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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 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]

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 major goal of the article is to use the generalized exponential rational function method to seek abundant closed-form solutions and dynamics of solitary wave profiles to the (3+1)-dimensional modified KdV–Zakharov–Kuznetsov (MKdV–ZK) equation. We also apply the Painlevé analysis with Kruskal simplification to investigate the integrability of the said equation. The (3+1)-dimensional MKdV–ZK model is a significant model with great advantages in various disciplines of physics, such as nonlinear optics, fluid dynamics, plasma physics, mathematical physics, and quantum mechanics. By taking advantage of the generalized exponential rational function technique, we extracted various soliton solutions, including exponential form, trigonometric form, and hyperbolic form function solutions. From both mathematical and physical points of view, it is important to obtain bright–dark soliton forms and other solitonic wave profiles. Furthermore, the results have been graphically revealed using 3D, 2D, and contour plots to demonstrate the underlying dynamics of the interactive waves. The obtained findings indicate the method's efficacy and dependability, allowing it to be widely applied in a variety of advanced nonlinear evolution equations. • The (3+1)-dimensional modified KdV–Zakharov–Kuznetsov (MKdV–ZK) equation is investigated. • By taking the advantage of generalized exponential rational function method, we obtain numerous soliton solutions. • The derived solutions could be more useful in the dynamics of wave propagation. • The dynamic behavior of the generated soliton solutions is exhibited using 3D, 2D, and contour plots. • Wolfram Mathematica is used to carry out symbolic computations. [ABSTRACT FROM AUTHOR]