Lie Symmetry Research Articles

On the generalized time fractional reaction–diffusion equation: Lie symmetries, exact solutions and conservation laws

In this paper, Lie symmetry analysis method is applied to the generalized time fractional reaction–diffusion equation. We obtain a conditional symmetric group and some conservation laws of the governing equation. The obtained Lie symmetries are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative. Furthermore, we obtained asymptotic stable solutions and convergent power series solutions for the reduced equations. The dynamic behavior of these exact solutions is presented graphically.

Lie symmetry analysis for fractional evolution equation with $$\zeta $$-Riemann–Liouville derivative

Lie symmetry analysis for fractional evolution equation with $$\zeta $$-Riemann–Liouville derivative

Dynamics of the breather and solitary waves in plasma using the generalized variable coefficient Gardner equation

This paper presents an analytical approach to study the generalized variable coefficient Gardner equation, aiming to investigate the behaviour of plasma phenomena, which presents challenges due to the non-linear nature of the Gardner equation. The point symmetry group associated with the proposed equation is determined by Lie symmetry analysis. The Exp-function method is employed to construct exact travelling wave solutions and to obtain anomalous solutions with non-zero amplitude. Additionally, conservation laws are derived and analysed from the equation in the frame of Noether symmetry. Through simulation, breather, periodic, and localized solitary wave solutions are also presented in graphical form. Our results are expected to elucidate the impact of the equation in the study of the propagation of Alfvén waves with non-uniform density, temperature, and wave-particle interactions in a magnetized plasma.

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.

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.

First integrals, conserved vectors of nonlinear partial difference equations

We perform a symmetry analysis of some nonlinear partial difference equations (nP △ Es), where the discrete version is obtained using some discretization approach. The discrete versions of the wave, diffusion, Fisher and Huxley equations are the subject of this research. At first, the initial invariance approach is the Lie symmetry approach. The first integrals technique that Hydon introduced to be used with discrete ordinary difference equations (O △ Es) serves as our inspiration in this situation. We develop a similar technique for generating the first integral vectors of the nP △ Es without recourse to symmetry generators.

Revisitation of “implicit quiescent optical solitons with complex Ginzburg–Landau equation having nonlinear chromatic dispersion": generalized temporal evolution

AbstractThis paper is a revisitation to the study on the retrieval of quiescent optical solitons to the complex Ginzburg–Landau equation that is considered with generalized temporal evolution and nonlinear chromatic dispersion. The results are recovered with the application of Lie symmetry. Apart from a couple of self-phase modulation structures where the results are in quadratures, the integrals are with a range of special functions. Two new forms of self-phase modulation structures are addressed in the paper.

Investigation of Some Time–Space M-Truncated Partial Differential Equations: Lie Symmetry Analysis, Exact Solutions and Conservation Laws

Investigation of Some Time–Space M-Truncated Partial Differential Equations: Lie Symmetry Analysis, Exact Solutions and Conservation Laws

The Lie point symmetry criteria and formation of exact analytical solutions for Kairat-II equation: Paul-Painlevé approach

The renowned non-linear Kairat-II equation, which expresses the surface curves, is examined meticulously in this work. There has never been a study before that discussed conserved quantities, sensitivity analysis, Hamiltonian function, Lie symmetry invariance criteria, and invariant solutions of Kairat-II equation The Lie invariance criteria are taken into consideration by the symmetry generators. The suggested technique leads to in a four-dimensional Lie algebra, where the translation point symmetries in space and time correlate with the conservation of mass and the energy, respectively, and remaining point symmetries are dilation and scaling. Firstly, we use symmetry reduction of Lie subalgebras to obtain closed-form invariant solutions. In specific reduction cases, we transform the Kairat-II equation into a spectrum of non-linear ordinary differential equations, which have the benefit of being able to offer an extensive selection of closed-form solitary wave solutions. For this equation, the Cauchy problem cannot be solved by the inverse scattering transform, therefore, traveling wave exact solutions are generated using the analytical Paul-Painlevé approach. In both two and three dimensions, the graphical behavior of specific solutions is presented for specific quantities of physical factors of examined equation. Utilizing conservation laws multipliers, the study culminates by determining an extensive set of local conservation laws for the non-linear Kairat-II equation that are applicable to arbitrary constant coefficients. Instead of focusing on the physical aspects of conservation laws, this study computed the conserved quantities using a mathematical perspective that can be used to identify potential symmetries of partial differential equations, guiding readers toward the solutions of partial differential equations. The existence of the Hamiltonian function is presented. The model’s sensitivity to various starting conditions is highlighted by the sensitivity analysis.

Some more variety of analytical solutions to (2+1)-Bogoyavlensky-Konopelchenko equation

The goal of this study is to get analytical solutions to the Bogoyavlensky-Konopelchenko equation, which describes an interaction of a long-wavelength wave moving along the x-axis and a Riemann wave-form moving along the y-axis. The equation has been widely used in soliton theory, fluid dynamics, optics, biological systems, and differential geometry. It is a member of the Ablowitz-Kaup-Newell-Segur hierarchy of integrable systems. By exploiting the classical Lie symmetry approach, the equation is reduced to an ordinary differential equation. After solving the second similarity reduction, the authors derived a novel class of solutions for each case. Additionally, the derived solutions comprise some parameters, and various functions might be utilized to explore wave profiles like stationary, dark, and bright soliton, parabolic dark and bright soliton, and progressive nature. Solutions are compared with previous researches [–] to show novelty of results. Solutions can help with numerous applications in physics, such as fluid dynamics, plasma physics, and nonlinear optics.

Lie symmetries and conserved quantities of static Bertotti–Robinson spacetime

In this study, Lie symmetries and conserved quantities of the Bertotti–Robinson (B–R) spacetime were obtained, and classified according to this spacetime. Different classifications are discussed along with their Lie symmetries and conservation laws. The main objective of this research is to find all possible forms of Lie symmetry classifications for B–R spacetime and discuss their physical significance.

Lie symmetry, convergence analysis, explicit solutions, and conservation laws for the time-fractional modified Benjamin-Bona-Mahony equation

Lie symmetry, convergence analysis, explicit solutions, and conservation laws for the time-fractional modified Benjamin-Bona-Mahony equation

Qualitative analysis, exact solutions and symmetry reduction for a generalized (2＋1)-dimensional KP–MEW-Burgers equation

The objective of this manuscript is to examine the non-linear characteristics of the modified equal width-Burgers equation, known as the generalized Kadomtsive–Petviashvili equation, and its ability to generate a long-wave with dispersion and dissipation in a nonlinear medium. We employ the Lie symmetry approach to reduce the dimension of the equation, resulting in an ordinary differential equation. Utilizing the newly developed generalized logistic equation method, we are able to derive solitary wave solutions for the aforementioned ordinary differential equation. In order to gain a deeper understanding of the physical implications of these solutions, we present them using various visual representations, such as 3D, 2D, density, and polar plots. Following this, we conduct a qualitative analysis of the dynamical systems and explore their chaotic behavior using bifurcation and chaos theory. To identify chaos within the systems, we utilize various chaos detection tools available in the existing literature. The results obtained from this study are novel and valuable for further investigation of the equation, providing guidance for future researchers.

Lie symmetries of Lemaitre–Tolman–Bondi metric

The aim of this paper is to investigate Lie symmetries including Killing, homothetic and conformal symmetries of Lemaitre–Tolman–Bondi (LTB) spacetime metric. To find all LTB metrics admitting these three types of symmetries, we have analyzed the set of symmetry equations by a Maple algorithm that provides some restrictions on the functions involved in LTB metric under which this metric admits the three mentioned symmetries. The solution of symmetry equations under these restrictions leads to the explicit form of symmetries. The stress–energy tensor is calculated for all the obtained metrics in order to discuss their physical significance. It is noticed that most of these metrics satisfy certain energy conditions and correspond to anisotropic fluids.

Implicit quiescent optical solitons For Lakshmanan–Porsezian–Daniel model having nonlinear chromatic dispersion and power-law of self-phase modulation by lie symmetry

AbstractThis paper recovers implicit quiescent optical solitons for the Lakshmanan–Porsezian–Daniel equation that is studied with nonlinear chromatic dispersion and power-law of self-phase modulation. The Lie symmetry analysis has made this retrieval possible. An interesting observation has been made with the results that was not recoverable from the prequel paper.

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

The propagation of nonlinear solitary and shallow water waves can be examined by nonlinear evolution equations. The dynamics of nonlinear behavior are described by exact solutions having parametric functions. The integrability of nonlinear equation discloses various characteristics of real phenomenon with continuous and fluctuating background. This work aims to generate symmetry reductions and exact solutions of KdV‐mKdV equation, which is a completely integrable system of nonlinear equations and describes combined solitary wave effects in inhomogeneous medium. The infinitesimal generators and their commutative relations are deduced under invariance property of Lie groups. The invariance property provides a symmetry reduction of governing system. Therefore, a continuous process of reductions transmutes the KdV‐mKdV equation to an equivalent system of ODEs. This system of ODEs leads to the exact solutions involving several arbitrary constants and functions. Such obtained results are generalized than earlier works and never reported in literature. Furthermore, adjoint equation and conserved quantities associated with Lagrangian formulation are constructed under Lie symmetries. To reveal the significance of governing phenomenon, the exact solutions are traced via numerical simulation and thus the graphical structures demonstrate line multisoliton, single soliton, lumps, parabolic and stationary wave profiles.

Spectral problem for the complex mKdV equation: singular manifold method and Lie symmetries

This article addresses the study of the complex version of the modified Korteweg-de Vries equation using two different approaches. Firstly, the singular manifold method is applied in order to obtain the associated spectral problem, binary Darboux transformations and $\tau$-functions. The second part concerns the identification of the classical Lie symmetries for the spectral problem. The similarity reductions associated to these symmetries allow us to derive the reduced spectral problems and first integrals for the ordinary differential equations arising from such reductions.

A mathematical model for two solutes transport in a poroelastic material and its applications

Using well-known mathematical foundations of the elasticity theory, a mathematical model for two solutes transport in a poroelastic material (soft tissue is a typical example) is suggested. It is assumed that molecules of essentially different sizes dissolved in fluid and are transported through pores of different sizes. The stress tensor, the main force leading to the material deformation, is taken not only in the standard linear form but also with an additional nonlinear part. The model is constructed in 1D space and consists of five nonlinear equations. It is shown that the governing equations are integrable in stationary case, therefore all steady-state solutions are constructed. The obtained solutions are used in an example for healthy and tumour tissue, in particular, tissue displacements are calculated and compared for parameters taken from experimental data in cases of the linear and nonlinear stress tensors. Since the governing equations are non-integrable in non-stationary case, the Lie symmetry analysis is used in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with non-trivial Lie symmetries are identified. As a result, multiparameter families of exact solutions are constructed including those in terms of special functions(hypergeometric and Bessel functions). A possible application of the solutions obtained is demonstrated.

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.

Lie symmetry based neural networking analysis for Powell–Eyring fluid flow with heat and mass transfer effects

Heat exchangers, drying, dehydration, pollutant dispersion, separation processes, and biological individualities to name a few are significant phenomena in numerous fields and industries subject to both mass transfer and heat transfer aspects. Owing to such importance the aim of the present study is to offer a machine learning remedy for mass transfer rate at the surface subject to the Powell–Eyring (PE) fluid model. The flow field is rooted with suction and injection effects. The heat transfer aspects are considered by using the energy equation. The developed differential system is reduced by using Lie symmetry and the shooting scheme is used to get numerical data corresponding to four inputs namely suction parameter, PE fluid parameter, Schmidt number, and power law index. Levenberg–Marquardt backpropagation-based neural networking model is developed along with MSE and regression analysis to approximate the ShD number. It is observed that the mass transfer rate shows inciting values towards the suction parameter, Schmidt number, and concentration power law index. Further, the magnitude of Powell–Eyring fluid concentration is higher in the case of injection.