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The current study is important from two perspectives. Firstly, in this article, we suggest a novel analytical technique for creating the exact solutions to nonlinear partial differential equations (NLPDEs). In order to study the dynamical behaviors of various wave phenomena, we can construct the several exact solutions in the form of Jacobi elliptic solutions, hyperbolic solutions, trigonometric solutions, and exponential solutions by using this method. Secondly, we consider a more generalized form of the (2+1)‐dimensional Korteweg–de Vries (KdV) and modified Korteweg‐de Vries (mKdV) equations that plays an important role in describing the shallow water. We successfully extract several soliton solutions for the examined equation. By choosing the appropriate parameters, some graphs of the discovered solutions have been represented in the figures. Every obtained solution has been demonstrated to satisfy the corresponding equation. The findings demonstrate that the method can be applied to solve a number of nonlinear evolution equations. The new solutions and the paper's findings could improve our understanding of how the waves move through shallow water and open up new research avenues. [ABSTRACT FROM AUTHOR]

The main goal of this research work is to explore novel soliton solutions and dynamics of solitonic structures to asymmetric Nizhnik–Novikov–Veselov (ANNV) equations by two methods, namely, the generalized exponential rational function (GERF) method and the modified extended tanh expansion (METE) method. These techniques are the most effective and reliable tools for solving nonlinear equations with partial derivatives. The used methods provide a wide range of soliton solutions that can motivate applied scientists and researchers for these specific structures. These acquired solutions are in the form of exponential, trigonometric and hyperbolic functions including, tan , sin , sec , cot , csc , sinh , tanh , sech , cosh , coth , csch and of their combinations. It has also been observed that the obtained soliton solutions of the ANNV equations are bell-shape, anti-bell-shape, periodic solution, multisoliton and different types of soliton solutions. To illustrate the physical features and dynamics characteristics of some obtained solutions, three-dimensional (3D) and two-dimensional (2D) figures are exhibited through the Mathematica software. The methods applied in this paper are suitable to study the soliton solutions for the ANNV equation without producing the complexities in some other known analytical method. Finally, the employed methods can be earmarked easily for solving various classes of nonlinear evolution equations appearing in mathematical physics, fluid dynamics, plasma physics, nonlinear waves and nonlinear sciences. [ABSTRACT FROM AUTHOR]

This research studies the Kudryashov's equation by using the generalised exponential rational function (GERF) technique. This technique allows us to use the traveling wave reduction for Kudryashov's equation which provides the system of equations associated with the real and imaginary parts of the profile pulse for a complex function. First and foremost, we reduced these systems of equations into nonlinear ordinary differential equations (ODE) under the travelling wave transformations. The critical steps of the GERF technique are then applied to the resulting nonlinear ODE. With the assistance of the mentioned technique, we obtained different kinds of families of optical soliton solutions for Kudryashov's equation. The real part, imaginary part and modulus of some of the obtained solutions are investigated/explored by the three-dimensional visual representations under the appropriate choice of the involved arbitrary parameters with suitable range space, demonstrating that the solutions represent a series of solitons, bright solitons, dark solitons, singular solitons, the interaction of solitons with solitary wave profiles, the interaction of kink waves with solitons and the interaction of kink waves with solitons. [ABSTRACT FROM AUTHOR]

In this paper, we use two efficient mathematical approaches to obtain a variety of soliton solutions to the (3+1)-dimensional Schrödinger equation: the generalized Riccati equation mapping method and the newly proposed modified generalized exponential rational function method. These techniques extracted standard, illustrative, rich dynamical structures, and further comprehensive soliton solutions and traveling wave solutions involving hyperbolic form, trigonometric form, and exponential form. The obtained results have been verified by placing them back into the mentioned nonlinear partial differential equation via symbolic computation in Mathematica. Thereafter, the graphical demonstrations of some attained solutions are discussed for a better understanding of the physical phenomenon. We have portrayed the three-dimensional, contour plot, and two-dimensional graphs for different parametric values. The attained results demonstrate the generalized Riccati equation mapping method and modified generalized exponential rational function techniques for extracting soliton solutions to nonlinear partial differential equations are efficient, compatible, and reliable in nonlinear sciences, optical fibers, and engineering. [ABSTRACT FROM AUTHOR]

This paper studies the optical soliton solutions of the Biswas–Arshed equation with the help of two different techniques, such as the generalised exponential rational function (GERF) technique and Kudryashov's simplest equation technique. The GERF technique extracts distinct families of exact solitary wave solutions involving trigonometric function solutions, hyperbolic function solutions, rational function solutions, etc. After that, we apply Kudryashov's simplest equation method in the context of Bernoulli and Riccati equations to attain different kinds of families of exact soliton solutions. All the acquired solutions of the equation have numerous applications in many branches of nonlinear sciences such as plasma physics, superconductivity, nonlinear optics, biophysics, star formation, quantum mechanics, etc. and many more connected fields of nonlinear wave sciences. The exact solitary wave solutions obtained by GERF technique and Kudryashov's simplest equation technique are in more generalised form as they contained several arbitrary parameters. Subsequently, to understand the behaviour of deduced solutions, we graphically discuss the real part, imaginary part and modulus of these solutions by suitable choice of involved arbitrary parameters. [ABSTRACT FROM AUTHOR]

This paper studies the (2 + 1)-dimensional modified Heisenberg ferromagnetic (MHF) system, which describes the different nonlinear dynamical structures for long water-waves of small amplitude with weakly nonlinear restoring forces and high-frequency dispersion. This MHF system arises in the propagation of the magnetization vector of the isotropic ferromagnet and biological pattern formation. By employing the Lie symmetry analysis method, infinitesimal generators, Lie point symmetries, potential vector fields, commutation relations of infinitesimal vectors, and attractive symmetry reductions are derived. Based on the two phases of Lie similarity reductions, (2 + 1)-dimensional MHF system is reduced to various nonlinear ordinary differential equations (ODEs). Afterwards, with the help of symbolic computation, we solve the acquired ODEs and obtain a variety of exact closed-form solutions involving arbitrary independent functions and other constant parameters. The physical features of the obtained multi-wave soliton solutions are demonstrated to analyze the impact of the involved arbitrary independent functions on the dynamics of the solitary wave solutions via three-dimensional graphics. These exact solutions are accomplished in the shapes of single solitons, doubly solitons, multi-wave solitons, elastic behavior of multisoliton, oscillating multi-solitons, curved-shaped periodic solitons, and kink-type solitons, and so on. The newly constructed results show the trustworthiness, reliability, and efficiency of the Lie symmetry technique for obtaining the invariant closed-form solutions to nonlinear governing model. Moreover, conservation laws and self-adjoint systems have been obtained by implementing Noether's technique. By using Lie symmetry analysis, the achieved outcomes might be helpful to understand the physical formation of this model and confirm the effectiveness and authenticity of the mentioned method. [ABSTRACT FROM AUTHOR]

This investigation focuses on two novel Kadomtsev–Petviashvili (KP) equations with time-dependent variable coefficients that describe the nonlinear wave propagation of small-amplitude surface waves in narrow channels or large straits with slowly varying width and depth and non-vanishing vorticity. These two variable coefficients, Kadomtsev–Petviashvili (VCKP) equations in (2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique, we carry out infinitesimal generators, potential vector fields, and various similarity reductions of the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via two similarity reductions. The closed-form analytic solutions are achieved, including in the shape of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave solitons. All the obtained solutions are verified and validated by using back substitution to the original equation through Wolfram Mathematica. We analyze the dynamical behaviors of these obtained solutions with some three-dimensional graphics via numerical simulation. The obtained variable coefficient solutions are more relevant and useful for understanding the dynamical structures of nonlinear KP equations and shallow water wave models. [ABSTRACT FROM AUTHOR]

In this paper, the soliton solutions and dynamical wave structures for the generalized (3+1)-dimensional shallow water wave (SWW) equation, which is an important physical property in ocean engineering and hydrodynamics, are presented. The generalized exponential rational function (GERF) method is used to investigate the closed-form wave solutions of the generalized SWW equation, which is used to describe the evolutionary dynamics of SWW. We successfully archive a variety of soliton solutions such as exponential solutions, kink wave solutions, non-topological solutions, periodic singular solutions, and topological solutions. These newly established results are also important for understanding the wave-propagation and dynamics of exact solutions of the equation, which is of great significance in physical oceanography and chemical oceanography. Eventually, it is shown that the proposed GERF technique is effective, robust, and straightforward and is also used to solve other types of higher-dimensional nonlinear evolution equations. In our work, we have used Mathematica extensively for such complicated algebraic calculations. [ABSTRACT FROM AUTHOR]

The Dullin-Gottwald-Holm equation models the unidirectional propagation of shallow regime water waves. In this work, the Lie symmetry analysis of the generalised two-component modified weakly dissipative Dullin-Gottwald-Holm system is performed. Using symmetry reduction, the exact solutions are obtained in the form of power series and trigonometric functions. Also using multiplier approach, the conservation laws are obtained. The 3D graphical representations are also shown for obtained solutions. [ABSTRACT FROM AUTHOR]

In the present article, our main aim is to construct abundant exact solutions for the (2 + 1) -dimensional Kadomtsev–Petviashvili-Benjamin–Bona–Mahony (KP-BBM) equation by using two powerful techniques, the Lie symmetry method and the generalised exponential rational function (GERF) method with the help of symbolic computations via Mathematica. Firstly, we have derived infinitesimals, geometric vector fields, commutation relations and optimal system. Therefore, the KP-BBM equation is reduced into several nonlinear ODEs under two stages of symmetry reductions. Furthermore, abundant solutions are obtained in different shapes of single solitons, solitary wave solutions, quasiperiodic wave solitons, elastic multisolitons, dark solitons and bright solitons, which are more relevant, meaningful and useful to describe physical phenomena due to the existence of free parameters and constants. All these generated exact soliton solutions are new and completely different from the previous findings. Moreover, the dynamical behaviour of the obtained exact closed-form solutions is analysed graphically by their 3D, 2D-wave profiles and the corresponding density plots by using the mathematical software, which will be comprehensively used to explain complex physical phenomena in the fields of nonlinear physics, plasma physics, optical physics, mathematical physics, nonlinear dynamics, etc. [ABSTRACT FROM AUTHOR]

This paper is devoted to obtaining some new types of exact solutions of the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation utilizing the Lie symmetry method. All the Lie symmetries, infinitesimal generators, and possible geometric vector fields have been obtained by using the invariance condition of the group-theoretic method. Meanwhile, the Lie symmetry reductions and explicit exact solutions are obtained by a one-dimensional (1D) optimal system. All the obtained exact solutions are absolutely new and completely different from the earlier established results in the literature. Moreover, the dynamical behavior of obtained solitons like doubly solitons, dark solitons, kink wave, curved shaped multi-solitons, parabolic waves, solitary waves, and annihilation of elastic multi-soliton profiles is depicted graphically via interesting 3D-shapes. That will be widely used to provide many more attractive complex physical phenomena in the fields of plasma physics, statistical physics, fiber optics, fluid dynamics, condensed matter physics, and so on. Finally, we have verified all the achieved soliton solutions through symbolic computations with Mathematica. [ABSTRACT FROM AUTHOR]

In the present article, Lie group of point transformations method is successfully applied to study the invariance properties of the (2 + 1) -dimensional Pavlov equation. Applying the Lie symmetry method, we strictly obtain the infinitesimals, vector fields, commutation relation and several interesting symmetry reductions of the equation. The explicit exact solutions are derived under some limiting conditions imposed on the infinitesimals ξ , ϕ , τ and η . Then, the Pavlov equation is transformed into a number of nonlinear ODEs through several symmetry reductions. These new exact solutions are more general and entirely different from the work of Kumar et al (Pramana – J. Phys.94: 28 (2020)). The obtained invariant solutions are examined analytically as well as physically through numerical simulation by giving free alternative values of arbitrary functions and constants. Consequently, graphical representations of all these solutions are studied and demonstrated in 3D-graphics and the corresponding contour plots. Interestingly, the solution profiles show the annihilation of three-dimensional parabolic profile, doubly soliton and elastic multisolitons and nonlinear wave nature form. [ABSTRACT FROM AUTHOR]

This paper propounds the Lie group analysis method for finding exact solutions of Born–Infeld (BI) equation arising in nonlinear electrodynamics. We obtain generators of infinitesimal transformations, commutator table of Lie algebra, the complete geometric vector field, group symmetries and reduction equations. For the set of geometric vector field, we find an optimal system of the vector fields. Each element in this system helps to reduce the main equation into an ordinary differential equation, which provides analytical solution to the BI equation. We perform numerical simulation to obtain an appropriate visual appearance and dynamic behavior of the traced solutions. The nature of the solutions is investigated both analytically and physically through their evolutionary profile by considering appropriate choices of arbitrary constants. [ABSTRACT FROM AUTHOR]

In present article, we establish the exact solutions of non-linear (2+1)-Dimensional Calogero Degasperis(CD) equation by using Lie group of transformation method. The properties of the solutions of CD equation such as travelling waves, single solitons, singular periodic and singular kink, depending upon appropriate arbitrary functions γ(t) and λ(t) have been discussed. [ABSTRACT FROM AUTHOR]

In this paper, we have considered the variable-coefficients Gardner (vc-Gardner) equation. We have check the Painlevé properties of vc-Gardner by using the Painlevé analysis and the symmetries for the equations are obtained using Lie group analysis. The exact solutions obtained from the symmetries and Painlevé analysis are presented. [ABSTRACT FROM AUTHOR]

The purpose of this work is to seek various innovative exact solutions using the new Kudryashov approach to the nonlinear partial differential equations (NLPDEs). This technique obtains novel exact solutions of soliton types. Moreover, several 3D and 2D plots of the higher dimensional Klein-Gordon, Kadomtsev-Petviashvili, and Boussinesq equations are demonstrated by considering the relevant values of the aforementioned parameters to exhibit the nonlinear wave structures more adequately. The new Kudryashov technique is an effective, and simple technique that provides new generalized solitonic wave profiles. It is anticipated that these novel solutions will enable a thorough understanding of the development and dynamic nature of such models. [ABSTRACT FROM AUTHOR]

Some new infinite-dimensional generalised Lie symmetries of Einstein’s field equations for perfect fluid distribution are found by using the Lie symmetry analysis. The reduced ordinary differential equations are solved to obtain new non-trivial exact solutions. The software MAPLE is used for computation and MAPLE code is given to facilitate the research in this field. [ABSTRACT FROM AUTHOR]

In this study, the improved (℘ ′ / ℘) expansion method and Exp a function method are used to construct the newly closed-form exact solutions for the deoxyribonucleic acid (DNA) model which includes hyperbolic, trigonometric and exponential solutions. These solutions include a wealth of information regarding the dynamical behavior of homogeneous long elastic rods with circular cross-sections. These rods comprise a pair of polynucleotide rods of the DNA molecule that are connected by an elastic diaphragm, demonstrating the involvement of the hydrogen bond in this communication. The performance of these approaches demonstrates their use and efficacy in solving a variety of nonlinear evolution problems of integer and fractional order. The physical significance of the established and the obtained solutions has been shown via 3D shapes. It is worth noting that the solutions obtained here via the proposed schemes are more generalized and can be helpful to demonstrate the internal interaction of the DNA model arising in mathematical biology. The proposed method has been used to obtain exact traveling wave solutions for fraction nonlinear partial differential equations (NPDEs) arising in nonlinear sciences. [ABSTRACT FROM AUTHOR]

Abstract: The Lie-group formalism is applied to investigate the symmetries of the Benjamin–Bona–Mahony (BBM) equation with variable coefficients. We derive the infinitesimals and the admissible forms of the coefficients that admit the classical symmetry group. The ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained. [Copyright &y& Elsevier]