Group Of Symmetry Transformations Research Articles

Soliton generation and conservation laws for vector light pulses propagating in weakly birefringent waveguides

The principal algebra Lp of coupled nonlinear Schrödinger equations describing the propagation of polarized optical pulses and involving four-wave mixing terms is obtained in terms of the three arbitrary labelling parameters of the system of equations. This algebra is found to be five-dimensional, indicating the richness in symmetries of the system. The most general symmetry group transformation by generators of Lp is found and it is shown that this transformation preserves the boundedness of solutions, and an example of transformation of a soliton solution into a completely new soliton of a different type is presented. Moreover, it is explained how an infinite sequence of bounded solutions can thus be generated, yielding new solitons. Several other important properties of solutions and symmetry group transformations of the system are also demonstrated. As the system of Schrödinger equations under study turns out to be of Lagrange type, conservation laws associated with all variational symmetries of Lp are constructed and interpreted. Some symmetry reductions of the system are also derived.

Study of optimal subalgebras, invariant solutions, and conservation laws for a Verhulst biological population model

AbstractIn this research, the (2+1)‐dimensional normal biological population model, incorporating the Verhulst law for population growth, is employed to explore species population dynamics. Employing Lie symmetry analysis, we address a nonlinear degenerate parabolic partial differential equation, yielding much‐improved results. This analysis includes computing one‐dimensional optimal subalgebras, reduced ordinary differential equations, and obtaining invariant solutions with a visual depiction of the physical behavior of the Verhulst biological population model through symmetry group transformations. Additionally, the multiplier method leads to novel conservation laws and potential systems not locally connected to the governing partial differential equation (PDE). These findings have significant implications for understanding and controlling biological populations, offering insights for applications in ecology and the environment.

The Geodesics for Poincaré’s Half-Plane: A Nonstandard Derivation

Constants of motion in Mechanics are usually inferred from groups of symmetry transformations of the given system, as, for example, a Lagrangian function that is time-invariant implies the conservation of energy. Here we wish to show that useful properties of a mechanical system can sometimes be deduced from a family of Noether-like transformations that are not inspired by any symmetry whatsoever. The sample system we concentrate on is the Lagrangian interpretation of Poincaré’s half plane of hyperbolic geometry, and the properties we will derive in a new way are the shape and the time parameterization of its geodesics.

Optimal subalgebras and conservation laws with exact solutions for biological population model

In the present study, we focus on the (2+1) dimensional normal biological population model (NBPM), which describes the population migration of species. We employ Lie symmetry analysis to the given nonlinear degenerate parabolic partial differential equation (PDE), which shows substantial advancement and upgraded results over other analytical techniques in determining some classes of exact solutions. Using the symmetry group of transformations, we construct the one-dimensional and two-dimensional optimal subalgebras for the NBPM. Further, we present the reduced ordinary differential equation(ODE) for each one-dimensional optimal subalgebras and construct some exact solutions for the physical model. Furthermore, we illustrate the physical behaviour of the model graphically through the obtained exact solutions. Lastly, applying the multipliers method, we develop some new conservation laws yielding some potential systems which are nonlocally related to the given PDE.

Poisson's Ratio of f.c.c. Hard‐Sphere Crystals with Cubic Supercells Containing Four Nanochannels Filled by Hard Spheres of Another Diameter

It is known that a face centered cubic (f.c.c.) crystal of identical hard spheres containing a periodic array of nanochannels, oriented along one of the principal crystallographic directions and filled by hard spheres of another diameter, shows a significant decrease of its Poisson's ratio with respect to pristine crystal. This indicates an essential enhancement of auxetic properties by such nanoinclusions. The present work describes studies of the f.c.c. hard‐sphere crystal containing four nanochannel arrays, oriented along the direction and its images with respect to cubic symmetry group transformations and selected in such a way as to not affect the symmetry of the crystal. The elastic properties of the current model are analyzed under different thermodynamic conditions and for different sizes of inclusions. These studies reveal that the auxetic properties of the system vanish when the diameters of inclusion spheres are greater than the ones forming the matrix f.c.c. crystal and, in the case of the widest nanochannels, also when these diameters are smaller. The studies are performed by Monte Carlo computer simulations in the isothermal–isobaric (NpT) ensemble, using the Parrinello–Rahman approach.

A study on dual solutions for water-based hybrid nanofluids flowing through convergent channel with dissipative heat transport

Over the years, the fluid flows in conjunction with thermal transport between non-parallel surfaces having converging nature is of great significance due to their broad spectrum of applications, which include fluid flows through nozzles in petroleum engineering, blood flow in arteries, lubrication systems, automobile radiators, thermal pumps, and water purification processes. Additionally, hybrid nanofluid is a prolific topic because of its thermal properties and potentials which provide a better performance even compared with common nanofluid in optimizing heat transfer. Therefore, this article presents a numerical simulation to investigate the heat transport characteristics of hybrid nanofluids in Jeffery-Hamel flow through a convergent channel. The considered hybrid nanofluids are composed of Copper (Cu) and Graphene-oxide (Go) as suspended nanoparticles and water as base fluid. This analysis further includes the impacts of viscous dissipation and magnetic field. A mathematical model for fluid flow and heat transfer are constructed with the help of cylindrical polar coordinates. The governing equations are converted into a system of ordinary differential equations (ODEs) by Lie symmetry group transformation. A MATLAB code is exercise to get the numerical solutions for flow and thermal distributions. An interesting phenomenon is that dual solutions are obtained in the computation. Thus, a comprehensive discussion is included on the dual solutions for various involved variables. The current findings may be employed in petroleum science, r biomedical scientists, polymer industry, etc.

MHD boundary layer flow of couple-stress fluid over a bidirectional moving surface with non-Fourier heat flux

This work is addressed to the study of MHD boundary layer couple-stress fluid flow considering the non-Fourier Cattaneo-Christov hyperbolic heat transfer model over a bidirectional movable surface. The group of continuous symmetry transformations is adopted to determine invariant solutions of governing equations for boundary layer fluid flow with heat transfer. Using Lie symmetry analysis, the governing equations having boundary conditions are converted into self-similar equations under a group of transformations only when initial prescribed velocity distributions are in linear form. The similar velocity and temperature profiles are obtained and the effects of the magnetic field and hyperbolic heat-flux are explored. The study revealed that the flow boundary layer thickness becomes thicker due to the increase of the couple-stress parameter, while the thermal boundary layer thickness becomes thinner for increasing relaxation time of heat-flux. The couple-stress parameter acts in the flow field in opposite style to the magnetic parameter. Also, for an increase in the couple-stress parameter and the thermal relaxation-time parameter, the higher wall heat transfer is exhibited.

Lie Symmetries and Conservation Laws for the Viscous Cahn-Hilliard Equation

In this paper, we study a viscous Cahn–Hilliard equation from the point of view of Lie symmetries in partial differential equations. The analysis of this equation is motivated by its applications since it serves as a model for many problems in physical chemistry, developmental biology, and population movement. Firstly, a classification of the Lie symmetries admitted by the equation is presented. In addition, the symmetry transformation groups are calculated. Afterwards, the partial differential equation is transformed into ordinary differential equations through symmetry reductions. Secondly, all low-order local conservation laws are obtained by using the multiplier method. Furthermore, we use these conservation laws to determine their associated potential systems and we use them to investigate nonlocal symmetries and nonlocal conservation laws. Finally, we apply the multi-reduction method to reduce the equation and find a soliton solution.

Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds

As the space of solutions of the first-order Hamiltonian field theory has a presymplectic structure, we describe a class of conserved charges associated with the momentum map, determined by a symmetry group of transformations. A gauge theory is dealt with by using a symplectic regularization based on an application of Gotay’s coisotropic embedding theorem. An analysis of electrodynamics and of the Klein–Gordon theory illustrate the main results of the theory as well as the emergence of the energy–momentum tensor algebra of conserved currents.

Self-similar analysis of Eyring-Powell fluid in boundary layer without simplification

The paper focuses on a theoretically study of fluid flow and heat transfer of Eyring-Powell fluid in a boundary layer over a flat surface. For the first time, groups of symmetry transformations were obtained for the full rheological law of Eyring-Powell fluid. This enabled obtaining self-similar forms of nonlinear differential equations different from the classical Blasius forms, which describe the limiting case for the large values of the parameter A. At the same time, the self-similar Blasius forms correctly describe the transition from the velocity profiles for a Newtonian fluid to linear profiles for Eyring-Powell fluid. The velocity and temperature profiles become less full due to the increasing parameter A, so that finally these profiles exhibit a linear shape. The friction coefficients and the Nusselt numbers decrease with the increasing parameter А.

Dispersionless Davey–Stewartson system: Lie symmetry algebra, symmetry group and exact solutions

Lie symmetry algebra of the dispersionless Davey–Stewartson (dDS) system is shown to be infinite dimensional. The structure of the algebra turns out to be Kac–Moody–Virasoro one, which is typical for integrable evolution equations in $$2+1$$ dimensions. Symmetry group transformations are constructed using a direct (global) approach. They are split into both connected and discrete ones. Several exact solutions are obtained as an application of the symmetry properties.

Lorentz Symmetry Group, Retardation and Energy Transformations in a Relativistic Engine

In a previous paper, we have shown that Newton’s third law cannot strictly hold in a distributed system of which the different parts are at a finite distance from each other. This is due to the finite speed of signal propagation which cannot exceed the speed of light in vacuum, which in turn means that when summing the total force in the system the force does not add up to zero. This was demonstrated in a specific example of two current loops with time dependent currents, the above analysis led to suggestion of a relativistic engine. Since the system is effected by a total force for a finite period of time this means that the system acquires mechanical momentum and energy, the question then arises how can we accommodate the law of momentum and energy conservation. The subject of momentum conservation was discussed in a pervious paper, while preliminary results regarding energy conservation where discussed in some additional papers. Here we give a complete analysis of the exchange of energy between the mechanical part of the relativistic engine and the field part, the energy radiated from the relativistic engine is also discussed. We show that the relativistic engine effect on the energy is 4th-order in 1c and no lower order relativistic engine effect on the energy exists.

On quantum determinants in integrable quantum gravity

Einstein–Rosen waves with two polarizations are cylindrically symmetric solutions to vacuum Einstein equations. Einstein equations in this case reduce to an integrable system. In 1971, Geroch has shown that this system admits an infinite-dimensional group of symmetry transformations known as the Geroch group. The phase space of this system can be parametrized by a matrix-valued function of spectral parameter, called monodromy matrix. The latter admits the Riemann–Hilbert factorization into a pair of transition matrices, i.e. matrix-valued functions of spectral parameter such that one of them is holomorphic in the upper half-plane, and the other is holomorphic in the lower half-plane. The classical Geroch group preserves the determinants of transition and monodromy matrices by construction. The algebraic quantization of the quadratic Poisson algebra generated by transition matrices of Einstein–Rosen system was proposed by Korotkin and Samtleben in 1997. Paternain, Peraza and Reisenberger have recently suggested a quantization of the Geroch group, which can be considered as a symmetry of the quantum algebra of observables. They have shown that commutation relations involving quantum monodromy matrices are preserved by the action of the quantum Geroch group. In present paper we introduce the notion of the determinant of the quantum monodromy matrix. We derive a factorization formula expressing the quantum determinant of the monodromy matrix as a product of the quantum determinants of the transition matrices. The action of the quantum Geroch group is extended from the subalgebra generated by the monodromy matrixonto the full algebra of observables. This extension is used to prove that the quantum determinant of the quantum monodromy matrix is invariant under the action of quantum Geroch group.

Energy conservation of nanofluids from a biomagnetic needle in the presence of Stefan blowing: Lie symmetry and numerical simulation

Thermal energy management associated with the transmission of heat is one of the main problems in many industrial setups (e.g. pharmaceutical, chemical and food) and bio-engineering devices (e.g. hospital ventilation, heating, cooling devices, heat exchanger and drying food, etc). The current study aims to examine thermo-bioconvection of oxytactic microorganisms taking place in a nanofluid-saturated needle with the magnetic field. Stefan-blowing is applied. The leading equations of continuity, momentum and energy, species transport equations for oxygen concentration and population density of microorganisms are reduced dimensionless and Lie symmetry group transformations are used to generate appropriate invariant transformations. The resulting similarity boundary value problem (in which the blowing parameter is coupled with concentration) have been simulated using MATLAB (2015a) bvp5c built in function. The impact of the emerging factors on the nondimensional velocity, temperature, nanoparticle concentration and motile microorganism density functions and their slopes at the wall, are pictured and tabulated. Justification with published results are included. It is found that all physical quantities decrease with Stefan blowing and increase with power law index parameter. With elevation in magnetic field parameter i.e., Lorentzian drag force, the friction factor reduces while the local Nusselt number, local Sherwood number, and the local motile microorganism density wall gradient increase. Present study could be used in food and pharmaceutical industries, chemical processing equipment, fuel cell technology, enhanced oil recovery, etc.

Exact Solutions of Newell-Whitehead-Segel Equations Using Symmetry Transformations

In this article, Lie and discrete symmetry transformation groups of linear and nonlinear Newell-Whitehead-Segel (NWS) equations are obtained. By using these symmetry transformation groups, several group invariant solutions of considered NWS equations have been constructed. Furthermore, some more group invariant solutions are generated by using discrete symmetry transformation group. Graphical representations of some obtained solutions are also presented.

Lie Symmetry Group for Unsteady Free Convection Boundary-Layer Flow over a Vertical Surface

The Lie symmetry group transformation method was used to investigate the partial differential equations that model the motion of a natural convective unsteady flow past to a non-isothermal vertical flat surface. The one-parameter Lie group transformation was applied twice consecutively to convert the motion governing equations into a system of ordinary differential equations. The obtained system of ordinary differential equations was solved numerically using the Lobatto IIIA formula (implicit Runge–Kutta method). The effect of the Prandtl number on the temperature and velocity profiles is illustrated graphically.

Effects of thermal slip and chemical reaction on free convective nanofluid from a horizontal plate embedded in a porous media.

We consider a two-dimensional, uniform, incompressible and free convection flow of a nano-fluid along a plane. The plate is located facing upward about the porous medium. Throughout the investigation, thermal slip, chemical reaction, heat emission/absorption is considered. In the modeling of nano-fluid we have considered the dynamic effect along with the Brownian and thermophoresis. In obtaining the governing equations, including the boundary conditions, an appropriate scaling is applied. The governing momentum equations, including thermal energy and nanoparticles equations are translated into a group of nonlinear ODEs by using Lie symmetry group transformation. The transformed equations are then solved numerically using the Runge-Kutta-Fehlberg fourth-fifth order. The numerical results of velocity, temperature, and nanoparticle volume fraction profiles for varied physical parameters will be discussed and analyzed at the end. The discussion also includes the local Nusselt and the local Sherwood numbers against several of the systems' physical parameters. It is found that the velocity and temperature decrease with thermal slip and heat absorption whilst it increases by increasing heat generation and chemical reaction order. Our present results will be compared with similar existing literature results.

MHD mixed convection on an inclined stretching plate in Darcy porous medium with Soret effect and variable surface conditions

AbstractThis work is concerned with a steady 2D laminar MHD mixed convective flow of an electrically conducting Newtonian fluid with low electrical conductivity along with heat and mass transfer on an isothermal stretching semi-infinite inclined plate embedded in a Darcy porous medium. Along with a strong uniform transverse external magnetic field, the Soret effect is considered. The temperature and concentration at the wall are varying with distance from the edge along the plate, but it is uniform at far away from the plate. The governing equations with necessary flow conditions are formulated under boundary layer approximations. Then a continuous group of symmetry transformations are employed to the governing equations and boundary conditions which determine a set of self-similar equations with necessary scaling laws. These equations are solved numerically and similar velocity, concentration, and temperature for various values of involved parameters are obtained and presented through graphs. The momentum boundary layer thickness becomes larger with increasing thermal and concentration buoyancy forces. The flow boundary layer thickness decreases with the angle of inclination of the stretching plate. The concentration increases considerably for larger values of the Soret number and it decreases with Lewis number. The skin friction coefficient increases for increasing angle of inclination of the plate, magnetic and porosity parameters, however it decreases for rise of thermal and solutal buoyancy parameters. In this double diffusive boundary layer flow, Nusselt and Sherweed numbers increase for rise of thermal and solutal buoyancy parameters, Prandtl number, but they behave opposite nature in case of angle of inclination of the plate, magnetic and porosity parameters. The Sherwood number increases for increasing Lewis number but it decreases for increasing Soret number.

Unsteady MHD mixed convective Casson fluid flow over a flat surface in the presence of slip

In this paper, we address a two-dimensional unsteady MHD mixed convective boundary layer flow past a flat surface of an electrically conducting Casson fluids taking radiative heat transfer. The partial slip boundary condition is imposed in the presence of suction/blowing. By adopting the continuous group of symmetry transformations, the symmetries and scaling laws are determined for the governing nonlinear partial differential equations along with boundary conditions. Finally, the set of coupled self-similar nonlinear ODEs is obtained. These equations are solved numerically for finding similar velocity and temperature for different parameter values. The effects of physical parameters on velocity and temperature are determined. The main findings are that the velocity profiles grow concavely and after reaching a critical point, the profiles overshoot by decreasing the velocity. This is due to the change of stress behavior which is an intrinsic property of the Casson fluid model. The imposition of partial slip condition on the flat surface leads to enhancement of the velocity overshoot.

The Endless Beta Integrals

We consider a special degeneration limit $\omega_1\to - \omega_2$ (or $b\to {\rm i}$ in the context of $2d$ Liouville quantum field theory) for the most general univariate hyperbolic beta integral. This limit is also applied to the most general hyperbolic analogue of the Euler-Gauss hypergeometric function and its $W(E_7)$ group of symmetry transformations. Resulting functions are identified as hypergeometric functions over the field of complex numbers related to the ${\rm SL}(2,\mathbb{C})$ group. A new similar nontrivial hypergeometric degeneration of the Faddeev modular quantum dilogarithm (or hyperbolic gamma function) is discovered in the limit $\omega_1\to \omega_2$ (or $b\to 1$).