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This research article investigates the (2+1)-dimensional variable-coefficient Boiti–Leon–Manna–Pempinelli equation using the Lie classical method and the unified method. The Lie classical method is employed to deduce the Lie symmetry generators and associated symmetric vectors, shedding light on the symmetries and invariance properties of the equation. Through this method, we deepen our understanding of the (2+1)-dimensional variable-coefficient Boiti–Leon–Manna–Pempinelli equation. Additionally, the unified method is utilized to further explore the equation's properties, aiming to develop a comprehensive understanding of its mathematical properties and solutions. To enhance comprehension, graphical representations such as 3-dimensional plots, 2-dimensional plots, and contour plots are presented using the symbolic computation software Mathematica. Analysis of the graphics reveals various solution profiles, including single-peak, doubly-peaks, multi-peaks, sinusoidal waves, breather solitons, lump solitons, interactions of kinks and solitons, solitary waves, paraboloids, and more. Moreover, this research aims to bridge the gap between mathematical visualization and real-world applications. By advancing knowledge of the (2+1)-dimensional variable-coefficient Boiti–Leon–Manna–Pempinelli equation and its mathematical characteristics, this study contributes to a broader understanding of nonlinear equations and their practical implications. Furthermore, Lumps and multi-peaks also arise in a variety of mathematical and physical fields, including nonlinear dynamics, oceanography, water engineering, optical fibres, and other 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]

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]

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]

Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunk lining routing calls between telephone switching centers, computer network connections and high speed computer data buses. In this paper, we seek the solutions of a nonlinear transmission line model by applying the Lie symmetry method. Corresponding to the optimal system of Lie subalgebras, similarity reductions and a variety of new exact solutions in the form of trigonometric functions and hyperbolic functions are obtained. Further, power series solution is obtained, and the convergence of the power series solution is also shown. Corresponding to one similarity reduction, by bifurcation of dynamical system, the stable and unstable regions are determined, which shows the existence of soliton solutions from the nonlinear dynamics view point. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear transmission line model with the power law nonlinearity and the constant capacitance and voltage relationship is studied using Lie symmetry analysis. Corresponding to the infinitesimals obtained, using commutation relations, abelian and non-abelian Lie subalgebras are obtained. Also, using the adjoint table, a one-dimensional optimal system of subalgebra is presented. Based on this optimal system, the corresponding Lie symmetry reductions are obtained. Moreover, variety of new similarity solutions in the form of trigonometric functions, hyperbolic functions, are obtained. Corresponding to one similarity reduction, by bifurcation analysis of dynamical system, the stable and unstable regions are determined, which show the existence of soliton solutions from the nonlinear dynamics point of view. Some of the obtained solutions represented graphically and observations are also discussed. [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 work, the abundant exact closed-form solutions and dynamics of solitons for the double-chain deoxyribonucleic acid (DNA) model is obtained by utilizing the generalized exponential rational function (GERF) method. Deoxyribonucleic acid (DNA) retains the genetic information that creatures need to live and reproduce themselves. We obtained several novel exact soliton and exponential rational functional solutions in the shapes of dynamics of solitons like multi-solitons, breather-type solitons, abundant elastic interactions between multi-solitons, and nonlinear waves, oscillating multi-solitons, and Lump solitons. These derived solutions were never reported in the literature. The dynamical structures of some exact solitons are exhibited graphically by assigning suitable values to the free parameters via 3D figures. The generated solutions can be more useful and help to explain the internal interactions of the double-chain DNA model. The symbolic computational work and the obtained solutions show that the present proposed GERF method is effective, robust, and straightforward. Moreover, these types of higher-order NLEEs can be solved using the current technique. [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]

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]

The Biswas-Milovic equation in generalized form and with power law nonlinearity is analyzed for Lie symmetries. The classical Lie group method is applied to derive symmetries of this equation, and the ordinary differential equations deduced are further studied; and some exact solutions are obtained. [ABSTRACT FROM AUTHOR]

The Vakhnenko-Parkes (VP) equation with power law nonlinearity is analyzed for Painlevé test and for Lie symmetries. The Painlevé analysis of Vakhnenko-Parkes equation with power law nonlinearity is performed by the Kruskal approach to check the Painlevé property. The Lie group formalism is also applied to derive symmetries of this equation, the ordinary differential equations deduced are further studied, and some exact solutions are obtained. [ABSTRACT FROM AUTHOR]