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In this paper we analyze the wavefront solutions of parabolic partial differential equations of the type g (u) u τ + f (u) u x = (D (u) u x) x + ρ (u) , u (τ , x) ∈ [ 0 , 1 ] where the reaction term ρ is of monostable-type. We allow the diffusivity D and the accumulation term g to have a finite number of changes of sign. We provide an existence result of travelling wave solutions (t.w.s.) together with an estimate of the threshold wave speed. Finally, we classify the t.w.s. between classical and sharp ones. [ABSTRACT FROM AUTHOR]

The research outlined here focuses on the hyperbolic tangent (tanh) method for deriving analytical solutions for travelling wave to the two-dimensional stochastic Allen–Cahn equation with multiplicative noise. The novelty of our work is to derive these exact solutions since the previous studies focused on the solutions of numerical nature. The tanh approach, which employs a finite tanh power series, is particularly adept at modeling travelling wave profiles. A key area of interest in this study is the effect of multiplicative noise on these travelling waves dynamics, especially how high levels of noise can lead to propagation failure of waves. The results demonstrate that for weak noise, the propagation of the travelling wave is basically unaffected, while the wave fails to propagate in the strong noise regime. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a nonlocal and time-delayed reaction–diffusion vector-borne model with the saturated incidence with respect to infectious terms and derive the basic reproduction number R 0 , which serves as a threshold parameter that predicts whether the vector-borne disease will spread. Furthermore, the global attractivity of the endemic equilibrium is obtained by Lyapunov functionals if all the parameters are positive constants. Also, we can get the globally asymptotical stability for the corresponding time-delayed ODE system. In the case of an unbounded spatial habitat, we show the existence and non-existence of travelling wave solutions for the spatially homogeneous model. [ABSTRACT FROM AUTHOR]

The Schamel-Burgers equation, producing the shock-type traveling waves in magnificent physical cases, has lots of potential for analyzing ion-acoustic waves in plasma physics and fluid dynamics. Scientists have worked for a long time to explore the traveling wave solutions of such equations. Thus, in this article, some new traveling wave solutions of the Schamel-Burgers equation, different from those found in the literature, are generated. For this aim, the exp(-φ(ξ))- expansion method is implemented. We also provide the solutions through two- and three-dimensional figures. Generally, exact traveling wave solutions will be useful in the theoretical and numerical study of the nonlinear evolution equations. The obtained results are very supportive, which ensures a more effective mathematical instrument for examining exact traveling wave solutions of the nonlinear equations arising in the recent area of applied sciences and engineering [ABSTRACT FROM AUTHOR]

Nonlinear fractional evolution equations are important for determining various complex nonlinear problems that occur in various scientific fields, such as nonlinear optics, molecular biology, quantum mechanics, plasma physics, nonlinear dynamics, water surface waves, elastic media and others. The space-time fractional modified equal width (MEW) equation is investigated in this paper utilizing a variety of solitary wave solutions, with a particular emphasis on their implications for wave propagation characteristics in plasma and optical fibre systems. The fractional-order problem is transformed into an ordinary differential equation using a fractional wave transformation approach. In this article, the polynomial expansion approach and the sardar sub-equation method are successfully used to evaluate the exact solutions of space-time fractional MEW equation. Additionally, in order to graphically represent the physical significance of created solutions, the acquired solutions are shown on contour, 3D and 2D graphs. Based on the results, the employed methods show their efficacy in solving diverse fractional nonlinear evolution equations generated across applied and natural sciences. The findings obtained demonstrate that the two approaches are more effective and suited for resolving various nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

The Gilson–Pickering equation, which describes wave propagation in plasma physics and the structure of crystal lattice theory that is most frequently used. The discussed model is converted into a bilinear form utilizing the Hirota bilinear technique. Sets of case study are kink wave solutions; breather solutions; collision between soliton and periodic waves; soliton and periodic waves. The exp (- θ (α)) -function approach is employed to discover travelling wave solutions including five classes of solutions. In addition, it has been confirmed that the established findings are stable, and it has been helpful to validate the computations. The effect of the free variables on the behavior of obtained solutions to some plotted graphs for the exact cases is also explored depending upon the nature of nonlinearities. The exact solutions are utilized to demonstrate the physical natures of 3D, density, and 2D graphs. [ABSTRACT FROM AUTHOR]

In this article, the Bernoulli sub-ODE and generalized Kudryashov methods have been successfully used to look for travelling wave solutions for the coupled non-linear evolution equations including the coupled Higgs equation and the Maccari system. The aforementioned approaches provide more new broad solutions than the previous ones now in use and are straightforward to apply. In both techniques, a travelling wave transformation is employed to convert non-linear partial differential equations into an ordinary differential equation. The solitary wave solutions are also generated from the travelling wave solutions when the parameters are considered as unique values. Several soliton solutions are generated for various parameter combinations. Numerous innovative solutions have been created by using Bernoulli sub-ODE and generalized Kudryashov approaches, including the periodic wave, kink wave, bell shape, anti-bell shape, M-shape, V-shape, W-shape, Z-shape, bright, dark and singular soliton solutions. To demonstrate how the Bernoulli sub-ODE and generalized Kudryashov approaches can be used to uncover the analytical solutions of the coupled Higgs equation and the Maccari system, replicate many figures in computer software Mathematica 13.2 along with several 3D, 2D, and contour plots. The similarities, contrasts, advantages and disadvantages of the two analytical techniques are examined. It has been demonstrated that the travelling wave solutions generated by these two analytical techniques, each using a different basis equation, have unique characteristics. The outcomes show that the methods used may reliably identify wide-spectral stable travelling wave solutions to non-linear evolution equations that appear in a variety of scientific, technical, and engineering disciplines. [ABSTRACT FROM AUTHOR]

The new precise traveling wave solutions to the Geophysical KdV equation obtained by the (G ′ G 2) -expansion method and Sine-Cosine method reflect the results for hyperbolic, trigonometric, and rational functions. Different physical wave shapes are displayed by all solutions, including the periodic, bright, and singular soliton solutions. Compared to the (G ′ G 2) -expansion approach, the Sine-Cosine method is shown to be more straightforward, efficient, and involves less time-consuming symbolic computations, however the key factors of (G ′ G 2) -expansion approach is that the great capacity of solutions can be obtained as more number parameters are involved in the solution procedure. The obtained results demonstrate the propagation of nonlinear tsunami structures, their interaction, and the progress of solitons. The propagation of nonlinear tsunami waves are shown to be strongly influenced by the travelling wave’s velocity and the Coriolis parameter. [ABSTRACT FROM AUTHOR]

In this work, the (2+1)-dimensional Painlevé integrable Burgers equation is investigated. By applying a certain unified method, some analytical solutions, involving rational functions, trigonometric functions and hyperbolic functions, are achieved. In order to predict the wave dynamics, several three-dimensional and two-dimensional graphs and contour profiles are constructed. Bright, dark, periodic, kink, anti-kink, singular, singular periodic, bell-shaped waves are thus obtained. The dynamics of these solutions can be illustrated graphically by choosing appropriate values for the parameters involved. Due to the presence of arbitrary constants in these derived solutions, they can be used to explain a variety of qualitative traits present in wave phenomena. The approach is efficient to algebraic computation and it can be used to categorize a wide range of wave forms, as shown by the demonstrated soliton solutions. Travelling wave solutions are converted into solitary wave solutions when certain values are set for the parameters. Using the Wolfram program Mathematica, we sketch the figures for various values of the associated parameters in order to closely examine the obtained solitons. [ABSTRACT FROM AUTHOR]

In this article, we investigate a couple of nonlinear fractional models of eminent interests subsequently the conformable derivative sense is used to designate the fractional order derivatives. The given structures are transformed into nonlinear ordinary differential equations of integer order, and the extended simple equation technique is then employed to solve the resulting equations. Initially, the nonlinear space time fractional Klein–Gordon equation is considered emerging from quantum and classical relativistic mechanics, which have application in plasma physics, dispersive wave phenomena, quantum field theory, and optical fibres. Later, the (2 + 1)-dimensional time fractional Zoomeron equation is analysed which is convenient to explore the innovative phenomena related to boomerons and trappons. As a result, various new soliton solutions are successfully established. The reported results offer a key implementation for analysing the soliton solutions of nonlinear fractional models which are extremely encouraging arising in the recent era of science and engineering. The 3D simulations have been carried out to demonstrate dynamics of the various soliton solutions for a given set of parameters. [ABSTRACT FROM AUTHOR]

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

The development of an effective and efficient local discontinuous Galerkin algorithm for the solution of the coupled (1 + 1) -dimensional Broer–Kaup system is the primary focus of this research. In this algorithm, spatial discretization is attained using the local discontinuous Galerkin method, while temporal discretization is handled using the explicit total variation diminishing higher-order Runge-Kutta method. However, the ( G ′ G ) -expansion method is also implemented to produce the exact solutions for the coupled (1 + 1) -dimensional Broer–Kaup system. The obtained solutions are the traveling wave solutions. The efficiency and reliability of the proposed method are analyzed by comparing the generated numerical simulations to various traveling wave solutions using several tables and figures. The exact solutions and the numerical simulations of the coupled (1 + 1) -dimensional Broer–Kaup system are shown to correspond exceptionally well. [ABSTRACT FROM AUTHOR]

In this study, we investigate the travelling wave solutions of the (2+1)-dimensional new generalized Korteweg–de Vries equation by employing Lie group analysis along with various techniques which include direct integration, simplest equation method and Kudryashov’s method. The results obtained consists of periodic, kink, soliton and hyperbolic solutions. The symbolic computation software Maple is used to check the accuracy of all solutions obtained. Finally, 3D, density, and 2D plots of the derived solution are displayed to show the physical appearance of the model. Furthermore, we utilize the general multiplier technique and Ibragimov’s method to derive its conserved vectors. Conservation of energy and momentum, amongst others were found. Conservation laws have many significant uses with regards to integrability, linearization and analysis of solutions.

One of the fastest growing fields of study in nonlinear optics is optical solitons. This article investigates the optical soliton solutions of the coupled Kaup-Newell equation in birefringent fibers without four wave mixing. This model helps explain how modified structures, particularly Alfven waves, propagate in optical fiber and plasma physics. The results presented in this research clearly show how well the new mapping method handle nonlinear Kaup-Newell equation in coupled vector form, in birefringent fiber without four-wave mixing, to extract the analytical soliton solutions. Four-wave mixing expresses a nonlinear optical effect in which four waves interact each other as a result of the third order nonlinearity. As a valuable outcome of this article, a variety of solitons are discovered including bright, dark, periodic, singular bell-shaped, gap, flat-kink, dark-singular, and w-shaped. The existence criteria for these solitons are also obtained that are presented as constraint conditions. The constraint relations guarantee the existence of such solitons. Furthermore, graphs in 2D, 3D and contour are created to show the physical characteristics of the solutions that are produced. The new mapping method is clear and effective, and the solutions provide a wealth of avenues for additional research. The technique can also be functional to other sorts of nonlinear evolution equations in contemporary areas of research. The outcomes presented in this work are new versions which have never been observed before in existing literature.

In this work, we investigate a partial differential equation (PDE) derived from a Boussinesq system of two equations that describe the wave motion in shallow water when the wavelength is small compared to wave amplitude. The Lie point symmetries of the PDE are utilized to reduce it to an ordinary differential equation (ODE) the travelling wave solutions of which are obtained. These solutions are of trigonometric, hyperbolic and exponential types. Graphical representation of some solutions exhibiting solitary and cnoidal wave phenomena are presented.

The simplified modified Camassa Holm (SMCH) equation is an essential nonlinear model for classifying several wave phenomena in physical science. In this study, the new m + 1 / G ′ -expansion method has been applied to the SMCH equation. Based on the given method, we have obtained some typical analytical solutions, such as exponential and rational function and complex function solutions of the SMCH equation. The three-dimensional and contour graphs of the obtained results are drawn for appropriate values of parameters. Getting the results and graphic drawings have been made using the Wolfram Mathematica. All the obtained solutions are crucial for understanding the nonlinear phenomena of dispersive waves. The solutions have shown that the presented method is powerful, practical, and suitable for investigating nonlinear models. [ABSTRACT FROM AUTHOR]

Understanding protection from SARS-CoV-2 infection and severe COVID-19 induced by natural SARSCoV-2 infection versus vaccination is essential for informed vaccine mandate decisions. In this article, we construct a system of reaction-diffusion-integral equations to describe the development of vaccinated population, not previously infected, and pre-infected population, vaccinated or not, subject to continued exposure to coronavirus leading to possible reinfection. The model accounts for the differences in induced immunity in the two populations and the spread of infection due to movements of various populations in space. To realistically describe the nature of immunity, which has been found to decline with time following vaccination or infection, the rate of infection is expressed here as an integral of a function of the specific rate of infection that increases exponentially with time, depending on how long it is after the subjects have been infected with, or vaccinate against, the virus. The model is analyzed for its stability, and the contour plot is presented. The analytical solutions of the model system are derived in the form of traveling waves, using the modified extended hyperbolic tangent method. Inspection and interpretation of the different shapes of these plots yield valuable insights. [ABSTRACT FROM AUTHOR]

Direct spring operated pressure relief valves connected to a constantly charged vessel and a downstream pipe have a complex dynamics. The vessel-valve subsystem is described with an autonomous system of ordinary differential equations, while the presence of the pipe adds two partial differential equations to the mathematical model. The partial differential equations are transformed to a delay algebraic equation coupled to the ordinary differential equations. Due to a square root nonlinearity, the system is implicit. The linearized system can be transformed to a standard system of neutral delay differential equations (NDDEs) having more elaborated literature than the delay algebraic equations. First, the different forms of the mathematical model are presented, then the transformation of the linearized system is conducted. The paper aims at introducing this unusual mathematical model of an engineering system and inducing research focusing on the methodology to carry out bifurcation analysis in implicit NDDEs.

The Schamel-Korteweg-de Vries (S-KdV) equation including a square root nonlinearity is very important pattern for the research of ion-acoustic waves in plasma and dusty plasma. As known, it is significant to discover the traveling wave solutions of such equations. Therefore, in this paper, some new traveling wave solutions of the S-KdV equation, which arises in plasma physics in the study of ion acoustic solitons when electron trapping is present and also it governs the electrostatic potential for a certain electron distribution in velocity space, are constructed. For this purpose, the Bernoulli Sub-ODE and modified auxiliary equation methods are used. It has been shown that the suggested methods are effective and give different types of function solutions as: hyperbolic, trigonometric, power, exponential, and rational functions. The applied computational strategies are direct, efficient, concise and can be implemented in more complex phenomena with the assistant of symbolic computations. The results found in the paper are of great interest and may also be used to discover the wave sorts and specialities in several plasma systems.

In recent years, the modified Extended Direct Algebraic Method (mEDAM) has demonstrated to be an effective method for finding novel soliton solutions to nonlinear Fractional Partial Differential Equations that appear in the fields of science and engineering. In this study, mEDAM is used to explore special soliton solutions for the (3+1)-Fractional Modified Zakharov–Kuznetsov Equation (FMZKE) arising in electrical engineering which is first mathematically modeled through the implementation of Kirchhoff's Law to the nonlinear electrical transmission line circuit. The wave behaviours of various soliton solutions are graphically represented using three-dimensional (3D) graphs which provide a clear and thorough explanation of the usefulness and high performance of the suggested method. The acquired results offer helpful insights to the behavior and dynamics of the FMZKE, leading to a more deeply comprehending of the model and its applications in various fields. [ABSTRACT FROM AUTHOR]

The (3 + 1)-dimensional hyperbolic nonlinear Schrödinger equation (HNLS) is used as a model for different physical phenomena such as the propagation of electromagnetic fields, the dynamics of optical soliton promulgation, and the evolution of the water wave surface. In this paper, new and different exact solutions for the (3 + 1)-dimensional HNLS equation is emerged by using two powerful methods named the Riccati equation method and the F-expansion principle. The behaviors of resulting solutions are different and expressed by dark, bright, singular, and periodic solutions. The physical explanations for the obtained solutions are examined by a graphical representation in 3d profile plots. [ABSTRACT FROM AUTHOR]

In this paper, a comprehensive analysis of traveling wave solutions of two nonlinear Schrödinger type equations are carried out with help of three different integration techniques namely the tanh–coth, Kudryashov and sine–cosine methods. These equations include the (2 + 1)-dimensional perturbed nonlinear Schrödinger's equation and cubic–quintic nonlinear Schrödinger's equation. The obtained travelling wave solutions are in the form of rational function solutions, trigonometric function solutions, exponential function solutions and hyperbolic function solutions. Our proposed results showed that these techniques are reliable to study the nonlinear PDEs in fiber optics. The higher order cubic–quintic nonlinear Schrödinger equation (NLSE) explains the transmission of incredibly low signals and broadband communications that stretch into the spectral region, as well as the doping of optical fiber and the encryption of data in optical fibers. [ABSTRACT FROM AUTHOR]

In this study, the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients (vcBK) which describes the interaction of Riemann waves has been solved two times by using the balance technique. First, the balance procedure was used to reduce the vcBK equation into a fourth-order nonlinear ordinary differential equation and by its implementation again composed with the auxiliary method, many novel periodic, kink and solitary wave solutions were obtained for the vcBK equation. Second, the balance method was used to construct the Bäcklund transformation for the vcBK equation, by which n-soliton solutions were obtained. Finally, some figures were given to show the soliton interaction for different values of the variable coefficients.

A Hamiltonian reduction approach is defined, studied, and finally used to derive asymptotic models of internal wave propagation in density stratified fluids in two-dimensional domains. Beginning with the general Hamiltonian formalism of Benjamin (1986 J. Fluid Mech. 165 445–74) for an ideal, stably stratified Euler fluid, the corresponding structure is systematically reduced to the setup of two homogeneous fluids under gravity, separated by an interface and confined between two infinite horizontal plates. A long-wave, small-amplitude asymptotics is then used to obtain a simplified model that encapsulates most of the known properties of the dynamics of such systems, such as bidirectional wave propagation and maximal amplitude travelling waves in the form of fronts. Further reductions, and in particular devising an asymptotic extension of Dirac's theory of Hamiltonian constraints, lead to the completely integrable evolution equations previously considered in the literature for limiting forms of the dynamics of stratified fluids. To assess the performance of the asymptotic models, special solutions are studied and compared with those of the parent equations [ABSTRACT FROM AUTHOR]

This investigation proposes the novelty of the modified (G ′ G 2) -expansion method to look for new exact traveling wave solutions to two important nonlinear evolution equations such as the Konno–Oono equation and the Boussinesq equation. The simplicity and dependability of this approach make it advantageous for solving nonlinear issues. The technique involves wave transformation to get the nonlinear evolution equation down to the corresponding ordinary differential equations. The solutions include some new exact traveling solutions and are categorized into three classes of trigonometric, hyperbolic, and rational solutions. Numerical simulation is used to support the solutions and give them physical meaning. These results contain a large number of travelling wave solutions that are crucial for explaining certain scientific phenomena in fluid media. [ABSTRACT FROM AUTHOR]

The Schamel-Korteweg-de Vries (S-KdV) equation including a square root nonlinearity is very important pattern for the research of ion-acoustic waves in plasma and dusty plasma. As known, it is significant to discover the traveling wave solutions of such equations. Therefore, in this paper, some new traveling wave solutions of the S-KdV equation, which arises in plasma physics in the study of ion acoustic solitons when electron trapping is present and also it governs the electrostatic potential for a certain electron distribution in velocity space, are constructed. For this purpose, the Bernoulli Sub-ODE and modified auxiliary equation methods are used. It has been shown that the suggested methods are effective and give different types of function solutions as: hyperbolic, trigonometric, power, exponential, and rational functions. The applied computational strategies are direct, efficient, concise and can be implemented in more complex phenomena with the assistant of symbolic computations. The results found in the paper are of great interest and may also be used to discover the wave sorts and specialities in several plasma systems. [ABSTRACT FROM AUTHOR]

The Landau-Ginzburg-Higgs (LGH) equation explains the ocean engineering models, superconductivity and drift cyclotron waves in radially inhomogeneous plasma for coherent ion-cyclotron waves. In this paper, with a simple modification of the Ablowitz-Kaup-Newell-Segur (AKNS) formalism, the integrability of LGH equation is proved by deriving the Lax pair. Hence for that, the inverse scattering transformation (IST) is applied, and the travelling wave solutions are obtained and graphically represented in 2d and 3d profiles. [ABSTRACT FROM AUTHOR]

We use the m + 1 G ′ -expansion method in reaching the exact solution of the (3+1)-dimensional B-type Kadomtsev-Petviashvili-Boussinesq, Newel-Whitehead-Segel, and Zeldovich equations. New solutions in form of the kink, complex and singular solutions are reported. On the other hand, the Adomian decomposition method is employed to find approximate solutions to the the (3+1)-dimensional B-type Kadomtsev-Petviashvili-Boussinesq equation. The three-dimensional figures and their corresponding contour plots for the reported solutions are drawn. Also, a table is presented for the approximate solutions. The reported results may be useful in studying physical features of various nonlinear mathematical models. [ABSTRACT FROM AUTHOR]

In this article, two methods are proposed to solve the fractional Cahn-Hilliard equation. This model describes the process of phase separation with nonlocal memory effects. Cahn-Hilliard equations have numerous applications in real-world scenarios, e.g., material sciences, cell biology, and image processing. Different types of solutions have been obtained. For this, the fractional complex transformation has been used to convert fractional differential equation to ordinary differential equation of integer order. As a result, these solutions are new solutions that do not exist in the literature.

The equations in which time derivative appears in the highest order derivative term are called pseudo-parabolic equations. In this article, we will propose new singular and dark soliton solutions to a few such equations. We first find the symmetry algebra of such equations, and then by invariant reduction we obtain the exact solutions. Then, we classify the solutions by using their Lie subalgebras. Also, we will discuss the travelling wave solutions by using tanh–coth method. Moreover, we will discuss reciprocal Bäcklund transformations of the conservation laws.

In this study, the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients (vcBK) which describes the interaction of Riemann waves has been solved two times by using the balance technique. First, the balance procedure was used to reduce the vcBK equation into a fourth-order nonlinear ordinary differential equation and by its implementation again composed with the auxiliary method, many novel periodic, kink and solitary wave solutions were obtained for the vcBK equation. Second, the balance method was used to construct the Bäcklund transformation for the vcBK equation, by which n-soliton solutions were obtained. Finally, some figures were given to show the soliton interaction for different values of the variable coefficients. [ABSTRACT FROM AUTHOR]

This article examines the coupled KdV equations and the coupled system of variant Boussinesq equations with beta time derivative and explores their travelling wave solutions. This work explains the evolution of waves with fractional parameter. The simple ansatz approach produces a variety of novel solutions in terms of hyperbolic and periodic functions. Graphical representation of solutions are also presented.

In this paper, we use an effective method which is the rational sine-Gordon expansion method to present new wave simulations of a governing model. We consider the (1+1)-dimensional conformable Fisher equation which is used to describe the interactive relation between diffusion and reaction. Various types of solutions such as multi-soliton, kink, and anti-kink wave soliton solutions are obtained. Finally, the physical behaviours of the obtained solutions are shown by 3D, 2D, and contour surfaces. [ABSTRACT FROM AUTHOR]

We consider the following nonlinear parabolic equation (F (v)) x + (G (v)) τ = (D (v)) x x + ρ (v) , v ∈ [ α , β ] where F , G are generic C 1 -functions in [ α , β ] , D ∈ C 1 [ α , β ] ∩ C 2 (α , β) is positive inside (α , β) (possibly vanishing at the extreme points), and finally ρ is a monostable reaction term. We investigate the existence and the properties of travelling wave solutions for such an equation and provide their classification between classical and sharp solutions, together with an estimate of the minimal wave speed. [ABSTRACT FROM AUTHOR]

This study focuses on investigating the modified Benjamin-Bona-Mahony equation that is used to model the long wave in nonlinear dispersive media of the optical illusion field. Two effective techniques, the variational direct method and He's frequency formulation method, are employed to seek the travelling wave solutions. Using these two techniques, abundant exact solutions such as the bright wave, bright-dark wave, bright-like wave, kinky-bright wave and periodic wave solutions, are obtained. The 3-D contours and 2-D curves are drawn to present the dynamic physical behaviors of the solutions by assigning the proper parameters. It shows that the proposed methods are effective but simple and only need one or two steps to construct the exact solutions, which are expected to provide some new insights to study the travelling wave solutions of the PDEs arising in physics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the local fractional Kundu–Mukherjee–Naskar (LFKMN) equation in (2+1) dimensional case. The Yang's local fractional calculus tool has fulfilled a significant character in defining the fractal behaviours in a fractal space or microgravity space that arise in applied nonlinear sciences. The travelling wave transformation of the non-differentiable type is introduced and we retrieve successfully the non-differentiable exact traveling wave solutions (soliton pulses in (2+1)-dimensions) of LFKMN equation with aid of generalized exp-function method in the form of generalized functions described on Cantor sets. With the help of Mathematica package program, 3D graphs were drawn for the special values of the parameters in the solutions, and the physical structures of the solutions obtained in this way were also observed. The solutions obtained can be used in the explanation of physical phenomena occurring in propagation of rogue waves in oceans and higher order optical solitons in optical fibers in current-like nonlinearities.The deduced explicit solutions will cause a new pathway of the nonlinear wave theory by the help of local fractional derivative. The proposed approach is demostrated to ensure a beneficial tool to solve the local fractional nonlinear evolution equations in applied nonlinear sciences. [ABSTRACT FROM AUTHOR]

Based on two algorithm integrations, such as the exp (- Φ (ξ)) -expansion method and the hyperbolic function method, we build dark, bright and trigonometric function solution to the Klein–Gordon equations with M-fractional derivative of order α . By adopting the travelling-wave transformation, the constraint condition between the model coefficients and the travelling-wave frequency coefficient for the existence of soliton solutions is also obtained. Moreover, miscellaneous soliton solutions obtained is depicted in 3D and 2D. [ABSTRACT FROM AUTHOR]

From point of view of two distinct various techniques accurate solutions for the thin-film ferroelectric materials equation which plays vital role in optics are implemented which represent haw utilized waves propagate through ferroelectric materials. The first one is the modified simple equation method which surrender to the balance rule and gives closed form analytical solution for all applicable problems while the second has personal profile named theas Riccati-Bernoulli Sub-ODE method which not surrender to the balance rule and has special effective properties in calculations. These methods can be used perfectly to achieve the exact solutions for different types of nonlinear problems arising in various branches of science. Via giving the appearing variables definite values, 2D and 3D- impressive graphs of some achieved solutions are drawled. [ABSTRACT FROM AUTHOR]

In this article, we will implement the(G'=G)-expansion method which is used for the first time to obtain new optical soliton solutions of the thin-film ferroelectric materials equation (TFFME). Also, the numerical solutions of the suggested equation according to the variational iteration method (VIM) are demonstrated effectively. A comparison between the achieved exact and numerical solutions has been established successfully. [ABSTRACT FROM AUTHOR]

In this article, the Paul–Painleve approach (PPA) which was formulated recently and built on the balance role has been used perfectly to achieve new impressive solitary wave solutions to the nanosoliton of ionic waves (NSOIW) propagating along the microtubules in the living cells. In addition, variational iteration method (VIM) has been applied in the same vein and parallel to establish numerical solutions of this model. [ABSTRACT FROM AUTHOR]

Based on the classical theory of simple materials of differential type and the results on the analytical form of constitutive models consistent with the laws of thermodynamics, we introduce a very general response function for the Cauchy stress tensor of a dispersive hyperelastic solid. Next, by focusing on the propagation of localised waves in slightly dispersive quasi incompressible solids, we prove the existence of a rich variety of solitary wave solutions as well as kink wave solutions. Our analysis and results can be easily specialised to shape memory alloys. [Display omitted] • We develop a mathematical model for the stress tensor of dispersive hyperelastic solids within the theory of simple materials of differential type. • We study the wave propagation in dispersive hyperelastic solids. • In the small but finite regime, we find exact solutions representing solitary and kink waves. [ABSTRACT FROM AUTHOR]

One of the fastest growing fields of study in nonlinear optics is optical solitons. This article investigates the optical soliton solutions of the coupled Kaup-Newell equation in birefringent fibers without four wave mixing. This model helps explain how modified structures, particularly Alfven waves, propagate in optical fiber and plasma physics. The results presented in this research clearly show how well the new mapping method handle nonlinear Kaup-Newell equation in coupled vector form, in birefringent fiber without four-wave mixing, to extract the analytical soliton solutions. Four-wave mixing expresses a nonlinear optical effect in which four waves interact each other as a result of the third order nonlinearity. As a valuable outcome of this article, a variety of solitons are discovered including bright, dark, periodic, singular bell-shaped, gap, flat-kink, dark-singular, and w-shaped. The existence criteria for these solitons are also obtained that are presented as constraint conditions. The constraint relations guarantee the existence of such solitons. Furthermore, graphs in 2D, 3D and contour are created to show the physical characteristics of the solutions that are produced. The new mapping method is clear and effective, and the solutions provide a wealth of avenues for additional research. The technique can also be functional to other sorts of nonlinear evolution equations in contemporary areas of research. The outcomes presented in this work are new versions which have never been observed before in existing literature. [ABSTRACT FROM AUTHOR]

The current research manifests kink wave answers, mixed singular optical solitons, the mixed dark-bright lump answer, the mixed dark-bright periodic wave answer, and periodic wave answers to the conformable fractional ZK model, including power law nonlinearity by plugging the revised $(G'/G) $-expansion process. The constraint requirements for the occurrence of substantial solitons are provided. Under the selection of proper values of a, b, n, t, λ, μ and α, the 2D and 3D pictures to a few of the recorded answers are sketched. From our obtained solutions, we might decide that the investigated procedure is hugely muscular, sincere, and essential in rendering various new soliton solutions of distinct nonlinear conformable fractional evolution equations and accordingly, we shall bring it up in our future investigations.

This article applied the Riccati–Bernoulli (RB) sub-ODE method in order to get new exact solutions for the long–short-wave interaction (LS) equations. Namely, we obtain deterministic and random solutions, since we consider the proposed method in deterministic and random cases. The RB sub-ODE technique gives the travelling wave solutions in forms of hyperbolic, trigonometric and rational functions. It is shown that the proposed method gives a robust mathematical tool for solving nonlinear wave equations in applied science. Furthermore, some bi-random variables and some random distributions are used in random case corresponding to the LS system. The stability for the obtained solutions in random case is considered. In addition, there is a display of several numerical simulations, which helps to understand the physical phenomena of these soliton wave solutions.

In this research, we constructed the exact travelling and solitary wave solutions of the Kudryashov–Sinelshchikov (KS) equation by implementing the modified mathematical method. The KS equation describe the phenomena of pressure waves in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. Our new obtained solutions in the shape of hyperbolic, trigonometric, elliptic functions including dark, bright, singular, combined, kink wave solitons, travelling wave, solitary wave and periodic wave. We showed the physical interpretation of obtained solutions by three-dimensional graphically. These new constructed solutions play vital role in mathematical physics, optical fiber, plasma physics and other various branches of applied sciences.

This paper poses the Riccati–Bernoulli sub-ODE method in order to find the exact (random) travelling wave solutions for the (2+1)-dimensional cubic nonlinear Klein–Gordon (cKG) equation and the (2+1)-dimensional nonlinear Zakharov–Kuznetsov modified equal width (ZK-MEW) equation. The obtained travelling wave solutions are expressed by the hyperbolic, trigonometric and rational functions. Indeed, these solutions reflect some interesting physical interpretation for nonlinear phenomena. We discuss our method in deterministic case and in a random case. Additionally, we can show and discuss this method under some random distributions. Finally, some three-dimensional graphics of some solutions have been illustrated.

By using bifurcation theory of planar dynamical systems, we classify all bounded travelling wave solutions of the general Burgers-Boussinesq equation, and we give their corresponding phase portraits. In different parametric regions, different types of trav- elling wave solutions such as solitary wave solutions, cusp solitary wave solutions, kink(anti kink) wave solutions and periodic wave solutions are simulated. Also in each parameter bifurcation sets, we obtain the exact explicit parametric representation of all travelling wave solutions.

The travelling wave solutions and conservation laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) equation are considered in this paper. Under the travelling wave frame, the BKK equation is transformed to a system of ordinary differential equations (ODEs) with two dependent variables. Therefore, it happens that one dependent variable u can be decoupled into a second order ODE that corresponds to a Hamiltonian planar dynamical system involving three arbitrary constants. By using the bifurcation analysis, we obtain the bounded travelling wave solutions u, which include the kink, anti-kink and periodic wave solutions. Finally, the conservation laws of the BBK equation are derived by employing the multiplier approach. [ABSTRACT FROM AUTHOR]

In this article we will utilize new perception of the Benjamin-Bona-Mahony equation that represents developed stretch for the Korteweg-de Varies equation which denotes to the unidirectional propagation of small amplitude long waves on the surface of hydro magnetic and acoustic waves in channel significantly for shallow water. The main idea of this research realizing new exact and hence solitary solutions of this new perception with its spatial and temporal variables and fractional order whose called 3D-fractional Wazwaz-Benjamin-Bona-Mahony model (3D -FWBBM). Two important different techniques are invited for the first time to proposed this new perception of the exact solutions, the first one is the modified simple equation method (MSEM) and the second is the Riccati-Bernolli Sub-ODE method (RBSOM) whose are used effectively to obtain accurate description of the wave solutions to this equation. [ABSTRACT FROM AUTHOR]