Lie Symmetries of Klein-Gordon and Schrodinger Equations

In this paper, we investigate the Lie point symmetries of Klein-Gordon equation and Schr?dinger equation by applying the geometric concept of Noether point symmetries for the below defined Lagrangian. Moreover, we organize a strong relationship among the Lie symmetries related to Klein-Gordon as well as Schr?dinger equations. Finally, we utilize the consequences of Lie point symmetries of Poisson and heat equations within Riemannian space to conclude the Lie point symmetries of Klein-Gordon equation and Schr?dinger equation within universal Riemannian space.

Lie symmetries and conserved quantities for generalized Birkhoff system

To study Lie symmetry and the conserved quantity of a generalized Birkhoff system with additional terms,the determining equations of the Lie symmetry of the system is derived.A conserved quantity of Hojman's type and a Noether's conserved quantity are deduced by the Lie symmetry under some conditions.One example is given to illustrate the application of the result.

Conservation laws,Lie symmetries,self adjointness,and soliton solutions for the Selkov–Schnakenberg system

In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous soliton solutions like dark,bright,periodic,rational,Jacobian elliptic function,Weierstrass elliptic function,and hyperbolic solutions of the SS system will be achieved by a well-known technique called sub-ordinary differential equations.All these results are displayed graphically by 3D,2D,and contour plots.

Lie Symmetries of Quasihomogeneous Polynomial Planar Vector Fields and Certain Perturbations

In this work we study Lie symmetries of planar quasihomogeneous polynomial vector fields from different points of view, showing its integrability. Additionally, we show that certain perturbations of such vector fields which generalize the so-called degenerate infinity vector fields are also integrable.

Lie symmetry analysis and invariant solutions for the(3+1)-dimensional Virasoro integrable model

Lie symmetry analysis is applied to a(3+1)-dimensional Virasoro integrable model and the corresponding similarity reduction equations are obtained with the different infinitesimal generators.Invariant solutions with arbitrary functions for the(3+1)-dimensional Virasoro integrable model,including the interaction solution between a kink and a soliton,the lump-type solution and periodic solutions,have been studied analytically and graphically.

Multiple Darboux–B?cklund transformations via truncated Painleve′ expansion and Lie point symmetry approach

For a given truncated Painleve′ expansion of an arbitrary nonlinear Painleve′ integrable system, the residue with respect to the singularity manifold is known as a nonlocal symmetry, called the residual symmetry, which is proved to be localized to Lie point symmetries for suitable prolonged systems. Taking the Korteweg–de Vries equation as an example, the n-th binary Darboux–Ba¨cklund transformation is re-obtained by the Lie point symmetry approach accompanied by the localization of the n-fold residual symmetries.

Lie symmetries and conserved quantities of fractional nonconservative singular systems

In this paper,according to the fractional factor derivative method,we study the Lie symmetry theory of fractional nonconservative singular Lagrange systems in a configuration space.First,fractional calculus is calculated by using the fractional factor,and the fractional equations of motion are derived by using the differential variational principle.Second,the determining equations and the limiting equations of Lie symmetry under an infinitesimal group transformation are obtained.Furthermore,the fractional conserved quantity form of singular Lagrange systems caused by Lie symmetry is obtained by constructing a gauge-generating function that fulfills the structural equation,which conforms to the Noether criterion equation.Finally,we present an example of a calculation.The results show that the Lie symmetry condition of nonconservative singular Lagrange systems is more strict than conservative singular systems,but because of increased invariance restriction,the nonconservative forces do not change the form of conserved quantity;meanwhile,the fractional factor method has high natural consistency with the integral calculus,so the theory of integer-order singular systems can be easily extended to fractional singular Lagrange systems.

Einstein’s Dark Energy via Similarity Equivalence, ‘tHooft Dimensional Regularization and Lie Symmetry Groups

Realizing the physical reality of ‘tHooft’s self similar and dimensionaly regularized fractal-like spacetime as well as being inspired by a note worthy anecdote involving the great mathematician of Alexandria, Pythagoras and the larger than life man of theoretical physics Einstein, we utilize some deep mathematical connections between equivalence classes of equivalence relations and E-infinity theory quotient space. We started from the basic principles of self similarity which came to prominence in science with the advent of the modern theory of nonlinear dynamical systems, deterministic chaos and fractals. This fundamental logico-mathematical thread related to partially ordered sets is then applied to show how the classical Newton’s kinetic energy E = 1/2mv2 leads to Einstein’s celebrated maximal energy equation E = mc2 and how in turn this can be dissected into the ordinary energy density E(O) = mc2/22 and the dark energy density E(D) = mc2(21/22) of the cosmos where m is the mass;v is the velocity and c is the speed of light. The important role of the exceptional Lie symmetry groups and ‘tHooft-Veltman-Wilson dimensional regularization in fractal spacetime played in the above is also highlighted. The author hopes that the unusual character of the analysis and presentation of the present work may be taken in a positive vein as seriously attempting to propose a different and new way of doing theoretical physics by treating number theory, set theory, group theory, experimental physics as well as conventional theoretical physics on the same footing and letting all these diverse tools lead us to the answer of fundamental questions without fear of being labelled in one way or another.

Perturbation to Lie Symmetry and Adiabatic Invariants for General Holonomic Mechanical Systems

Based on the concept of adiabatic invariant,the perturbation to the Lie symmetry and adiabatic invariantsfor general holonomic mechanical systems are studied.The exact invariants induced directly from the Lie symmetryof the system without perturbation are given.The perturbation to the Lie symmetry is discussed and the adiabaticinvariants that have the different form from that in[Act.Phys.Sin.55(2006)3236(in Chinese)]of the perturbedsystem,are obtained.

On Form Invariance,Lie Symmetry and Three Kinds of Conserved Quantities of Generalized Lagrange's Equations

In this paper,the form invariance and the Lie symmetry of Lagrange's equations for nonconservativesystem in generalized classical mechanics under the infinitesimal transformations of group are studied,and the Noether'sconserved quantity,the new form conserved quantity,and the Hojman's conserved quantity of system are derived fromthem.Finally,an example is given to illustrate the application of the result.

Birkhoff’s Theorem and Lie Symmetry Analysis

Three dimensional space is said to be spherically symmetric if it admits SO(3) as the group of isometries. Under this symmetry condition, the Einstein’s Field equations for vacuum, yields the Schwarzschild Metric as the unique solution, which essentially is the statement of the well known Birkhoff’s Theorem. Geometrically speaking this theorem claims that the pseudo-Riemanian space-times provide more isometries than expected from the original metric holonomy/ansatz. In this paper we use the method of Lie Symmetry Analysis to analyze the Einstein’s Vacuum Field Equations so as to obtain the Symmetry Generators of the corresponding Differential Equation. Additionally, applying the Noether Point Symmetry method we have obtained the conserved quantities corresponding to the generators of the Schwarzschild Lagrangian and paving way to reformulate the Birkhoff’s Theorem from a different approach.

Time-fractional Davey–Stewartson equation:Lie point symmetries,similarity reductions,conservation laws and traveling wave solutions

As a celebrated nonlinear water wave equation,the Davey–Stewartson equation is widely studied by researchers,especially in the field of mathematical physics.On the basis of the Riemann–Liouville fractional derivative,the time-fractional Davey–Stewartson equation is investigated in this paper.By application of the Lie symmetry analysis approach,the Lie point symmetries and symmetry groups are obtained.At the same time,the similarity reductions are derived.Furthermore,the equation is converted to a system of fractional partial differential equations and a system of fractional ordinary differential equations in the sense of Riemann–Liouville fractional derivative.By virtue of the symmetry corresponding to the scalar transformation,the equation is converted to a system of fractional ordinary differential equations in the sense of Erdélyi–Kober fractional integro-differential operators.By using Noether’s theorem and Ibragimov’s new conservation theorem,the conserved vectors and the conservation laws are derived.Finally,the traveling wave solutions are achieved and plotted.

Application of canonical coordinates for solving single-freedom constraint mechanical systems

This paper introduces the canonical coordinates method to obtain the first integral of a single-degree freedom constraint mechanical system that contains conservative and non-conservative constraint homonomic systems. The definition and properties of canonical coordinates are introduced. The relation between Lie point symmetries and the canonical coordinates of the constraint mechanical system are expressed. By this relation, the canonical coordinates can be obtained. Properties of the canonical coordinates and the Lie symmetry theory are used to seek the first integrals of constraint mechanical system. Three examples are used to show applications of the results.

Quantum Dark Energy from the Hyperbolic Transfinite Cantorian Geometry of the Cosmos

The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.

Streamlines in the Two-Dimensional Spreading of a Thin Fluid Film: Blowing and Suction Velocity Proportional to the Height

The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the thin film satisfies a nonlinear diffusion equation with a source/sink term. The Lie point symmetries of the nonlinear diffusion equation are derived and exist, which provided the fluid velocity at the base, vn satisfies a first order linear partial differential equation. The general form has algebraic time dependence while a special case has exponential time dependence. The solution in which vn is proportional to the height of the thin film is studied. The width of the base always increases with time even for suction while the height decreases with time for sufficiently weak blowing. The streamlines of the fluid flow inside the thin film are plotted by first solving a cubic equation. For sufficiently weak blowing there is a dividing streamline, emanating from the stagnation point on the centre line which separates the fluid flow into two regions, a lower region consisting of rising fluid and dominated by fluid injection at the base and an upper region consisting of descending fluid and dominated by spreading due to gravity. For sufficiently strong blowing the lower region expands to completely fill the whole thin film.

From Witten’s 462 Supercharges of 5-D Branes in Eleven Dimensions to the 95.5 Percent Cosmic Dark Energy Density behind the Accelerated Expansion of the Universe

The measured 95.5% dark energy density of the cosmos presumed to be behind the observed accelerated cosmic expansion is determined theoretically based upon Witten’s five branes in eleven dimensions theory. We show that the said dark energy density is easily found from the ratio of the 462 states of the five dimensional Branes to the total number of states, namely 528 minus the 44 degrees of freedom of the vacuum, i.e. , almost exactly as found in WMAP and Type 1a supernova measurements.

A symmetry-preserving difference scheme for high dimensional nonlinear evolution equations

In this paper,a procedure for constructing discrete models of the high dimensional nonlinear evolution equations is presented.In order to construct the difference model,with the aid of the potential system of the original equation and compatibility condition,the difference equations which preserve all Lie point symmetries can be obtained.As an example,invariant difference models of the(2+1)-dimensional Burgers equation are presented.

Hyperbolic Monge-Ampère Equation

In this paper, based on the Lie symmetry method, the symmetry group of a hyperbolic Monge-Ampère equation is obtained first, then the one-dimensional optimal system of the obtained symmetries is given, and finally the group-invariant solutions are investigated.

Some New Nonlinear Wave Solutions for a Higher-Dimensional Shallow Water Wave Equation

In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended F-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.

Lie symmetry analysis, optimal system and conservation laws of a new(2+1)-dimensional KdV system

In this paper, Lie point symmetries of a new(2+1)-dimensional KdV system are constructed by using the symbolic computation software Maple. Then, the one-dimensional optimal system,associated with corresponding Lie algebra, is obtained. Moreover, the reduction equations and some explicit solutions based on the optimal system are presented. Finally, the nonlinear selfadjointness is provided and conservation laws of this KdV system are constructed.