Lie Symmetry Research Articles

Lie symmetries and invariant solutions for three generalized short pulse equations

Lie symmetries and invariant solutions for three generalized short pulse equations

Lie symmetries, exact solutions and conservation laws of time fractional coupled (2+1)-dimensional nonlinear Schrödinger equations

ABSTRACT In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem.

Lie symmetries, exact solution and conservation laws of (2 + 1)-dimensional time fractional Kadomtsev–Petviashvili system

Abstract In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied.

Lie symmetry method to discuss the optimal system of Lie subalgebras and similarity solutions for shock propagation in a perfectly conducting dusty gas with magnetic field in rotating medium

In this paper, the similarity solutions for the shock wave propagation in a perfectly conducting dusty gas under the effect of axial or azimuthal magnetic field in the case of rotating medium by using Lie symmetry method have been discussed. The optimal system of one-dimensional Lie subalgebras related to the hyperbolic system of nonlinear partial differential equations had been constructed by using general invariant function and general adjoint transformation matrix. The optimal system is the collection of inequivalent optimal classes, which provides crucial information regarding different types of the similarity solutions. On the basis of the optimal system, we have discussed all possible similarity solutions for inequivalent optimal classes individually. Also, we have found that the similarity solutions exist in two cases for power law shock path and in one case with exponential law shock path. The shock wave decay is due to an increment in the physical parameter such as shock Cowling number or non-idealness parameter or adiabatic exponent or rotational parameter. For smaller value of the ratio of density of solid particles to initial species density of the gas i.e. [Formula: see text], the shock strength decreases with an increment in the mass fraction of solid particles in the mixture [Formula: see text], but the shock strength decreases as [Formula: see text] increases for [Formula: see text]. Also, the shock strength increases with an increment in [Formula: see text] at constant [Formula: see text]. It is observed that the shock wave is stronger in the presence of axial magnetic field as compared to azimuthal magnetic field.

Lie-symmetry analysis of a three dimensional flow due to unsteady stretching of a flat surface with non-uniform temperature distribution

Various mathematical frameworks have been developed to study dynamics of the fluid flow in stretching films. In this work, a three–dimensional flow induced by an unsteady stretching of an infinite flat sheet is analyzed through the Lie symmetry approach. Twelve Lie point symmetries for the nonlinear partial differential equations describing the considered flow and heat transfer phenomena are derived. Invariants for these Lie symmetries are obtained to construct a new class of similarity transformations. With the deduced similarity transformations, the flow equations are converted to ordinary differential equations which are solved using the Homotopy analysis method. The deduced analytic solution enables an exploration of the effects of various parameters on the flow and heat transfer, which have not been revealed previously using the Lie method as per the authors’ knowledge. The influences of the stretching rate, parameters that maintain the non-uniformity and unsteadiness of sheet temperature, and the Prandtl number, are demonstrated with the help of graphs and tables. These results show that for certain values of the system parameters, heat transfer reverses its direction and occurs from the fluid to the sheet. At these values, maximum heat transfer does not occur at the sheet but rather slightly above it.

Nonlinear waves for a variable-coefficient modified Kadomtsev–Petviashvili system in plasma physics and electrodynamics

In this paper, a variable-coefficient modified Kadomtsev–Petviashvili (vcmKP) system is investigated by modeling the propagation of electromagnetic waves in an isotropic charge-free infinite ferromagnetic thin film and nonlinear waves in plasma physics and electrodynamics. Painlevé analysis is given out, and an auto-Bäcklund transformation is constructed via the truncated Painlevé expansion. Based on the auto-Bäcklund transformation, analytic solutions are given, including the solitonic, periodic and rational solutions. Using the Lie symmetry approach, infinitesimal generators and symmetry groups are presented. With the Lagrangian, the vcmKP equation is shown to be nonlinearly self-adjoint. Moreover, conservation laws for the vcmKP equation are derived by means of a general conservation theorem. Besides, the physical characteristics of the influences of the coefficient parameters on the solutions are discussed graphically. Those solutions have comprehensive implications for the propagation of solitary waves in nonuniform backgrounds.

Lie Group Classification of a Class of Variable Coefficient Boiti–Leon–Manna–Pempinelli Equations

The Boiti–Leon–Manna–Pempinelli (BLMP) equation with coefficients being functions of time is considered. Since the coefficient functions are arbitrary, we have a class of BLMP equations. Symmetry analysis is carried out for this class. We derive the equivalence group admitted by the class and we present the enhanced Lie group classification. Lie symmetries are used to construct similarity reductions. Reduction operators that are not equivalent to Lie ones are also constructed.

Exploring concentration-dependent transport properties on an unsteady Riga plate by incorporating thermal radiation with activation energy and gyrotactic microorganisms

Abstract The aim of this study is to examine the entropy generation (EG) associated with the transfer of mass and heat in a concentration-dependent fluid with thermal radiation and activation energy, specifically in the context of an unsteady Riga Plate with gyrotactic microorganism. It is important to solve the ordinary differential equations generated from the controlling partial differential equations using Lie symmetry scaling to verify their quality and reliability. The system’s anticipated physical behavior is compared to Mathematica’s Runge–Kutta–Fehlberg numerical solution. Source parameters are essential for validation since they offer accurate results. Methodically change these values as a percentage to determine how they affect the unsteady fluid’s density, mass, and heat transfer over the Riga plate. Velocity, temperature, nanoparticle concentration and microorganism concentration profiles decrease with varying values of the unsteadiness parameter. EG increases with increasing values of concentration difference, thermal radiation, and Reynold number parameters. The Nusselt number experiences a 26.11% rise as a result of radiation when the unsteadiness parameter is A = − 0.25 A=-0.25 , in comparison with the scenario without radiation. Mass transfer upsurges with increasing values of the Brownian motion parameter and reduces with increasing values of thermophoresis parameter. To verify our conclusions, we compare calculated data, specifically the skin friction factor, to theoretical predictions. Tabular and graphical data can show how physical limits affect flow characteristics.

On the Limited Role of Conservation Laws in the Double Reduction Routine for Partial Differential Equations

The double reduction method for finding invariant solutions of a given partial differential equation (PDE) provides for the reduction of a $ q $-th order PDE that admits a nontrivial Lie symmetry and an associated nontrivial conservation law to an ordinary differential equation (ODE) of order $ q-1. $ In all the articles we have seen where the method has been used, the algorithm has involved writing the conservation law in canonical variables determined by the associated symmetry. In this paper, we illustrate that it is not necessary to use or even have the associated conservation law. It is enough to know that there exists a conservation law associated with a given Lie symmetry. Canonical variables derived from the symmetry are sufficient to achieve double reduction. In the canonical variables, the PDE is transformed after routine calculations into an ODE of order one less than that of the PDE. We have outlined steps involved in this variation of the double reduction method and illustrated the routine using five PDEs.

Lie symmetries, exact solutions and conservation laws of time fractional Boussinesq–Burgers system in ocean waves

In this paper, the Lie symmetry analysis method is applied to the time-fractional Boussinesq–Burgers system which is used to describe shallow water waves near an ocean coast or in a lake. We obtain all the Lie symmetries admitted by the system and use them to reduce the fractional partial differential equations with a Riemann–Liouville fractional derivative to some fractional ordinary differential equations with an Erdélyi–Kober fractional derivative, thereby getting some exact solutions of the reduced equations. For power series solutions, we prove their convergence and show the dynamic analysis of their truncated graphs. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission

In this paper, the combined upshot of Soret and Dufoue of a convective Maxwell nanofluid on a porous perpendicular surface with nonlinear thermal emission was investigated. In the present work, the impact of permeable stretching sheet, nonlinear thermal emission, heat sour sink, Dufour and Soret effect, chemical reaction, Brownian motion and thermophoresis in a convective Maxwell nanofluid flow is widely discussed. The governing equations derived for the problem are highly nonlinear coupled partial differential equations. The governing equations were transformed into ordinary differential equations using Lie symmetry group alterations. The BVP4C MATLAB solver was employed to solve the ordinary differential equations numerically after validating the convergence of the method with existing results in the literature. The numerical results were established and discussed using tables and graphs. It was found that variations in porosity parameter (K), Dufour (Du) and Soret (Sr) improves velocity, temperature and concentration profiles respectively and the present of nonlinear thermal radiation and heat source emit more heat for the flow. Also, it is exciting to report that both porosity (K) and Dufour (Du) parameters has a strong impact on the flow of skin frictions, Nusselt number and Sherwood number. However, the current results may present applications in the areas of petroleum reservoir, heat exchangers, steel industries, cooling applications, nuclear waste disposal and so on.

Resonant multi-wave, positive multi-complexiton, nonclassical Lie symmetries, and conservation laws to a generalized Hirota bilinear equation

This paper explores the evolutionary behavior of some specific waves for a Generalized Hirota Bilinear (GHB) equation with applications in modern sciences. More precisely, the linear superposition principle is first adopted to construct the resonant multi-wave of the GHB equation. Through the resonant N-wave and some particular computations, positive multi-complexiton to the governing model is then extracted with the use of computational packages. Furthermore, after deriving nonclassical Lie symmetries and their corresponding invariant solutions, Conservation Laws (CLs) for the GHB equation are formally constructed based on a general method developed by Ibragimov. The propagation dynamics of resonant double- and triple-waves as well as positive single- and double-complexitons are examined in detail by considering several case studies. The present results demonstrate how adjusting the nonlinear parameter can be used to control specific waves in nonlinear systems.

Auto-Bäcklund transformation and exact solutions for a new integrable (2+1)-dimensional shallow water wave equation

Shallow water waves (SWWs) are often used to describe water flow and wave movement in shallow water areas. The article introduces a novel (2 + 1)-dimensional SWW equation. We prove that the equation is integrable and obtain an auto-Bäcklund transformation by truncating Painlevé expansion. Using the bilinear form of the equation, a new auto-Bäcklund transformation and some exact solutions are obtained. Besides, a convergent power series solution is derived using Lie symmetry method. These exact solutions can enrich mathematical modeling and help us understand nonlinear wave phenomena. Finally, conserved vectors are derived.

A New (3+1)-Dimensional Extension of the Kadomtsev–Petviashvili–Boussinesq-like Equation: Multiple-Soliton Solutions and Other Particular Solutions

In this study, we focus on investigating a novel extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KPB-like) equation. Initially, we utilized the Lie symmetry method to determine the symmetry generator by considering the Lie invariance condition. Subsequently, by similar reduction, the equation becomes ordinary differential equations (ODEs). Exact analytical solutions were derived through the power series method, with a comprehensive proof of solution convergence. Employing the (G′/G2)-expansion method enabled the identification of trigonometric, exponential, and rational solutions of the equation. Furthermore, we established the auto-Bäcklund transformation of the equation. Multiple-soliton solutions were identified by utilizing Hirota’s bilinear method. The fundamental properties of these solutions were elucidated through graphical representations. Our results are of certain value to the interpretation of nonlinear problems.

Lie Symmetry Analysis, Closed-Form Solutions, and Conservation Laws for the Camassa–Holm Type Equation

Lie Symmetry Analysis, Closed-Form Solutions, and Conservation Laws for the Camassa–Holm Type Equation

Lie symmetries, exact solutions and conservation laws of (2+1)-dimensional time fractional cubic Schrödinger equation

Abstract In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) {(2+1)} -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) {(2+1)} -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) (1+1) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators.

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.

In search of hidden symmetries

Abstract This paper exemplifies the importance of finding hidden symmetries of differential equations that are models of physical phenomena. The hidden symmetries (Lie symmetries) may be determined by either linking together different equations for certain values of their parameters or transforming the original model into another equivalent system of equations that may have more symmetries. Therefore, hidden symmetries may help to solve the original model or yield its hidden properties, e.g. linearity and conservation laws. Moreover Noether symmetries are shown to be preserved by going from classical to quantum mechanics, namely from Lagrangian systems to the corresponding time-dependent Schrödinger equation.

Analysis of a general reaction–diffusion model using Lie symmetries and conservation laws

Analysis of a general reaction–diffusion model using Lie symmetries and conservation laws

Exact solution of a mathematical model for human muscular motion

An ordinary differential equation (ODE) which models human muscular movement is considered. A functional form of the model parameter is specified through the Lie symmetry approach, yielding a different expression from the one derived in the previous study (Kosugi et al., 2019). The Lie point symmetries corresponding to the model parameter are employed for derivation of exact solution.