABSTRACT FROM AUTHOR]

We looked at the (3+1)-dimensional fractional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which comes up in fluid dynamics, plasma physics, physics, and superfluids, as well as when connecting the optical model and hydrodynamic domains. Furthermore, unlike the Kadomtsev–Petviashvili equation (KPE), which permits the modeling of waves traveling in both directions, the zero-mass assumption, which is required for many scientific applications, is not required by the KP-B equation. In several applications in engineering and physics, taking these features into account allows researchers to acquire more precise conclusions, particularly in studies pertaining to the dynamics of water waves. The foremost purpose of this manuscript is to establish diverse solutions in the form of exponential, trigonometric, hyperbolic, and rational functions of the (3+1)-dimensional fractional (KP-B) via the application of four analytical methods. This KP-B model has fruitful applications in fluid dynamics and plasma physics. Additionally, in order to better explain the potential and physical behavior of the equation, the relevant models of the findings are visually indicated, and 2-dimensional (2D) and 3-dimensional (3D) graphics are drawn. [ABSTRACT FROM AUTHOR]

In this work, we aim to explore new exact traveling wave solutions for the reaction–diffusion equation, which describes complex nonlinear phenomena such as cell growth and chemical reactions in nature. Obtaining exact solutions to this equation is crucial for understanding aspects such as reaction activity and the diffusion coefficient. We solve the reaction–diffusion equation by using the Riccati equation as an auxiliary equation. By controlling the parameters in the Riccati equation, we obtained a large number of traveling wave solutions, many of which were not formerly recorded in other documents. Numerical simulations demonstrate the evolution of various traveling waves of the reaction–diffusion equation in time and space. These rich exact solutions and wave phenomena help to expand our knowledge of this equation. [ABSTRACT FROM AUTHOR]

This article studied the new traveling wave solutions of the cascaded model with higherorder dispersion effects combined with the effects of spatiotemporal dispersion and multiplicative white noise. In the process of exploring traveling wave solutions, a clever combination of the polynomial complete discriminant system was used to discover more forms of traveling wave solutions for this equation. In order to better observe and analyze the propagation characteristics of traveling wave solutions, we used Maple and Matlab software to provide two-dimensional and three-dimensional visualization displays of the equation solutions. Meanwhile, we also analyzed the internal mechanism of nonlinear partial differential equations using planar dynamical systems. The research results indicated that there are differences in the results of different forms of soliton solutions affected by external random factors, which provided more beneficial references for people to better understand the cascaded model with higher-order dispersion effects combined with the effects of spatiotemporal dispersion and multiplicative white noise, and helped people to more comprehensively understand the propagation characteristics of optical solitons. The solution method in this article was also applicable to the study of other nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

This paper discusses the newly discovered acoustic waves that arise from the equilibrium between the inertia of dust particles and plasma pressure. The propagation of these waves is demonstrated to be non-linear, with supersonic solitons exhibiting either a positive or negative electrostatic potential, and linear, with normal modes in a dusty plasma. Implementing the unified method and the generalized projective Ricatti equation technique, we generate a variety of optical wave structures for the governing framework. Using 3D and 2D dimensional illustrations, the fundamental behavior of the derived solutions is visually analyzed by choosing appropriate values for arbitrary parameters that satisfy the specified constraints. The employed methodologies are acknowledged to be a powerful and effective mechanism for creating solitary wave solutions for non-linear evolution models.

The cubic–quartic perturbed Biswas–Milovic equation, which contains Kudryashov's nonlinear form and two generalized nonlocal laws, has been explored qualitatively and quantitatively, as demonstrated in the present work. The research methods used include the complete discrimination system for polynomial method and the trial equation method. The results show that the Hamiltonian has the conservation property, and the global phase diagrams obtained via the bifurcation method reveal the existence of periodic and soliton solutions. Furthermore, we fully classify all the single traveling wave solutions to substantiate our findings, covering singular solutions, solitons, and Jacobian elliptic function solutions. We analyze their topological stabilities and present two‐dimensional graphs of solutions. We also delve deeper into the dynamic system by incorporating the perturbation item to explore the chaotic phenomena associated with the equation. These outcomes are valuable for studying the propagation of high‐order dispersive optical solitons and have potential applications in optimizing optical communication systems to improve efficiency. [ABSTRACT FROM AUTHOR]

The Manakov model, a framework of coupled non-linear Schrödinger equations, reflects an essential integrable model in non-linear physics, particularly used to analyze soliton behavior and wave interactions in multiple physical realms such as optical theory and plasma physics. Its distinct integrability features make it a treasure trove for investigating complex interactions between wave propagation and non-linear systems. With an emphasis on the stochastic Manakov model, this research article explores the field of stochastic partial differential equations (SPDE). This writing aims to lower the cost and time frame for experiments by using particular integration approaches to generate a wide variety of optical solitons to satisfy the said model. We complete our thrust using two innovative approaches: the Unified Ricatti Equation Algorithm which depends on the Riccati equation and its special solutions, and the New Extended Auxiliary Equation Method which builds several, more general traveling wave solutions to nonlinear partial differential equations using a more straightforward and cohesive approach. The combination of these two not only improves our capability to handle complicated SPDEs but also provides fresh perspectives on how soliton structures and stochastic effects connect within the governing model. By employing these techniques, we obtain various solitary and traveling wave solutions, including dark, singular, and bright solitons. To ensure the validity of the results, the density, contour, 3D, and 2D profiles of various attained solutions are numerically generated by selecting appropriate values for several proposed framework parameters. The presence of multiplicative white noise does not propagate to other components; it just affects the phase portion of the solitons. Research conducted on the mentioned framework involving multiplicative white noise encounters significant applications in optical communication systems. By comprehending soliton behavior and effects, we can optimize the transmission of information in optical fibers, strengthen mode-locked lasers, and advance laser-based material handling. This exclusive blend of challenging model and methodologies led to the discovery of numerous unique soliton solutions and their corresponding behaviors. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE) is the most significant physical model to explain the fluctuations of optical soliton proliferation in optical fiber theory. Optical soliton propagation in nonlinear fibers is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. In this article, the time-fractional perturbed NLSE that demonstrates the super fast wave propagation in optical fibers is investigated analytically. To better understand the underlying mechanisms for these kinds of nonlinear systems, the results are displayed using 3D, 2D, and contour graphics. Furthermore, it is confirmed that the established results are stable, and the modulation instability for the governing model is also studied. The computational intricacies and results highlight the clarity, efficacy, and simplicity of the approaches, pointing to the applicability of these methods to various sets of dynamic and static nonlinear equations governing evolutionary phenomena in computational physics, as well as to other practical domains and a variety of research fields. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation dynamics of a three-species nonlocal dispersal predator–prey system with two weak competing aborigine preys. Precisely, our main concern is the invading process of an alien predator into the habitat of two aborigine preys, when two preys are weak competitors in the absence of predator. This invasion process can be characterized by the spreading speed of the predator as well as the traveling wave solutions connecting the predator-free state to other states. First, by applying Schauder’s fixed point theorem with the help of two pairs of upper and lower solutions, we prove that there exists a positive number c∗, when the wave speed is large than or equals to c∗, the system admits a traveling wave solution connecting the predator-free state to the co-existence state, which indicates the occurrence of invasion-coexistence phenomenon. Then the invading speed of the predator is investigated by using the comparison argument. Our results show that (i) the weak interspecific competition between two preys will not affect the phenomenon of invasive-coexistence; (ii) the invading speed of the predator coincides with the minimal wave speed c∗. [ABSTRACT FROM AUTHOR]

The optical soliton and exact solutions of the stochastic concatenation model are obtained by using the modified extended direct algebraic method. This method provides a wide variety of stochastic solutions, including stochastic bright solitons, stochastic dark solitons, stochastic dark combo solitons, stochastic singular combo solitons, stochastic periodic solutions, stochastic rational solutions, stochastic Jacobi elliptic functions solutions and stochastic Weierstrass elliptic functions solutions. For a variety of nonlinear partial differential equations, this method offers a practical and effective method for determining exact solutions. The impact of the noise is illustrated graphically using examples of some of the retrieved solutions with various noise strengths. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and nonexistence of traveling wave solutions for a discrete diffusive mosquito-borne epidemic model with general incidence rate and constant recruitment. It is observed that whether the traveling wave solutions exist or not depend on the so-called basic reproduction ratio R 0 of the corresponding kinetic system and the critical wave speed c ∗ . More precisely, when R 0 > 1 and c ≥ c ∗ , the system admits a nontrivial traveling wave solution by constructing an invariant cone in a bounded domain with initial functions being defined on, and employing the method of upper and lower solution, Schauder’s fixed point theorem and a limiting approach. Moreover, the asymptotic behavior of traveling wave solutions at positive infinity is obtained by constructing a suitable Lyapunov functional. When 0 < c < c ∗ or R 0 ≤ 1 , the system has no nontrivial traveling wave solution by using a contradictory approach and two-sided Laplace transforms. [ABSTRACT FROM AUTHOR]

This study deals with the Lonngren wave equation, which is used to model the transmission of electrical signals in a type of semiconductor material called a tunnel diode. The aim of this study is to produce dark soliton, singular soliton, trigonometric, complex rational solutions to the Lonngren wave equation by using the sub-equation method. Graphs representing the Lonngren wave are drawn by giving different values to the variables in the solutions obtained. Graphs are presented using a special package program. In the light of the data obtained as a result of the study, it is thought that this new method used in the research of the solution of the Lonngren wave equation will contribute to the literature. [ABSTRACT FROM AUTHOR]

The Fordy–Gibbons equation is a nonlinear differential equation. Physically, the motion of a damped oscillator with a more complex potential than in basic harmonic motion is described by the Fordy–Gibbons equation. For the equation under consideration, numerous novel families of precise analytical solutions are being successfully found. The soliton solutions are represented as rational and exponential functions. To further illustrate the potential and physical behavior of the equation, the findings are also stated visually. Three approaches are suggested in this paper for solving the Fordy–Gibbons equation. These solutions are new solutions. [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]

This work is concerned with full information about the traveling wave solutions for a diffusive tuberculosis model with fast and slow progression. After relaxing the rigid ordered restriction on the diffusion rates typically employed in the previous classical study, and overcoming some significant technical obstacles to verify the boundedness of the infected components, we first obtain the existence of super-critical traveling waves connecting disease-free equilibrium to endemic equilibrium. Then, on the basis of the uniform boundedness for sequences of super-critical traveling waves and further delicate analysis, the existence of critical traveling waves is thoroughly explored. Finally, the nonexistence of non-negative bounded traveling waves is discussed in two cases. [ABSTRACT FROM AUTHOR]

It is well known that propagation of waves in homogeneous linearized elastic materials of infinite extent is not dispersive. Motivated by the work of Rubin, Rosenau, and Gottlieb, we develop a generalized continuum model for the response of strain-limiting materials that are dispersive. Our approach is based on both a direct inclusion of Rivlin–Ericksen tensors in the constitutive relations and writing the linearized strain in terms of the stress. As a result, we derive two coupled generalized improved Boussinesq-type equations in the stress components for the propagation of pure transverse waves. We investigate the traveling wave solutions of the generalized Boussinesq-type equations and show that the resulting ordinary differential equations form a Hamiltonian system. Linearly and circularly polarized cases are also investigated. In the case of unidirectional propagation, we show that the propagation of small-but-finite amplitude long waves is governed by the complex Korteweg–de Vries (KdV) equation. [ABSTRACT FROM AUTHOR]

The dynamic behavior of the Vakhnenko-Parkes equation is examined in this manuscript. This is an important subject because of its implications for comprehending intricate mathematical models describing traveling wave phenomena and solitons. The construction of traveling wave solutions for the Vakhnenko-Parkes equation in closed form is the main issue addressed in the study. The modified auxiliary equation approach and the extended (G ′ G 2) -expansion method are used to address this because they are effective in producing precise solutions of a large class of nonlinear partial differential equations. A visual component to comprehending the behavior of the equation is added by employing 3D-surface graphs, 2D-line graphs, and contour plots to explore these solutions graphically. A variety of traveling wave behavior is observed from the obtained solutions. These results imply that the Vakhnenko-Parkes equation and its solutions are complex, offering important insights into the underlying dynamics. The proposed techniques are applied for the first time to study the considered model in this work. A comparison of the obtained results with the previous works is presented to confirm the significance and novelty of the reported results. [ABSTRACT FROM AUTHOR]

In this paper, the exact explicit traveling wave solutions to the simplified modified Camassa–Holm (SMCH) equation. The SMCH equation is a significant nonlinear evolution equation because it can be used to describe certain physical processes in oceanography and fluid dynamics. The SMCH equation has important applications in mathematics, physics, and engineering. Using the modified auxiliary equation method and the extended G ′ G 2 -expansion method, we achieve precise solutions with traveling wave behavior. The proposed methods are applied for the first time in this work to examine the considered mathematical model. The solutions are extracted in terms of trigonometric, hyperbolic, and rational functions. The obtained results not only confirm the previously reported solutions of SMCH equation but also offer some new results. To clearly illustrate the physical implications of the examined equation, we provide graphical representations using 3D, 2D and contour plots for certain values of the parameters. The illustrations help to clarify the diverse characteristics and behaviors, associated with the solutions, providing useful insights for researchers as well as practitioners across a variety of scientific and technical disciplines. [ABSTRACT FROM AUTHOR]

In this article, new traveling wave solutions are retrieved for two space-time fractional models: the resonant nonlinear Schrödinger equation and the Biswas–Milovic equation which illustrates the dynamics of optical soliton promulgation in optical fibers, both featuring Kerr law nonlinearity. These equations are explored via an efficient method namely, the extended simple equation method. The fractional derivative is used in the conformable sense to accomplish this analysis. The extracted solutions show dark, periodic, singular, and singular-periodic solitons behaviors, which are depicted graphically by using line, surface, and contour plots. The reported solutions are unique and novel. The proposed method distinguishes itself by its simplicity, reliability, and ability to generate novel soliton solutions for nonlinear Schrödinger equations within the realm of mathematical physics. [ABSTRACT FROM AUTHOR]

This study covers the traveling wave analysis of the two-mode Cahn–Allen (TMCA) equation. The TMCA model plays the role of transmitting information into two different locations and preserving its physical properties. A wave transformation converts the considered model into an ordinary differential equation (ODE). By applying the two analytical techniques, the extended tanh - coth technique and the extended G ′ G 2 -expansion scheme, we have obtained a new type of the wave structures like singular, kink and periodic. To present the significance of the acquired outcomes, we have shown the graphical behaviors of the results by taking some suitable values of the involved parameters. The acquired ODE can be transformed into a dynamical system by applying the Galilean transformation. To examine the sensitive behavior, the dynamical system is expressed in a non-autonomous form, and the effects are analyzed by considering various initial conditions. The considered model admits the two-dimensional Lie algebra. The nonlinear self-adjointness of the considered model is checked using a new conservation theorem. The results show that the considered model is not self-adjoint and is made self-adjoint by computing a new dependent variable. Finally, the conserved quantities corresponding to each symmetry generator are computed. [ABSTRACT FROM AUTHOR]

The present study investigates the fractional Dullin-Gottwald-Holm equation by using the Riccati-Bernoulli sub-optimal differential equation method with the Bãcklund transformation. By employing a well-established criterion, the present study reveals novel cusp soliton solutions that resemble peakons and offers valuable insights into their dynamic behaviors and mysterious phenomena. The solution family encompasses various analytical solutions, such as peakons, periodic, and kink-wave solutions. Furthermore, the impact of both the time- and space-fractional parameters on all derived solutions' profiles is examined. This investigation's significance lies in its contribution to understanding intricate dynamics inside physical systems, offering valuable insights into various domains like fluid mechanics and nonlinear phenomena across different physical models. The computational technique's straightforward, effective, and concise nature is demonstrated through introduction of some graphical representations in two- and three-dimensional plots generated by adjusting the related parameters. The findings underscore the versatility of this methodology and demonstrate its applicability as a tool to solve more complicated nonlinear problems as well as its ability to explain many mysterious phenomena. [ABSTRACT FROM AUTHOR]

Finding new, more widely applicable accurate traveling wave solutions to nonlinear partial differential equations is the aim of this work. The Konno–Oono equation and the Kadomtsev–Petviashvili modified equal width equation have various possible applications in mathematical physics and engineering disciplines, and their exact solutions are presented in this article. Using the first integral approach as an analytical method and the traveling wave transformation, we explore the precise traveling wave solutions. This approach is effective and yields unique exact solutions that fall into two categories: solutions in the form of trigonometric functions and solutions in the form of hyperbolic functions. Furthermore, graphical illustrations in two and three dimensions are showcased to offer a thorough explanation of their dynamic nature. In addition, it is critical to emphasize the significance of mastering the Konno–Oono equation and the KP–MEW equation, both of which have applications in a variety of fields Our findings are also crucial for understanding the many oceanographic applications that involve ocean gravity waves, offshore rigs in the water, energy from moving ocean waves, and several other related phenomena. This approach requires less computing work and is straightforward and succinct. That is its main advantage.

The present study investigates the fractional Dullin-Gottwald-Holm equation by using the Riccati-Bernoulli sub-optimal differential equation method with the Bäcklund transformation. By employing a well-established criterion, the present study reveals novel cusp soliton solutions that resemble peakons and offers valuable insights into their dynamic behaviors and mysterious phenomena. The solution family encompasses various analytical solutions, such as peakons, periodic, and kink-wave solutions. Furthermore, the impact of both the time- and space-fractional parameters on all derived solutions' profiles is examined. This investigation's significance lies in its contribution to understanding intricate dynamics inside physical systems, offering valuable insights into various domains like fluid mechanics and nonlinear phenomena across different physical models. The computational technique's straightforward, effective, and concise nature is demonstrated through introduction of some graphical representations in two- and three-dimensional plots generated by adjusting the related parameters. The findings underscore the versatility of this methodology and demonstrate its applicability as a tool to solve more complicated nonlinear problems as well as its ability to explain many mysterious phenomena.

In this article, we use the modified extended direct algebraic method (mEDAM) to explore and analyze the traveling wave phenomena embedded in the quintic conformable Benney-Lin equation (CBLE) that regulates liquid film dynamics. The proposed transformation-based approach developed for nonlinear partial differential equations (PDEs) and fractional PDEs (FPDEs), efficiently produces a plethora of traveling wave solutions for the targeted CBLE, capturing the system's nuanced dynamics. The methodically determined traveling wave solutions are in the form of rational, exponential, hyperbolic and trigonometric functions which include periodic waves, bell-shaped kink waves and signal and double shock waves. To accurately depict the wave phenomena linked to these solutions, we generate 2D, 3D, and contour graphs. These visualizations not only improve understanding of the CBLE model's dynamics, but also provide a detailed way to examine its behavior. Moreover, the use of the proposed techniques contributes to a better understanding of the other FPDEs' distinct characteristics, enhancing our comprehension of their underpinning dynamics.

The present article depicts the research being done on the microwave effect realized by the often-used Burger–Fisher equation. A particular view is on the emergence of multi-stability and the bifurcations associated with microwave engineering systems. A novel rational, trigonometric, and hyperbolic form of the traveling waves is furnished by the resolution of the non-linear issue with the aid of the advanced exp (−Ψ(η))-expansion function parameterization. Firstly, we will solve the exact form of the Burger–Fisher equation, which clarifies the behavior of the equation in different conditions. Subsequently, bifurcation analysis techniques will be employed to study the nuanced relationship between the system parameters and the emergence of multistability phenomena. Through our results, we exposed the complicated essential features of microwave physics and established crucial parameters that determine a system’s behavior. Next, we cover control principle issues and practical applications in microwave engineering problems. This model accommodates the essential features of multi-stable dynamic systems and provides an important framework for creating microwave devices and circuits.The main aim of this research work is to analyze the phenomenon of multi-stability and bifurcation in microwave engineering systems by applying Burger–Fisher equation. The study intends to obtain new solutions to the PDEs through the application of a new method, the exp (−Ψ(η))-expansion function method that uses rational, trigonometric, and hyperbolic traveling wave solutions. Existing research of the Burger–Fisher equation is not explicit and by solving the exact form the research aims at exploring the different conditions under which the equation behaves and studying the delicate interactions between the parameters of the multistable system.

This study proposed a modified G′/G2-expansion method to seek the new exact traveling wave solutions of the Newell–White–Head–Segel (NWHS) Model. This is an amplitude equation utilized for distributing temperature within a rod that is infinitely thin and long, or determining the flow velocity of a fluid through a pipe that is infinitely long but has a small diameter. The modified G′/G2-expansion method is used to extract the new exact solutions. The solutions of this model are categorized in hyperbolic, trigonometric, and rational forms. Moreover, we compare our results with the new auxiliary equation method. The 3D, line and corresponding contour representation of these solutions are depicted by choosing the different values of parameters. [ABSTRACT FROM AUTHOR]

The Novikov equation is a peakon equation with cubic nonlinearity, which, like the Camassa--Holm and the Degasperis--Procesi, is completely integrable. In this paper, we study the spectral and linear stability of peakon solutions of the Novikov equation. We prove spectral instability of the peakons in L²(ℝ). To do so, we start with a linearized operator defined on H²(ℝ) and extend it to a linearized operator defined on weaker functions in L²(ℝ). The spectrum of the linearized operator in L²(ℝ) is proven to cover a closed vertical strip of the complex plane. Furthermore, we prove that the peakons are spectrally unstable on W²,∞(ℝ) and linearly and spectrally stable on H²(ℝ). The result on W²,∞(ℝ) is in agreement with previous work about linear instability and our result on H²(ℝ) is in line with past work on orbital stability. [ABSTRACT FROM AUTHOR]

The FitzHugh–Nagumo model has been used empirically to model certain types of neuronal activities. It is also a non-linear dynamical system applicable to chemical kinetics, population dynamics, epidemiology and pattern formation. In the literature, many approaches have been proposed to study its dynamics. In this paper, initially, we have employed cutting-edge tools from discrete dynamics for discretization and fixed points. It has been proven that an exact discrete scheme exists for this paradigm. This project also considers the phase space and integral surfaces of these evolutionary equations. In addition, it carries out a thorough symmetry analysis of this reaction diffusion system to find equivalent systems. Moreover, steady-state solutions are obtained using ansatzes for traveling wave solutions. The existence of infinite traveling wave solutions has also been proven. Yet again, this investigation establishes the potential of symmetry methods to unravel non-linearity. Finally, singular perturbation theory has been employed to obtain analytical approximations and to study stability in different parameter regimes. [ABSTRACT FROM AUTHOR]

In this paper, we analyze Riemann–Liouville (R-L) time-fractional (2 + 1) dimensional extended quantum Zakharov–Kuznetsov (EQZK) equation by using the Lie symmetry method which arises in hydrodynamic that describes the nonlinear propagation of the quantum ion-acoustic waves. By using its symmetry, we convert the equation under consideration to a fractional order non-linear ordinary differential equation (ODE). In this reduced ODE, we use a special type of derivative which is known as Erdélyi–Kober (EK) derivative. This enables us to obtain explicit solutions with convergence analysis of the considered problem. By using Ibragimov's conservation laws theorem, we compute the conservation laws of the problem under investigation. Moreover, by employing the two potent methods explicit power series and ( 1 G ′ )-expansion technique, we get the explicit solutions to the problem under discussion. This analysis leads to the derivation of various key findings, including the identification of symmetries, the establishment of similarity reductions involving the EK fractional differential operator, the determination of exact solutions, and the formulation of conservation laws for the considered equation. We have confidence that these remarkable findings can provide valuable insights and contribute to the exploration of additional evolutionary mechanisms associated with the studied equation. [ABSTRACT FROM AUTHOR]

In this article, we use the modified extended direct algebraic method (mEDAM) to explore and analyze the traveling wave phenomena embedded in the quintic conformable Benney-Lin equation (CBLE) that regulates liquid film dynamics. The proposed transformation-based approach developed for nonlinear partial differential equations (PDEs) and fractional PDEs (FPDEs), efficiently produces a plethora of traveling wave solutions for the targeted CBLE, capturing the system's nuanced dynamics. The methodically determined traveling wave solutions are in the form of rational, exponential, hyperbolic and trigonometric functions which include periodic waves, bell-shaped kink waves and signal and double shock waves. To accurately depict the wave phenomena linked to these solutions, we generate 2D, 3D, and contour graphs. These visualizations not only improve understanding of the CBLE model's dynamics, but also provide a detailed way to examine its behavior. Moreover, the use of the proposed techniques contributes to a better understanding of the other FPDEs' distinct characteristics, enhancing our comprehension of their underpinning dynamics. [ABSTRACT FROM AUTHOR]

The complex Hirota-dynamical model (CHDM) has applications in the study of plasma physics, the investigation of fusion energy, astrophysical research, and space studies. The CHDM may also be used to investigate turbulent flows to study shocks and other nonlinear phenomena, and light waves venturing through the fibers. Nowadays, plasma physics, fusion energy, astrophysical research, and space studies are very interesting topics in the modern research. So, we need to shed light on this model as a good application in these fields. For deep investigation of these physical problems, we need to find their analytical solutions. In this study, we explore a variety of soliton solutions with different geometrical structures for the CHDM via the double variable expansion method. By means of this method, we have obtained three types of soliton solutions, namely, hyperbolic, trigonometric, and rational function solutions. The graphical interpretation of these solutions gives us some popular shapes such as singular-periodic, kink, bell, and singular shapes. The performed method is an efficient technique to execute and provides reliable analytical soliton solutions which are very important to further advanced investigation of the mention equation. [ABSTRACT FROM AUTHOR]

This analysis examines the time-fractional mixed hyperbolic-elliptic p-system of conservation laws by applying the new extended direct algebraic method. The p-system with generalized cubic van der Waals flux, and potential applications in the field of compressible isothermal viscosity-capillarity fluids, is investigated. In particular, this issue describes the longitudinal isothermal motion in elastic bars or fluids. A diverse periodic, kink, and singular soliton structures are extracted. The 3D dynamical behaviors and corresponding contour profiles of some obtained solitons are displayed. The fractional effects in the sense of Beta, M-truncated, and modified Riemann–Liouville, are discussed and illustrated. The method shows the straightforward, reliability, and efficiency for solving complex physical phenomena that is modeled by nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

This article aims to study the generalized exponential rational function method to solve the differential-difference Burgers' equation. Our approach is an efficient method for solving nonlinear partial differential equations, and it can be used for a specific type of nonlinear differential-difference equations. To recognize diverse singular soliton and multi-soliton wave structures, we displayed the 3-D and contour graphs associated with the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the bifurcation, phase portrait and the traveling wave solutions of the coupled fractional Lakshmanan–Porsezian–Daniel equation by using the dynamical system bifurcation theory approach. Based on phase portrait, we obtain some new traveling wave solutions, which include Jacobi elliptic function solutions, soliton solutions, torsion wave solutions and periodic wave solutions. What’s more, we plot three-dimensional diagrams, contour plots and two-dimensional diagrams with the help of Maple, which provide a more visual demonstration of the section of this equation. The investigations are innovative and unexplored, and they can be employed to elucidate the physical phenomena that have been simulated, providing insights into their transient dynamic characteristics. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear space–time fractional form of Cahn–Allen equation (FCAE) with beta and M-truncated derivatives. Cahn–Allen equation (CAE) is commonly used in many problems of physics and engineering, such as, solidification problems, phase separation in iron alloys and others. We apply the improved tan (Ψ (η) 2) -expansion method (ITEM). We obtain four types of traveling wave solutions, including, trigonometric, hyperbolic, rational and exponential function solutions. We demonstrate some of the extracted solutions using definitions of the beta (BD) and M-truncated derivatives (MTD) to understand their dynamical behavior. We observe the fractional effects of the aforementioned derivatives on the related physical phenomena up to possible extent. [ABSTRACT FROM AUTHOR]

The aim of this article is to investigate the existence of traveling waves of a diffusive model that represents the transmission of a virus in a determined population composed of the following populations: susceptible (S) , infected (I) , asymptomatic (A) , and recovered (R). An analytical study is performed, where the existence of solutions of traveling waves in a bounded domain is demonstrated. We use the upper and lower coupled solutions method to achieve this aim. The existence and local asymptotic stability of the endemic ( E e ) and disease-free (E 0) equilibrium states are also determined. The constructed model includes a discrete-time delay that is related to the incubation stage of a virus. We find the crucial basic reproduction number R 0 , which determines the local stability of the steady states. We perform numerical simulations of the model in order to provide additional support to the theoretical results and observe the traveling waves. The model can be used to study the dynamics of SARS-CoV-2 and other viruses where the disease evolution has a similar behavior. [ABSTRACT FROM AUTHOR]

In this article, we employ the Mikhailov–Novikov–Wang integrable equation (MNWIE) appearing by means of the perturbatives symmetry approach to the rating of integrable non‐evolutionary PDEs. The new exact soliton solutions of this equation which were not achieved before have been realized for the first time in the framework of the (G′/G)‐expansion method. In the same vein and parallel, the corresponding numerical solutions of this equation have been established using the variational iteration method (VIM). [ABSTRACT FROM AUTHOR]

The (2+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff equation appears in the mathematical description of physical phenomena in plasma physics, fluid dynamics and nonlinear optics. In this article, extended trial and modified auxiliary equation methods are utilized to observe the dynamical structures exhibiting the solitary wave behavior of the considered model. The traveling wave hypothesis is employed to extract explicit closed-form solution expressions. The presented methods show reliability and robust computational capabilities to investigate solitary waves. In order to investigate the physical behavior of these solutions, 3D, 2D and density plots are drawn for different values of parameters. The graphical observations depict kink soliton, dark–bright singular soliton and periodic wave solutions. The comparison of the outcomes of the proposed results with those obtained using prior techniques is made to confirm the usefulness of the proposed techniques. The presented study will be helpful to explain the wave propagation in many problems of plasma physics and fluid dynamics. Moreover, the reported solutions may help to understand optical wave transmission and aid in the development of optical devices. The results given in this study will contribute in the understanding of the behavior of waves in the higher-dimensional governing models. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of traveling wave solutions for a nonlocal evolution equation with delayed nonlocal response and sign-changing kernel. By constructing a new pair of upper-lower solutions and applying Schauder’s fixed point theorem, we first prove that there exists a number c # > 0 such that when c > c # , the nonlocal evolution equation admits a semi-wave solution with wave speed c, which connects the trivial equilibrium 0 at negative infinity. Then, we analyze the asymptotic behavior of wave profile at positive infinity and obtain the existence of a traveling wave solution with speed c and connecting the trivial equilibrium 0 and the positive equilibrium 1, when the wave speed c is large. [ABSTRACT FROM AUTHOR]

In this work, we investigate the existence and non-existence of traveling wave solutions for a chemotaxis model with two chemoattractants. To prove our main results, we apply the dynamical systems theory by constructing a positively invariant set in the four-dimensional space. Particularly, we analyze the monotonicity of traveling wave solutions. [ABSTRACT FROM AUTHOR]

The Boussinesq equation has drawn significant interest in models for coastline and oceanic engineering, as it can simulate various phenomena such as shallow water waves and harbors, tsunami transmission, and near-shore wave mechanisms. This study examines different approaches for solving the (3+1)-dimensional integrable Boussinesq equation. For this purpose, the Bäcklund transformation is derived by utilizing the Hirota bilinear representation. The understanding of the equation is improved by this transformation, which yields solutions for exponential functions. Furthermore, the model’s bilinear form is used to construct its two-, three-, and multi-wave solutions. The features and behavior of the wave solutions to the equation are clarified by this investigation. Additionally, the concerned equation is transformed into an ordinary differential equation by means of a traveling wave transformation, and the results consisting of solutions for rational and polynomial functions are extracted by means of the unified technique. The graphical representations are an essential visual assistance for comprehending the intricate dynamics and behaviors displayed by the governing equation’s solutions.

Abstract In this manuscript, our primary objective is to delve into the intricacies of an extended nonlinear Schrödinger equation. To achieve this, we commence by deriving a dynamical system tightly linked to the equation through the Galilean transformation. We then employ principles from planar dynamical systems theory to explore the bifurcation phenomena exhibited within this derived system. To investigate the potential presence of chaotic behaviors, we introduce a perturbed term into the dynamical system and systematically analyze the extended nonlinear Schrödinger equation. This investigation is further enriched by the presentation of comprehensive two- and 3D phase portraits. Moreover, we conduct a meticulous sensitivity analysis of the dynamical system using the Runge–Kutta method. Through this analytical process, we confirm that minor fluctuations in initial conditions have only minimal effects on solution stability. Additionally, we utilize the complete discrimination system of the polynomial method to systematically construct single traveling wave solutions for the governing model.