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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]

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 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]

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]

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]

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]