Group Invariant Solutions Research Articles

Lie symmetries in higher dimensional charged radiating stars

This study investigates the Lie symmetry properties of matter variables, spacetime dimension, and kinematic quantities in spherically symmetric radiating stars undergoing gravitational collapse in higher dimensional spacetimes. Our analysis explores how variations in functions and dependent variables influence the Lie algebra dimensionality and Lie symmetries of the boundary condition. Our study reveals the explicit dependence of charge and spacetime dimension in the Lie symmetry generators for the first time. We examine models with various kinematical features, such as shear, shear-free, geodesic, and non-expanding conditions with different matter variables. Using the Lie symmetries we obtain group invariant solutions to the models and observe how the kinematical features and matter variables affect the Lie symmetries. Additionally, we demonstrate the appearance of a linear equation of state. We recover known four-dimensional models from our results.

Signature of conservation laws and solitary wave solution with different dynamics in Thomas–Fermi plasma: Lie theory

We propose a Lie group method to discuss the modified KP equation appearing in Thomas–Fermi (TM) plasma, characterised by cold and hot electrons. The Lie method facilitates the identification of similarity reductions, infinitesimal symmetries, group-invariant solutions, and novel analytical solutions for nonlinear models. The similarity reduction method is carried out to transform the nonlinear partial differential equations (NLPDE) into the nonlinear ordinary differential equations (NLODE). This study focuses on solitary wave profiles due to their usefulness in various engineering applications, including monitoring public transportation systems, managing coastlines, and mitigating disaster risks. It also addresses the conservation laws associated with the modified KP equation. The generalised auxiliary equation (GAEM) scheme is used to compute the new solitary wave patterns of the modified KP equation, which explains the dynamics of nonlinear waves in Thomas–Fermi plasma. The idea of nonlinear self-adjointness is used to compute the conservation laws of the examined model. The graphical behaviour of some solutions is represented by adjusting the suitable value of the parameters involved.

(α+2)-Dimensional fractional evolution equation: group classification, symmetries, reduction and conservation laws

ABSTRACT A preliminary group classification based on symmetry operators is applied to study invariance properties of the time-fractional ( α + 2 ) -dimensional fractional evolution equation. The concepts of Riemann–Liouville and Caputo fractional derivatives are used in this study. The similarity variables obtained from symmetries and one-dimensional optimal systems of constructed Lie algebras are used in order to find the group-invariant solutions of the equation. Finally conservation laws of the equation are derived via a modified version of Noether’s theorem.

Invariant analysis of the multidimensional Martinez Alonso–Shabat equation

Abstract This present study is concerned with the group-invariant solutions of the (3 + 1)-dimensional Martinez Alonso–Shabat equation by using the Lie symmetry method. The Lie transformation technique is used to deduce the infinitesimals, Lie symmetry operators, commutation relations, and symmetry reductions. The optimal system for the obtained Lie symmetry algebra is obtained by using the concept of the adjoint map. As for now, the considered model equation is converted into nonlinear ordinary differential equations (ODEs) in two cases in the symmetry reductions. The exact closed-form solutions are obtained by applying constraint conditions on the symmetry generators. Due to the presence of arbitrary functional parameters, these group-invariant solutions are displayed based on suitable numerical simulations. The conservation laws are obtained by using the multiplier method. The conclusion is accounted for toward the end.

Optimal systems of Lie subalgebras: A computational approach

Lie groups of symmetries of differential equations constitute a fundamental tool for constructing group-invariant solutions. The number of subgroups is potentially infinite and so the number of invariant solutions; thus, it is crucial to obtain a classification of subgroups in order to have an optimal system of inequivalent solutions from which all other solutions can be derived by action of the group itself. Since Lie groups are intimately connected to Lie algebras, a classification of inequivalent subgroups induces a classification of inequivalent Lie subalgebras, and vice versa. A general method for classifying the Lie subalgebras of a finite–dimensional Lie algebra uses inner automorphisms that are obtained by exponentiating the adjoint groups. In this paper, we present an effective algorithm able to automatically determine optimal systems of Lie subalgebras of a generic finite–dimensional Lie algebra abstractly assigned by means of its structure constants, or realized in terms of matrices or vector fields, or defined by a basis and the set of non-zero Lie brackets. The algorithm is implemented in the computer algebra system Wolfram Mathematica™; some meaningful and non-trivial examples are considered.

Group analysis and invariant solutions of the (3+1)-dimensional defocusing Gardner-KP equation

In this paper, the (3+1)-dimensional defocusing Gardner-KP equation is investigated again with the help of Lie symmetry method since the results are either incorrect, or incomplete, or misleading in the literature. For the Lie algebra of infinitesimal symmetries spanned by eight vector fields, the one-dimensional optimal system of subalgebra is established by leveraging the fundamental invariants and general adjoint transformation representation. Meanwhile, by virtue of some infinitesimal generators in the optimal system, several symmetry reductions and exact solutions are presented.

Scattering of solutions with group invariance for the fourth-order nonlinear Schrödinger equation

In this paper, we consider the focusing, L 2-supercritical and H˙2 -subcritical fourth-order nonlinear Schrödinger equations. We show the scattering of group-invariant solutions below the ground state threshold, under the hypothesis that the threshold for group-invariant solutions is less than a certain value.

Invariant analysis of the two-cell tumor growth model in the brain

In this study, we explore the invariance properties of a tumor growth model involving two distinct cell populations. These populations are characterized by different diffusion coefficients but share a common killing rate. This particular model serves as a representation of tumor growth within the brain. By employing the Lie group method, we unveil a two-dimensional symmetry algebra for cases where both diffusion coefficients are allowed to vary arbitrarily. Interestingly, this method reveals a nine-dimensional symmetry algebra when the diffusion coefficients are held constant. In both scenarios involving varying and constant diffusion coefficients, we conduct similarity reductions to deduce group invariant solutions, thus elucidating the model’s behavior. Notably, our findings demonstrate that the tumor’s growth remains exponential irrespective of the presence or absence of a killing rate. This remarkable phenomenon holds for various configurations of diffusion coefficients. To validate our observations, we employ Mathematica simulations, which corroborate the model’s exponential growth behavior and emphasize the role of killing rates, diffusion coefficients, and growth rate parameters in driving this exponential trend. Also, the conserved flows and conserved quantities of the model are demonstrated.

Novel dynamical group-invariant solutions and conserved vectors of the Gilson–Pickering equation with applications in plasma physics

This work delves into the analysis of the Gilson–Pickering equation which governs the waves propagation in plasma physics by invoking Lie symmetry analysis. We commence by identifying the Lie point symmetries associated with the equation. These symmetries are then leveraged on to compute the commutator table and subsequently the adjoint representation, ultimately leading to the establishment of an optimal system of one-dimensional subalgebras. Each subalgebra within this system is subsequently utilized to perform symmetry reductions. Through these reductions, various forms of nonlinear ordinary differential equations are obtained, which are subsequently solved using the power series method and Kudryashov’s technique. The resulting solutions are given in terms of hyperbolic functions. To gain deeper insights into the behaviour of these solutions, three-dimensional and two-dimensional plots are presented. Furthermore, applying the Ibragimov’s theorem allows us to derive conserved vectors associated with the Gilson–Pickering equation.

Conserved vectors and symmetry solutions of the Landau–Ginzburg–Higgs equation of theoretical physics

This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation (LGHe), which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves. The LGHe finds applications in various scientific fields, including fluid dynamics, plasma physics, biological systems, and electricity-electronics. The study adopts Lie symmetry analysis as the primary framework for exploration. This analysis involves the identification of Lie point symmetries that are admitted by the differential equation. By leveraging these Lie point symmetries, symmetry reductions are performed, leading to the discovery of group invariant solutions. To obtain explicit solutions, several mathematical methods are applied, including Kudryashov’s method, the extended Jacobi elliptic function expansion method, the power series method, and the simplest equation method. These methods yield solutions characterized by exponential, hyperbolic, and elliptic functions. The obtained solutions are visually represented through 3D, 2D, and density plots, which effectively illustrate the nature of the solutions. These plots depict various patterns, such as kink-shaped, singular kink-shaped, bell-shaped, and periodic solutions. Finally, the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors. These conserved vectors play a crucial role in the study of physical quantities, such as the conservation of energy and momentum, and contribute to the understanding of the underlying physics of the system.

A similarity reduction of the generalized Grad–Shafranov equation

We extend previous work [Y. E. Litvinenko, Phys. Plasmas 17, 074502 (2010)] on a direct method for finding similarity reductions of partial differential equations such as the Grad–Shafranov equation, to the case of the generalized Grad–Shafranov equation (GGSE) with arbitrary incompressible flow. Several families of analytic solutions are constructed, the generalized Solovév solution being a particular case, which contain both the classical and non-classical group-invariant solutions to the GGSE. Those solutions can describe a variety of equilibrium configurations pertinent to toroidal magnetically confined plasmas and planetary magnetospheres.

Symmetry-based study and dynamics of casual bulk viscous matter-dominated universe

The dynamics of cosmological expansion is studied for a spatially flat matter-dominated universe filled with barotropic casual bulk viscous fluid. The evolution equation for the Hubble parameter is derived using Friedmann equations and conservation law for the viscous fluid, in which the viscous pressure is modeled by truncated version of the full Israel-Stewart theory of bulk viscosity. We use a Lie symmetry-based approach to study the evolution of the expansion rate of the Universe. The model equation possesses eight-parameter Lie symmetry generators and gives two different analytic group invariant solutions for the Hubble parameter. Expressions for various important cosmological parameters are obtained and values are estimated analytically for the explanation of the evolution dynamics of the Universe. The interesting features of the obtained symmetry-based solutions are that, one satisfies the present acceleration of the Universe, while the other could not support present accelerating phase, rather represents a universe dominated by non-viscous stiff fluid. We make useful checks on the observational constraints on the model parameters from the latest released observational data. The accelerating phase of the current universe is investigated using observational bounds and compared the model under investigation with the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM model. The evolutionary cosmic phases obtained from the analysis of the Lie symmetry-based solutions correspond to the association with the critical points in the equivalent phase space. These interesting features are investigated and analyzed using dynamical systems approach.

Nonclassical symmetries, optimal classification, and dynamical behavior of similarity solutions of (3+1)-dimensional Burgers equation

This paper is devoted to the study of symmetry group classification, and optimal subalgebras of the three-dimensional Burgers equation, which describes nonlinear wave motion exhibits nonlinear effects and enhanced dispersion in the context of nonlinear dynamics. The equation undergoes a Painlevé analysis to assess its integrability properties to verify the possibility that it passes the Painlevé test, followed by the derivation of nonclassical symmetries. Utilizing the invariance property of Lie groups, desirable infinitesimal symmetries of Lie algebra have been outlined for the equation. Relying on the invariance characteristic of adjoint transformation along with the newly generated similarity variables, an intensive and systematically organized approach is employed to achieve a one-dimensional optimal system. The entire set of group invariant solutions for each of the associated subalgebras has been established. The dynamical behavior of derived exact solutions is examined using numerical simulations, and numerous intriguing occurrences are discovered, such as flat sheet, periodic wave, doubly soliton, multisoliton, bright-dark soliton, breather soliton, line soliton type, which offer valuable insights into the dynamics of the system and can predict the behavior of waves in real-world scenarios.

Nonlocal symmetry and group invariant solutions of dissipative (2+1)-dimensional AKNS equation

In this paper, the nonlocal symmetry of dissipative (2+1)-dimensional AKNS equation is constructed by auxiliary equation method. In order to better study new group invariant solutions of AKNS equation with the help of nonlocal symmetry, we introduce an new auxiliary dependent variable, the (2+1) dimensional AKNS equation is extended to a new closed prolonged system. Therefore, the nonlocal symmetry of the original equation is the Lie point symmetry of the new prolonged system. Then, by solving the initial-value problems, the finite symmetry transformation for the prolonged system is derived. Furthermore, by using symmetry reduction method to the prolonged system, the interaction solution between cnoidal waves and solitary waves is given.

Lie symmetry analysis, multiple exp-function method and conservation laws for the (2+1)-dimensional Boussinesq equation

In this study, we take into account the (2 + 1)-dimensional Boussinesq equation, a nonlinear evolution partial differential equation that describes how gravity waves move across the surface of the ocean. The symmetry reductions and group invariant precise solutions are systematically determined using the Lie symmetry analysis. We derive the precise multiple wave solutions using the multiple exp-function method, and then, using the multiplier method, we give the conservation laws. The dynamics of complicated waves and their interplay are faithfully recreated by the findings.

(3+1)-Dimensional Gardner Equation Deformed from (1+1)-Dimensional Gardner Equation and its Conservation Laws

Through the application of the deformation algorithm, a novel (3+1)-dimensional Gardner equation and its associated Lax pair are derived from the (1+1)-dimensional Gardner equation and its conservation laws. As soon as the (3+1)-dimensional Gardner equation is set to be y or z independent, the Gardner equations in (2+1)-dimension are also obtained. To seek the exact solutions for these higher dimensional equations, the traveling wave method and the symmetry theory are introduced. Hence, the implicit expressions of traveling wave solutions to the (3+1)-dimensional and (2+1)-dimensional Gardner equations, the Lie point symmetry and the group invariant solutions to the (3+1)-dimensional Gardner equation are well investigated. In particular, after selecting some specific parameters, both the traveling wave solutions and the symmetry reduction solutions of hyperbolic function form are given.

Nonstationary one-dimensional flows of a two-temperature dissociating gas: Group analysis and invariant solutions

Modern problems such as detonation engines and hypersonic reentry are reviving interest in the point explosion theory, which gives simple gasdynamics models of similar processes. This work is devoted to the study, using group analysis methods of mathematical properties of gasdynamics equations with thermochemical nonequilibrium, which is a typical problem in the theory of a point explosion. For this purpose, the system of equations under study was slightly corrected without distorting the physical meaning. This made it possible to obtain self-similar solutions. Solutions of two different models describing: a second-order endothermic dissociation reaction with vibrational exiting and a first-order exothermic reaction with spontaneous decay, are given in the paper. The problem of strong explosion of linear charge was considered. As in the classical case, the solution of the problem is reduced to integrating a system of ordinary differential equations in self-similar variables. The calculations show that the solutions to the modified system of equations is physically consistent and adequately describes the known effects of thermochemical nonequilibrium behind the wave front.

A Generalized (3+1)-Dimensional Nonlinear Wave Equation in Liquid with Gas Bubbles: Symmetry Reductions; Exact Solutions; Conservation Laws

The analysis of a generalised (3+1)-dimensional nonlinear wave equation that simulates a variety of nonlinear processes that occur in liquids including gas bubbles will be performed. After some cosmetic adjustments to the underlying equation, this generalised (3+1)-dimensional nonlinear wave equation naturally degenerates into the (3+1)-dimensional Kadomtsev-Petviashvili equation, the (3+1)-dimensional nonlinear wave equation, and the Korteweg-de Vries equation. To completely investigate this fundamental equation, a clear and rigorous technique is used. In order to obtain innovative symmetry reductions, group invariant solutions, conservation laws, and eventually kink wave solutions, the Lie symmetry, multiplier, and simplest equation methods are used. Complex waves and their dealing dynamics in fluids can be well imitated by the verdicts.

Lie group analysis for obtaining the abundant group invariant solutions and dynamics of solitons for the Lonngren-wave equation

The Lonngren-wave equation (LW Equation), one of the many nonlinear evolution equations (N-EEs) that arise in the field of mathematical physics, is the subject of this study, which uses an extremely strong analytical technique known as “Lie group analysis” to create novel solutions. We obtain a five-dimensional optimal system based on four-dimensional Lie algebra. We compute the group invariant solutions via subalgebras. Our obtained solutions are based on the trigonometric, hyperbolic, and polynomial functions. By varying the parameters, the solutions exhibit wavelike properties that include bright, dark, singular, dark-singular-combined solitons, periodic singular and dark-bright-combined. The physical dynamics of the obtained solutions are explored by the 3D and 2D Mathematica simulations which are explaining the new properties of the model considered in this paper.

Exact Solutions and Conservation Laws of A (2+1)-dimensional Combined Potential Kadomtsev-Petviashvili-B-type Kadomtsev-Petviashvili Equation

This article investigates a sixth order integrable nonlinear partial differential equation model that fulfills the Hirota N-soliton. Space and time-dependent shift, rotation and space-dependent shift, time and space translations, and time and space dilations Lie point symmetries are presented methodically. Under a specific point symmetries, the Lie point symmetries lead to group invariant solutions. The significance of conservation laws of the underlying equation are shown. The results are quite accurate in recreating complex waves and the dynamics of their interactions.