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This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations. [ABSTRACT FROM AUTHOR]

In this paper, we use the Lie symmetry analysis to obtain the closed-form solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili Benjamin–Bona–Mahony (gKP-BBM) equation. We obtain the infinitesimal generators, commutator table of Lie algebra, adjoint representation of subalgebras, symmetry group, and similarity reduction for the gKP-BBM equation. Also, we evaluated the interactions of soliton solutions of the gKP-BBM equation. It is observed that multi-soliton complexes are self-localized in state wherein, several fundamental solitons including bright solitons and dark solitons are nonlinearly superimposed. A notable observation is the existence of topological defects in soliton interactions, where two solitons appear to be separated, with the adjoining structure being “out of phase” with each other. The kink, acting as a twist in the soliton’s value, overcomes topological defects, leading to a transition from one phase to another. This comprehensive analysis contributes to a deeper understanding of the dynamics and interactions of solitons in the (3+1)-dimensional gKP-BBM equation. [ABSTRACT FROM AUTHOR]

This study aims to employ the extended direct algebraic method (EDAM) to generate and evaluate soliton solutions to the nonlinear, space-time conformable Estevez Mansfield-Clarkson equation (CEMCE), which is utilized to simulate shallow water waves. The proposed method entails transforming nonlinear fractional partial differential equations (NFPDEs) into nonlinear ordinary differential equations (NODEs) under the assumption of a finite series solution by utilizing Riccati ordinary differential equations. Various mathematical structures/solutions for the current model are derived in the form of rational, exponential, trigonometric, and hyperbolic functions. The wide range of obtained solutions allows for a thorough analysis of their actual wave characteristics. The 3D and 2D graphs are used to illustrate that these behaviors consistently manifest as periodic, dark, and bright kink solitons. Notably, the produced soliton solutions offer new and critical insights into the intricate behaviors of the CEMCE by illuminating the basic mechanics of the wave's interaction and propagation. By analyzing these solutions, academics can better understand the model's behavior in various settings. These solutions shed light on complicated issues such as configuration dispersion in liquid drops and wave behavior in shallow water. [ABSTRACT FROM AUTHOR]

This investigation discusses the modified M-truncated form of the perturbed Chen–Lee–Liu (pCLL) dynamical equation. The pCLL equation is a generalization of the original CLL equation, which describes the propagation of optical solitons in optical fibers. The pCLL equation includes additional terms that account for various influences such as chromatic dispersion, nonlinear dispersion, inter-modal dispersion, and self-steepening. A new version of the generalized exponential rational function method is utilized to obtain multifarious types of soliton solutions. Moreover, the planar dynamical system of the concerned equation is created using a Hamiltonian transformation, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are added to the analytical results to better understand the dynamic behavior of these solutions. The study's findings could be extremely useful in solving additional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

Highly important is a three-dimensional nonlinear partial differential equation because for many physical systems, one can, subject to suitable idealizations, formulate a differential equation that describes how the system changes in time. Thus, this article comprehensively reveals the investigation carried out on a (3+1)-dimensional generalized fifth-order Zakharov–Kuznetsov equation with power-law as well as dual power-law nonlinearities analytically, where the fifth-order term involved is regarded as a dispersion perturbation term. We utilize the well-celebrated Noether's theorem to comprehensively construct conserved currents of the underlying equation. A detailed Lie group analysis of the understudied model consisting of power-law nonlinearities is further performed. This involves performing reductions of the underlying models using their Lie point symmetries. In consequence, various invariants are found. In addition, the equation reduces to diverse ordinary differential equations using its point symmetries and consequently diverse solutions of interest were achieved. Moreover, we derive some solitary wave solutions by invoking the newly introduced logistic function technique for some particular cases of the equation under consideration. In consequence, we achieve some exponential function solutions. In addition, the physical meaning of the results is put on the front burner by revealing the wave dynamics of these solutions via graphical depictions. Finally, the significance of the robust and detailed findings in the work are further corroborated with various real-world applications. [ABSTRACT FROM AUTHOR]

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

Despite the historical position of the F-expansion method as a method for acquiring exact solutions to nonlinear partial differential equations (PDEs), this study highlights its superiority over alternative auxiliary equation methods. The efficacy of this method is demonstrated through its application to solve the convective–diffusive Cahn–Hilliard (cdCH) equation, describing the dynamic of the separation phase for ternary iron alloys (Fe–Cr–Mo) and (Fe–X–Cu). Significantly, this research introduces an extensive collection of exact solutions by the auxiliary equation, comprising fifty-two distinct types. Six of these are associated with Weierstrass-elliptic function solutions, while the remaining solutions are expressed in Jacobi-elliptic functions. I think it is important to emphasize that, exercising caution regarding the statement of the term 'new,' the solutions presented in this context are not entirely unprecedented. The paper examines numerous examples to substantiate this perspective. Furthermore, the study broadens its scope to include soliton-like and trigonometric-function solutions as special cases. This underscores that the antecedently obtained outcomes through the recently specific cases encompassed within the more comprehensive scope of the present findings. [ABSTRACT FROM AUTHOR]

In this paper, the integrable (4 + 1)-dimensional Fokas equation is investigated. Exploiting a set of non-singular local multipliers, we present a set of local conservation laws for the equation. The nonlocally related partial differential equation (PDE) systems are found. Nine nonlocally related systems are discussed reveal thirty five interesting closed form solutions of the equation. These solutions contain different types of wave solutions, double soliton, multi-solitons, kink and periodic wave solutions. Some of the resulting solutions are graphically illustrated. Furthermore, we apply the sine-cosine method to find other traveling wave solutions for this equation. [ABSTRACT FROM AUTHOR]

The goal of this research is to utilize some ansatz forms of solutions to obtain novel forms of soliton solutions for the Benney–Luke equation. It is a mathematically valid approximation that describes the propagation of two-way water waves in the presence of surface tension. By using ansatz forms of solutions, with an appropriate set of parameters, the lump soliton, periodic cross-kink waves, multi-waves, breather waves, Ma-breather, Kuznetsov–Ma-breather, periodic waves and rogue waves solutions can be obtained. Breather waves are confined, periodic, nonlinear wave solutions that preserve their amplitude and shape despite alternating between compression and expansion. For some integrable nonlinear partial differential equations, a lump soliton is a confined, stable solitary wave solution. Rogue waves are unusually powerful and sharp ocean surface waves that deviate significantly from the surrounding wave pattern. They pose a threat to maritime safety. They typically show up in solitary, seemingly random circumstances. Periodic cross-kink waves are a particular type of wave pattern that has frequent bends or oscillations that cross at right angles. These waves provide insights into complicated wave dynamics and arise spontaneously in a variety of settings. In order to predict the wave dynamics, certain 2D, 3D and contour profiles are also analyzed. Since these recently discovered solutions contain certain arbitrary constants, they can be used to describe the variation in the qualitative characteristics of wave phenomena. [ABSTRACT FROM AUTHOR]

The nonlinear evolution equations (NLEEs) play a significant role in applied mathematics, including ordinary and partial differential equations, which are frequently used in many disciplines of applied sciences. The Drinfeld–Sokolov–Wilson (DSW) equation and the Burgers equation are the fundamental equations occurring in various areas of physics and applied mathematics, such as nonlinear acoustics and fluid mechanics. The new generalized (G ′ / G) -expansion method is an effective and more powerful mathematical tool for solving NLEEs arising in applied mathematics and mathematical physics. In this article, we investigate further exact solutions as well as soliton solutions to these couple of nonlinear evolution equations by executing the new generalized (G ′ / G) -expansion method. A large number of soliton solutions, including single soliton, bell-shaped soliton, kink-shaped soliton, singular kink soliton, singular soliton, periodic soliton, irregular periodic soliton solutions, and others, have been retrieved. Each of the derived solutions includes an explicit function of the variables in the equations under consideration. We provide some 3D plots to visualize and realize the characteristics of these solutions. It has been established that the suggested techniques are more potential and successful at obtaining soliton solutions for nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this study, the generalized perturbed-KdV partial differential equation is examined. Furthermore, symmetry generators address the Lie invariance criteria. The suggested approach produces the Lie algebra, where translation symmetries in space and time are associated with mass conservation and conservation of energy respectively, the other symmetries are scaling or dilation. The optimal system of the obtained system developed. By using Lie Group methods, the generalized perturbed-KdV partial differential equation is changed using suitable similarity transformations through a system of highly nonlinear ordinary differential equations. The new extended direct algebraic approach is applied to get the soliton solutions. As a result, a plane solution, periodic stumps, compacton, smooth soliton, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, peakon soliton, anti-peaked with decay, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions are developed. The behavior of certain solutions is shown in 3-D and 2-D for specific values of the physical components in the studied equation. The outcomes hold significance for elevating research to a more impactful and effective level. The whole set of local conservation laws for the generalized perturbed-KdV equation for any arbitrary constant coefficients is found by applying the conservation laws multiplier. These findings are pivotal for advancing the current understanding and pushing the boundaries of knowledge to new heights. [ABSTRACT FROM AUTHOR]

The primary focus of this paper is the determination of novel soliton solutions for the (2 + 1) -dimensional Heisenberg ferromagnetic spin chain (HFSC) problem. The suggested model may be thought of as a special case of the nonlinear Schrödinger equation. Exact solutions to the proposed equation have been found using efficient analytical approaches such as the singular manifold method and the tan (η 2) -expansion method. These proposed methods are significant mathematical tools to obtain the exact traveling wave solutions of nonlinear complex partial differential equations (PDEs). After decomposing the governing problem into its real and imaginary components, we analyze both the real and imaginary components separately. Soliton solutions are obtained using computational tools like Maple. Trigonometric function solutions, rational solutions and hyperbolic function solutions are found for Heisenberg ferromagnetic model by the application of the proposed techniques. Furthermore, the physical properties of the solutions, such as hyperbolic functions, are important. The hyperbolic tangent is used in the computation of magnetic moment and velocity in special relativity. As a result, we hypothesize that the obtained results have physically comparable meanings to those stated by these collocations. By selecting suitable values of arbitrary parameters, it has been observed graphically that the amplitude of obtained solutions can be changed. Graphical representations of few of the reported solutions are presented in both two and three dimensions by choosing appropriate values of parameters. The suggested approaches are very efficient and reliable for finding exact solutions of wide range of nonlinear PDEs in a complex medium. The advantage of these techniques is evident since they are not limited in their ability to locate such wave patterns. Back substitution in the original equation using Maple 18 validated all solutions found, and the physical significance of these results has been underlined. The constraint conditions for the existence of constructed solutions are also provided in this paper. [ABSTRACT FROM AUTHOR]

This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations.

Some exact solutions of boundary or initial conditions formulated for Bogomolny equations (derived by using the strong necessary conditions and associated with some ordinary equation and some partial differential equations) have been found. The solution obtained for the restricted baby Skyrme model, as well the density of energy for this solution, are localized. Moreover, it turns out that the densities of the ungauged Hamiltonian and the gauged Hamiltonian are correspondingly, non-zero and zero for the found solution of the Cauchy problem associated with the Bogomolny equation of the restricted baby Skyrme model. Hence, a degeneracy of the Hamiltonian for this model has been established. As such, one can see the breaking of some symmetry. [ABSTRACT FROM AUTHOR]

In this paper, the main motive is to mathematical explore the thin-film ferroelectric material partial differential equation which addresses the Ferroelectrics, that are being examined as key materials for applications in piezoelectric, pyroelectric electrostrictive, linear, and nonlinear optical systems. Thin ferroelectric films are used in a variety of modern electrical devices because they are both nonlinear ferroelectric and dielectric materials. This article appropriates the fractional travelling wave transformation allowing the partial differential equation to be changed into an ordinary differential equation. The considered fractional model is explored through employing the combo of G ′ G 2 − expansion method and new extended direct algebraic methodology. As an outcome, numerous types of soliton solutions like, Periodic pattern with anti-peaked crests and anti-troughs, singular solution, mixed complex solitary shock solution, mixed singular solution, mixed shock singular solution, mixed trigonometric solution, mixed periodic, periodic solution and mixed hyperbolic solution obtain via Mathematica. In addition, the G ′ G 2 − expansion technique produces singular, trigonometric, and hyperbolic solutions with different soliton families. The revealed solution will improve the mathematical analysis of this model and the associated physical phenomenon's. In order to visualize the graphical propagation of the obtained fractional soliton solutions, 3D, 2D, and contour graphics are generated by choosing appropriate parametric values. The impact of fractional parameter β is also graphically displayed on the propagation of solitons. [ABSTRACT FROM AUTHOR]

In this manuscript, a new form of the generalized q-deformed Sinh-Gordon equation is investigated which could model physical systems with broken symmetries and to incorporate phenomena involving amplification or dissipation. The proposed model is explored based on the Lie symmetry approach. Using similarity reduction, the partial differential equation is transformed into an ordinary differential equation. By employing the generalized auxiliary equation approach, precise results for the derived equation are obtained. The solutions are graphically depicted as 3D, 2D, and contour plots. Furthermore, the qualitative analysis of the considered model is investigated by employing the concepts of bifurcation and chaos. The phase profiles are displayed for different sets of the parameters. Additionally, by applying an external periodic strength, quasi-periodic and chaotic behaviors are documented. Various tools for detecting chaos are discussed, including 3D and 2D phase patterns, time series, and Poincaré maps. Additionally, a sensitivity analysis is conducted for various initial conditions. The obtained findings are unique and indicate the viability and efficacy of the suggested strategies for evaluating soliton solutions and phase illustrations for various nonlinear models. [ABSTRACT FROM AUTHOR]

In this research, we concentrated on the dispersive long wave system in two horizontal directions for dispersive nonlinear waves on the shallow water of an open sea or a wide channel of finite depth. We investigated this governing system by using two different methodologies, namely the generalized exponential rational function (GERF) method, and the new modified generalized exponential rational function (MGERF) method. The GERF method was first introduced by Ghanbari and Inc (Eur. Phys. J. Plus 133:-142, 2018) for finding the soliton solutions for highly nonlinear partial differential equations (NLPDEs). This technique is very reliable and straightforward and reduces the NLPDEs into ordinary differential equations (ODEs) under the wave transformation. Being motivated by the GERF technique, we proposed a newly modified generalized exponential rational function (MGERF) method under wave transformation. We obtained a diverse set of solutions involving trigonometric forms, hyperbolic forms, rational forms, and so on, which have a broad application spectrum in fields such as plasma physics, nonlinear optics, optical fibers, and nonlinear sciences by utilizing these methods. Due to the presence of various arbitrarily chosen constants, these solutions exhibit extensive and rich dynamical behavior. Based on the dynamical behaviors, we discovered that the soliton solutions were collisions of solitons, breather-like solitons, line-form solitons, multi-solitons, solitary waves, lump-form solitons, and other forms. Consider a nonlinear system with dispersive and dispersion terms, a nonlinear Schrödinger equation, and fractional nonlinear evolution equations, which will yield additional interesting and more achievable results. Finding solutions to these equations in solitary wave solution forms will be a difficult task. [ABSTRACT FROM AUTHOR]

Purpose: The soliton solutions are obtained by using extended rational sin/cos and sinh-cosh method. The methods are powerful and have ease of use. Applying wave transformation to the nonlinear partial differential equations (NLPDEs) and the considered equation turns into a nonlinear differential equation (NODE). According to the methods, the solution sets of the NODE are supposed to the form of the rational terms as sinh/cosh and sin/cos and the trial solutions are substituted into the NODE. Collecting the same power of the trigonometric functions, a set of algebraic equations is derived. Design/methodology/approach: The main purpose of this paper is to obtain soliton solutions of the modified equal width (MEW) equation. MEW is a form of regularized-long-wave (RLW) equation that represents one-dimensional wave propagation in nonlinear media with dispersion processes. This is also used to simulate the undular bore in a long shallow water canal. Findings: Thus, the solution of the main PDE is reduced to the solution of a set of algebraic equations. In this paper, the kink, singular and singular periodic solitons have been successfully obtained. Originality/value: Illustrative plots of the solutions have been presented for physical interpretation of the obtained solutions. The methods are powerful and might be used to solve a broad class of differential equations in real-life problems. [ABSTRACT FROM AUTHOR]

The Complex Ginzburg-Landau equation (CGLE) has been suggested and investigated as a model for turbulent dynamics in nonlinear partial differential equations, It is a particularly intriguing model in this aspect since it is a dissipative form of the nonlinear SchrOdinger equation, The model is studied with quadratic-cubic law nonlinear fiber by using φ6-model expansion method, bright, dark, dark-singular, singular and dark-bright optical soliton solutions are obtained, The findings of this study might assist in comprehending some of the physical implications of various nonlinear physics models, The hyperbolic sine, for example, appears in the calculation of the Roche limit and gravitational potential of a cylinder, while the hyperbolic cosine function is the shape of a hanging cable (the so-called catenary), Finally, there are some discussions regarding new complex solutions, it is explored by giving physical meaning to the constants found in traveling wave solutions, which are both physically and mathematically significant, The three-dimensional, two-dimensional simulations and contour plots are used to enhance these discussions. [ABSTRACT FROM AUTHOR]

Exact solutions of the fractional Kraenkel–Manna–Merle system in saturated ferromagnetic materials have been studied. Using the fractional complex transforms, the fractional Kraenkel–Manna–Merle system is reduced to ordinary differential equations, (1 + 1)-dimensional partial differential equations and (2 + 1)-dimensional partial differential equations. Based on the obtained ordinary differential equations and taking advantage of the available solutions of Jacobi elliptic equation and Riccati equation, soliton solutions, combined soliton solutions, combined Jacobi elliptic function solutions, triangular periodic solutions and rational function solutions, for the KMM system are obtained. For the obtained (1 + 1)-dimensional partial differential equations, we get the classification of Lie symmetries. Starting from a Lie symmetry, we get a symmetry reduction equation. Solving the symmetry reduction equation by the power series method, power series solutions for the KMM system are obtained. For the obtained (2 + 1)-dimensional partial differential equations, we derive their bilinear form and two-soliton solution. The bilinear form can also be used to study the lump solutions, rogue wave solutions and breathing wave solutions. [ABSTRACT FROM AUTHOR]

In this work, a unified lattice Boltzmann model is proposed for the fourth order partial differential equation with time-dependent variable coefficients, which has the form u t + α (t) (p 1 (u)) x + β (t) (p 2 (u)) x x + γ (t) (p 3 (u)) x x x + η (t) (p 4 (u)) x x x x = 0 . A compensation function is added to the evolution equation to recover the macroscopic equation. Applying Chapman-Enskog expansion and the Taylor expansion method, we recover the macroscopic equation correctly. Through analyzing the error, our model reaches second-order accuracy in time. A series of constant-coefficient and variable-coefficient partial differential equations are successfully simulated, which tests the effectiveness and stability of the present model. [ABSTRACT FROM AUTHOR]

In the given research, our main goal is to explore new soliton solutions in (2 + 1) -dimensional Heisenberg ferromagnetic spin chain (HFSC) equation. This was achieved via the extended (G ′ / G 2) -expansion method. For this, we first analyse the given (2 + 1) -dimensional HFSC system of a partial differential equation (PDE) that is obtained by separating the equation into real and imaginary parts. The solution of our equation is determined as rapidly convergent sequences effectively computed by using Mathematica software. Furthermore, a special class of nonlinear wave solutions like periodic, straight, decreasing, dark-singular soliton and dark bright singular soliton were obtained in Heisenberg ferromagnetic dynamics. The amplitude of the soliton is determined by giving different values of parameters. With a suitable choice of parameters, 3D and 2D graphical illustrations are reported. The methodology applied is effective for obtaining exact solutions for various fractional partial differential equations in complex medium. The numerical module of the results obtained is studied, with fascinating figures showing the physical significance of the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we obtained the exact solutions of the (1 + 1) -dimensional nonlinear dispersive modified Benjamin–Bona–Mahony (DMBBM) and the seventh-order Sawada–Kotera–Ito (S-K Ito) equations with the help of the (w / g) -expansion method specially the (g ′ / g 2) -expansion and (g ′) -expansion methods. Soliton solutions found for the given equations are in the form of hyperbolic, trigonometric and rational solutions. All obtained solutions were checked. 3D and 2D graphs of some solutions were given and discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study a fourth-order nonlinear symmetric regularized long-wave equation which arises in several physical applications including ion sound waves in plasma. This model also describes nonlinear ion-acoustic and space-charge waves. Lie group analysis and the tanh-coth method is utilized for the construction of solitary wave solutions to the equation. Based on each symmetry, we reduce the considered partial differential equation into different ordinary differential equations. The obtained equations are solved and new solitary wave patterns are reported. To understand the physical interpretation, some solitary wave structures are exhibited graphically. Also, by using the multiplier approach, we constructed the local conservation laws of the considered equation. [ABSTRACT FROM AUTHOR]

The quest for exact solutions to nonlinear partial differential equations has become a remarkable research subject in recent years. In this study, we employ the Kudryashov method and sub-equation method to retrieve the bright and dark soliton solutions of the generalized nonlinear Schrödinger-Korteweg-de Vries equations. Other soliton-type solutions like the periodic, singular, and rational solutions are achieved as well. These coupled equations occur in phenomena of interactions between short and long dispersive waves which are significant in various fields of applied sciences and engineering. The solutions obtained in this study have been verified with the help of the Mathematica package software. Furthermore, we present graphical representations of the solutions of bright and dark solitons for a useful understanding of the behavior and physical structures of the coupled equations considered. [ABSTRACT FROM AUTHOR]

In this work, a unified lattice Boltzmann model is proposed for the fourth order partial differential equation with time-dependent variable coefficients, which has the form ut+α(t)(p1(u))x+β(t)(p2(u))xx+γ(t)(p3(u))xxx+η(t)(p4(u))xxxx=0. A compensation function is added to the evolution equation to recover the macroscopic equation. Applying Chapman-Enskog expansion and the Taylor expansion method, we recover the macroscopic equation correctly. Through analyzing the error, our model reaches second-order accuracy in time. A series of constant-coefficient and variable-coefficient partial differential equations are successfully simulated, which tests the effectiveness and stability of the present model.

The higher dimensional Fokas equation is the integrable expansion of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations. The Fokas model has an important role in wave theory, to describe the physical phenomena of waves on the surface and inside the water. This article deals with the (4 + 1) -dimensional conformable space-time fractional-order Fokas partial differential equation. Two efficient methods, namely the generalized exp (- ϕ (ξ)) -expansion and improved F-expansion methods, are formulated for conformable fractional-order partial differential equation and new wave structures of fractional order Fokas model are constructed. The different kinds of new solitons are achieved such as bright soliton, dark soliton, Kink and anti-kink solitons, periodic solitary waves, and traveling waves. These new soliton waves are constructed at some values of fractional order α and using different parametric values of the methods by using the software package Mathematica. Newly obtained soliton solutions are compared with the available soliton solutions with different fractional derivatives in the literature. Some of the achieved results are explained 2D and 3D graphically. The new results interpreting that these obtained solutions can be a part, to complete the family of solutions and considered methods are effective, simple, and easy to use. Furthermore, this paper gives an idea, how can reduce the conformable fractional order higher dimensional partial differential equation into an ODE of one variable to obtain the exact solutions. These results and methods can be help to investigate the other higher-dimensional conformable fractional-order models which appear in nonlinear wave theory such as optics, quantum gases, hydrodynamics, photonics, plasmas, and solid-state physics. [ABSTRACT FROM AUTHOR]

In this paper, we constructed the variational principles for Bogoyavlensky–Konopelchenko equation, the generalized (3+1)-dimensional nonlinear wave in liquid containing gas bubbles and a new coupled Kadomtsev–Petviashvili (KP) equation via He's semi-inverse method. Based on this formulation, we obtained the solitary wave solutions via Ritz method. We explained the properties of the soliton waves numerically by some figures. Finally, the physical interpretation for these solutions are obtained. [ABSTRACT FROM AUTHOR]

The extension of exponential rational function method is obtained to construct a series of exact solutions for higher-dimensional Broer–Kaup–Kupershmidt (BKK) dynamical system. New and general solutions are found. The solutions reported in this work are kink solutions, anti-kink solutions and bright solutions. They are expressed in terms of rational exponential functions. A confrontation of our results with the well-known results are done and it comes from this study that the solutions obtained here are new. The mathematical method applied to search for our solutions can be used for other nonlinear partial differential equations. The graphics of the obtained solutions in this paper are shown. [ABSTRACT FROM AUTHOR]

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

This article deal with finding travelling wave solutions for the 7th order Sawada- Kotera-Ito dynamical wave equation which describes the evolution of steeper waves of shorter wavelength than KdV equations using modified extended direct algebraic method. The new solutions derived have various physical structure, we also give graphic representation of the exact and stable solutions. [ABSTRACT FROM AUTHOR]

In this work, the new coupled non-linear partial differential equations (CNLPDE) getting the time evolution of the curvatures of the evolving curve are derived in the Galilean space. Exact solutions for these new CNLPDE are obtained. Finally, Lie symmetry analysis is performed on these new CNLPDE and the algebra of Lie point symmetries of these new equations is found. [ABSTRACT FROM AUTHOR]

Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin, we provide a direct method for computing soliton solutions of the focusing nonlinear Schrödinger equation on the half‐line. The integrable BCs at the origin are represented by constraints of the Lax pair, and our method lies on dressing the Lax pair by preserving those constraints in the Darboux‐dressing process. The method is applied to two classes of solutions: solitons vanishing at infinity and self‐modulated solitons on a constant background. Half‐line solitons in both cases are explicitly computed. In particular, the boundary‐bound solitons, which are static solitons bounded at the origin, are also constructed. We give a natural inverse scattering transform interpretation of the method as evolution of the scattering data determined by the integrable BCs in space. [ABSTRACT FROM AUTHOR]

The purpose of this research is to investigate the nonlinear partial differential equation known as Benjamin–Bona–Mahoney–Peregrine equation. Two versatile techniques, namely the extended ( G ′ G 2 ) -expansion method and modified auxiliary equation method is used to extract the exact solutions of the nonlinear Benjamin–Bona–Mahoney–Peregrine equation. Consequently, a diverse range of novel collections of exact traveling wave solutions are identified such as kink, bright, singular kink and periodic soliton solution. To provide a physical understanding, several of the discovered solutions are shown in figures. The resulting solutions are of the trigonometric, hyperbolic, and rational varieties and the stability analysis are also discussed. In addition to this, the graphical aspects of the produced solutions are investigated. Due to the vital necessity of this aspect, we make use of the suitable parameter values to place a priority on the physical aspects of the provided findings by utilizing 3 D and contour charts. The suggested procedures are of great use and can be extended to additional nonlinear evolutionary equations that reflect nonlinear physical frameworks in the nonlinear sciences. [ABSTRACT FROM AUTHOR]

The exploration of closed form solutions for nonlinear partial differential equations (NLPDEs) is being an attractive subject in the different branches of mathematical and physical sciences. In this paper, we have applied extended Exp-function method to calculate closed form solutions for NLPDEs; such as the nonlinear Maccari system which is very significant in engineering and mathematical physics. Plentiful closed form solutions with arbitrary parameters are successfully obtained by this method. It is shown that the obtained solutions are more general and fresh and can be helpful to analyze the NLPDEs in mathematical physics and engineering problems. [ABSTRACT FROM AUTHOR]

In this paper, multi-soliton solutions of the fifth-order Kaup-Kupershmidt (KK) equation have been derived via the auxiliary function in conjunction with the bilinear method. These solutions have not been previously obtained. Propagation and interactions of three solitons have been presented analytically. The direction of the soliton is related to the signs of the parameters . The distances of the solitons are related to the values of the parameters . [ABSTRACT FROM AUTHOR]

We study a nonlinear generalized Sawada-Kotera equation of fractional order via the exp(-φ(η))-expansion method and Shifted modified Chebyshev Wavelet technique. We obtain abundant exact solutions and approximate solution of the equation. The results of the study shows that the exp(-φ(η))-expansion method is very effective and proficient for solving nonlinear fractional partial differential equations. The solitary wave solutions are obtained through the hyperbolic, trigonometric, exponential and rational functions. Graphical representations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very expedient for fractional PDEs, and could be extended to other physical problems. Results of the proposed methods show an excellent conformity with the exact solution of the considered problem. [ABSTRACT FROM AUTHOR]

In this paper, the Sakovich equation is extended for the first time to a new equation with variable-coefficients on the time variable. The infinitesimal generators are obtained by performing a Lie symmetry analysis of the equation. Following this, the optimal system of one-dimensional subalgebras of the equation is analyzed. The (2+1)-dimensional variable-coefficients Sakovich equation is then reduced to (1+1)-dimensional partial differential equations by similarity reductions. The reduced partial differential equations are simplified to ordinary differential equations by the traveling wave transform. The new periodic wave solution and two-soliton interaction solution are obtained, and some exact solutions are obtained using the extended tanh-function method and the extended sech-function method. Finally, different types of soliton solutions are obtained by assigning different functions to the coefficient functions, and 3-dimensional figures are used to visualize the structural features of the soliton solutions. • This paper presents the new extended equation for Sakovich with variable coefficients for the first time. • Give out the new optimal system and some new exact solutions of variable coefficients Sakovich equation. • The 3D images of variable coefficients Sakovich equation with different coefficient functions are drawn. [ABSTRACT FROM AUTHOR]

Some special travelling wave solutions for the 2D Ricci flow model under grav-ity, in the conformal gauge, are constructed in this paper, by using of an uniform algoritmic method. The proposed method is able to generate soliton solutions and periodic wave solutions at the same time. These special solutions are compared to the ones obtained by making use of other known and powerful methods for solving nonlinear partial differential equations. In fact, this method is readily applicable to a large variety of nonlinear dynamical models. [ABSTRACT FROM AUTHOR]

By using the Inverse Scattering Transform we construct an explicit soliton solution formula for the Hirota equation. The formula obtained allows one to get, as a particular case, the N -soliton solution, the breather solution and, most relevantly, a new class of solutions called multipole soliton solutions. We use these exact solutions to study the motion of a vortex filament in an incompressible Euler fluid with nonzero axial velocity. [ABSTRACT FROM AUTHOR]

The simplest (2 + 1) -dimensional extension of the nonlinear Schrödinger equation is called Fokas system. In the current article, Fokas system has been discussed which describes the nonlinear pulse propagation in monomode optical fibers. Soliton and solitary wave solutions of Fokas system have been retrieved using two algorithmic schemes namely, generalized projective Riccati equation method and the two variables G ′ G , 1 G -expansion method. The proposed methods have been employed to extract various types of solutions such as dark solitons, bright solitons and periodic solutions. The acquired solutions are visualized by 3D plots and corresponding 2D contour and density plots. The obtained results demonstrate that both suggested schemes are productive to retrieve wave solutions for many partial differential equations appearing in optics. [ABSTRACT FROM AUTHOR]

In this work, we apply the binary Bell polynomial approach to coupled Burgers system. In other words, we investigate possible integrability of referred system. Bilinear form and soliton solutions are obtained, some figures related to these solutions are given. We also get Bäcklund transformations in both binary Bell polynomial form and bilinear form. Based on the Bäcklund transformation, Lax pair is obtained. Namely, this is a study in which integrabilitiy of coupled burgers system is investigated. [ABSTRACT FROM AUTHOR]

The dynamics of solitons waves associated with higher-order nonlinear partial differential equations that model situations in physical systems obtainable in the fields of science and engineering help to understand the physical meaning of various soliton solutions obtained for these nonlinear differential equations. Thus, in this paper, we analytically investigate an extended Kadomtsev-Petviashvili-like equation existent in three dimensions. The robust technique of the Lie group theory of differential equation was invoked to achieve analytic solutions to the equation. This technique is used in a systematic way to generate the Lie point symmetries of the equation under study. Consequently, an optimal system of Lie subalgebra related to the equation is obtained. Thereafter, we engage the various gained subalgebras to reduce the Kadomtsev-Petviashvili-like equation to ordinary differential equations for the possibility of obtaining relevant closed-form solutions. Fortunately, various soliton solutions were found. These include different complex soliton solutions consisting of dark, bright, topological kink and singular. Some other solutions achieved are logarithmic, periodic and those which contain arbitrary functions. Therefore, to understand the physical meaning of these solutions, we depict them graphically. This exposed us to various wave structures which were later analyzed and applied. Moreover, we highlighted the significance of these solutions in various fields of science and engineering. [ABSTRACT FROM AUTHOR]

It is known that the Kadomtsev-Petviashvili (KP) equation can be decomposed into the first two members of the coupled Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy by the binary non-linearization of Lax pairs. In this paper, we construct the N-th iterated Darboux transformation (DT) for the second- and third-order m-coupled AKNS systems. By using together the N-th iterated DT and Cramer's rule, we find that the KPII equation has the unreduced multi-component Wronskian solution and the KPI equation admits a reduced multi-component Wronskian solution. In particular, based on the unreduced and reduced two-component Wronskians, we obtain two families of fully-resonant line-soliton solutions which contain arbitrary numbers of asymptotic solitons as y → ∓∞ to the KPII equation, and the ordinary N-soliton solution to the KPI equation. In addition, we find that the KPI line solitons propagating in parallel can exhibit the bound state at the moment of collision. [ABSTRACT FROM AUTHOR]

Abstract: Based on the Pfaffian derivative formula and Hirota bilinear method, the Pfaffian solutions to (3+1)-dimensional Jimbo–Miwa equation are obtained under a set of linear partial differential condition. Moreover, we extend the linear partial differential condition and proved that (3+1)-dimensional Jimbo–Miwa equation has extended Pfaffian solutions. As examples, special exact two-soliton solution and three-soliton solution are computed and plotted. Our results show that (3+1)-dimensional Jimbo–Miwa equation has Pfaffian solutions like BKP equation. [Copyright &y& Elsevier]

In this paper, the Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation is ana-lytically investigated using the Hirota bilinear method. Based on the bilinear form of the CDGKS equation, its Af-soliton solution in explicit form is derived with the aid of symbolic computation. Besides the soliton solutions, several integrable properties such as the Bácklund transformation, the Lax pair and the nonlinear superposition formula are also derived for the CDGKS equation. [ABSTRACT FROM AUTHOR]

Abstract: We observe that the form of the Kadomstev–Petviashvili equation studied by Yu (2011) [S. Yu, N-soliton solutions of the KP equation by Exp-function method, Appl. Math. Comput. (2011) doi:10.1016/j.amc.2010.12.095] is incorrect. We claim that the N-soliton solutions obtained by means of the basic Exp-function method and some of its known generalizations do not satisfy the equation considered. We emphasize that Yu’s results (except only one) cannot be solutions of the correct form of the Kadomstev–Petviashvili equation. In addition, we provide some correct results using the same approach. [Copyright &y& Elsevier]

Non-linear processes, waves and oscillations occupy a special place in modern physics, radio physics and the wavelet analysis is a promising tool for describing non-linear wave processes. The mathematical methods based on it can be applied to the detection of new features of such non- linear processes. Many non-linear physical phenomena modelled by non-linear partial differential equations have soliton solutions and these solutions have wavelet features. Moreover, there exists a strong connectivity between wavelets, solitons and probability distributions. With this motivation, in this study, the wavelet analysis of solitons arising especially as the solution of the sine-Gordon equation is carried out using continuous wavelet transform and the connectivity between wavelets, solitons and probability distributions is discussed. [ABSTRACT FROM AUTHOR]