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This paper focuses on obtaining a large number of exact solutions to the highly nonlinear (3+1)-dimensional Yu–Toda–Sassa–Fukuyama (YTSF) equation, which is important in fluid dynamics, plasma physics, nonlinear sciences, and weakly dispersive media, among other things. The primary goal of this research is to obtain a variety of exact soliton solutions and different dynamical wave-forms of the YTSF equation using two different techniques: Lie symmetry and generalized Kudryashov (GK) methods. These techniques reduce the considered partial differential equation (PDE) into ordinary differential equations (ODEs). Hereafter, a variety of exact solutions are derived with the help of the associated subalgebras and wave transformation by the Lie symmetry technique and GK technique, respectively. The obtained exact solutions involve various arbitrary constants, parameters, and arbitrary functionals that enhance the dynamic representation of the derived solutions in the research area. We use numerical simulations to generate 3-dimensional graphics and contour plots to determine the dynamical behavior of the obtained exact solutions. The physical interpretation of interaction soliton solutions demonstrates a rich diversity of attained solutions, such as solitons, kink waves, kink wave interaction with solitons, and others. • A highly nonlinear (3+1)-dimensional Yu–Toda–Sassa–Fukuyama (YTSF) equation is investigated. • Lie symmetries, infinitesimal generators, various symmetry reductions and diverse exact solutions are constructed. • Lie symmetry analysis and the generalized Kudryashov method are used to generate several analytical soliton solutions. • Through 3D graphics, an improved dynamical investigation of some interaction soliton solutions is discussed. [ABSTRACT FROM AUTHOR]

In the current study, the generalized (G ′ / G) -expansion method and a new modified simple equation method (NMSEM) have been utilized to secure the exact solutions of the Yu-Toda-Sasa-Fukuyama (YTSF) equation with time-dependent coefficients, which describes two layers of liquid (or in an elastic) media. By implementing these methods, we have secured the three different families of travelling wave solutions (such as hyperbolic, rational and trigonometric functions) of the independent variables and the arbitrary parameters of the equation. Further, the graphical representation of the solitons are done with the help of the software Maple. [ABSTRACT FROM AUTHOR]

The extended generalized G ′ G –expansion is well defined and an efficient technique, which is used to obtain the exact traveling wave solutions to the governing nonlinear equations with constant coefficients as well as variable coefficients. In this paper, the modified Korteweg–de Vries (mKdV) equation and Burgers equation with variable coefficients are investigated through the extended generalized G ′ G –expansion method, which are exceptional cases of the nonlinear evolution equations widely used in a two-layer fluid system, in fluid-filled elastic tubes, in an atmospheric and oceanic dynamical system, traffic flow, turbulence in fluid dynamics, dusty plasma, ion-acoustic waves in a plasma system. New families of exact closed-form solutions are obtained in hyperbolic, trigonometric, and rational function solutions with the available free constants. With the help of computerized symbolic computation work, the newly formed closed-form solutions are validated by back substituting them into the equations using the computational mathematical software. Furthermore, the graphical representations of all these obtained solutions are discussed and demonstrated by giving the suitable best values of arbitrary functions and constants via three-dimensional surface and density plots. The dynamics of the solution profiles demonstrate the annihilation of 3D kink-type soliton waves, shock waves, double solitons, and multi-soliton wave structures. [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]

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]

This paper investigates the exact invariant solutions and the dynamics of soliton solutions to the (2+1)-dimensional generalized Hirota–Satsuma–Ito (g-HSI) equations. By applying the Lie symmetry technique, infinitesimal vectors, the commutation relations, and various similarity reductions are derived from the g-HSI equations. Using the two stages of Lie symmetry reductions, the equation is transformed into various nonlinear ordinary differential equations (NLODEs). After that, by solving the various resulting ODEs, we obtain abundant explicit exact solutions in terms of the involved functional parameters. These closed-form invariant solutions are successfully presented in the form of distinct complex wave-structures of solutions like combo-form solitons, dark-bright solitons, W-shaped solitons, the interaction between multiple solitons, parabolic wave solitons, multi-wave structures, and curved-shaped parabolic solitons. Furthermore, using computerized symbolic computation and numerical simulation, the physical behaviors of some obtained solutions are displayed in three-dimensional graphics. The resulting solutions are found to be useful for understanding the dynamics of the exact closed-form solutions of this model and show the authenticity as well as the effectiveness of the proposed method. Therefore, the gained solutions and their dynamical wave structures are quite significant for understanding the propagation of the excitation waves in shallow water wave models. Furthermore, using the resulting symmetries, the conservation laws of g-HSI equations have been obtained by applying Ibragimov's theorem. • Exact invariant solutions are investigated for the generalized Hirota–Satsuma–Ito equations. • The Lie symmetry technique is applied to obtain abundant exact invariant solutions. • All the closed-form solutions are obtained, including arbitrary functional parameters. • The different dynamical structures of some invariant solutions are exhibited via 3D postures. • Such investigations are strongly advised in the field of advanced research and development. [ABSTRACT FROM AUTHOR]

• New exact solutions of Einstein-Maxwell field equations in pure magnetic fields, in correspond to the Bonnor metric are obtained. • Solutions are obtained in spherical coordinates with the use of Lie symmetry method. • Obtained solutions are accelerating static solutions of Einstein-Maxwell equations for a non-conformally flat semi-Riemannian manifold. • The graphical representations in 3D are also shown. The Bonnor field shed light on imposition of two magnetic poles, both with mass m , and strengths - m and m , that are situated at a coordinate distance 2 b on a symmetry axis. With the addition of an external magnetic field, these magnetic monopoles can be kept apart in an equilibrium. The new exact solutions of Einstein-Maxwell field equations in pure magnetic fields, in correspond to the Bonnor metric, are obtained in spherical coordinates with the use of Lie symmetry method. The obtained solutions are accelerating static solutions of Einstein-Maxwell equations in a more general form, for a semi-Riemannian manifold which is not conformally flat. The graphical representations in 3D are also shown. [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]