Conditional Symmetry Groups of Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

The generalized conditional symmetry and sign-invariant approaches are developed to study the nonlinear diffusion equations with x-dependent convection and source terms. We obtain conditions under which the equations admit the second-order generalized conditional symmetries and the first-order sign-invariants on the solutions. Several types of different generalized conditional symmetries and first-order sign-invariants for the equations with diffusion of power law are obtained. Exact solutions to the resulting equations are constructed.

Finite Symmetry Transformation Groups and Exact Solutions of Lax Integrable Systems

The general Lie point symmetry groups of the Nizhnik-Novikov-Vesselov (NNV) equation and the asymmetric NNV equation are given by a simple direct method with help of their weak Lax pairs.

Lie symmetry group transformation for MHD natural convection flow of nanofluid over linearly porous stretching sheet in presence of thermal stratification

The magnetohydrodynamics （MHD） convection flow and heat transfer of an incompressible viscous nanofluid past a semi-infinite vertical stretching sheet in the pres- ence of thermal stratification are examined. The partial differential equations governing the problem under consideration are transformed by a special form of the Lie symmetry group transformations, i.e., a one-parameter group of transformations into a system of ordinary differential equations which are numerically solved using the Runge-Kutta-Gill- based shooting method. It is concluded that the flow field, temperature, and nanoparticle volume fraction profiles are significantly influenced by the thermal stratification and the magnetic field.

Generating Lie Point Symmetry Groups of (2-10-Dimensional Broer-Kaup Equationvia a Simple Direct Method

Using the (2+1)-dimensional Broer-Kaup equation as an simple example, a new direct method is developed to find symmetry groups and symmetry algebras and then exact solutions of nonlinear mathematical physical equations.

Non-Lie Symmetry Groups of （2＋1）-Dimensional Nonlinear Systems

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known （2＋1）-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only speciai cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.

Symmetry Groups and New Exact Solutions to （2＋1）-Dimensional Variable Coefficient Canonical Generalized KP Equation

In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.

Finite Symmetry Transformation Groups and Some Exact Solutions to(2+1)-Dimensional Cubic Nonlinear Schrdinger Equantion

Making use of the direct method proposed by Lou et al. and symbolic computation, finite symmetry transformation groups for a （2＋ l）-dimensional cubic nonlinear Schrodinger （NLS） equation and its corresponding cylindrical NLS equations are presented. Nine related linear independent infinitesimal generators can be obtained from the finite symmetry transformation groups by restricting the arbitrary constants in infinitesimal forms. Some exact solutions are derived from a simple travelling wave solution.

Lie point symmetry algebras and finite transformation groups of the general Broer-Kaup system

Using a new symmetry group theory, the transformation groups and symmetries of the general Broer-Kaup system are obtained. The results are much simpler than those obtained via the standard approaches.

Relationship between group velocity and symmetry of photonic crystals

Group velocity （GV） of eigenmode is a crucial parameter to explain the extraordinary phenomena about light propagation in photonic crystals （PhCs）. To study relationships between group velocity and symmetry of PhCs, a new general expression of CV in PhCs made up of non-dispersive material is introduced. Based on this, the CVs of eigenmodes of PhCs, especially those of degenerate eigenmodes at highly symmetric points in the first Brillouin zone, are discussed. Some interesting results are obtained. For example, the summation of degenerate eigenmodes＇ CVs is invariant under the operations of wave vector K-group MK. In addition, some numerical results are presented to verify them.

Symmetry and general symmetry groups of the coupled Kadomtsev-Petviashvili equation

In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili （cKP） equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et alo From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.

Entanglement of E8E8 Exceptional Lie Symmetry Group Dark Energy, Einstein’s Maximal Total Energy and the Hartle-Hawking No Boundary Proposal as the Explanation for Dark Energy

The present note is concerned with two connected and highly important fundamental questions of physics and cosmology, namely if E8E8 Lie symmetry group describes the universe and where cosmic dark energy comes from. Furthermore, we reason following Wheeler, Hartle and Hawking that since the boundary of a boundary is an empty set which models the quantum wave of the cosmos, then it follows that dark energy is a fundamental physical phenomenon associated with the boundary of the holographic boundary. This leads directly to a clopen universe which is its own Penrose tiling-like multiverse with energy density in full agreement with COBE, WMAP and Type 1a supernova cosmic measurements.

Finite Symmetry Transformation Groups and Exact Solutions of Konopelchenko-Dubrovsky Equation

Based on the general direct method developed by Lou et al. [J. Phys. A： Math. Gen. 38 （2005） L129], the symmetry group theorem is obtained, from that both the Lie point groups and the non-Lie symmetry groups of the Konopelchenk-Dubrovsky （KD） equation are obtained. From the theorem, some exact solutions of KD equation are derived from a simple travelling wave solution and a multi-soliton solution.

Lie Symmetry Groups of(2+1)-Dimensional BKP Equation and Its New Solutions

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physicssystems.Applying the modified direct method to the well-known (2+1)-dimensional BKP equation we get its symmetry.Furthermore,the exact solutions of (2+1)-dimensional BKP equation are obtained through symmetry analysis.

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.

Finite symmetry transformation group of the Konopelchenko Dubrovsky equation from its Lax pair

Starting from a weak Lax pair, the general Lie point symmetry group of the Konopelchenko-Dubrovsky equation is obtained by using the general direct method. And the corresponding Lie algebra structure is proved to be a Kac-Mood-Virasoro type. Furthermore, a new multi-soliton solution for the Konopelchenko-Dubrovsky equation is also given from this symmetry group and a known solution.

Symmetry groups and Gauss kernels of Schrdinger equations

The relationship between symmetries and Gauss kernels for the SchrSdinger equation iut = uxx ＋ f（x）u is established. It is shown that if the Lie point symmetries of the equation are nontrivial, a classical integral transformations of the Gauss kernels can be obtained. Then the Gauss kernels of Schroedinger equations are derived by inverting the integral transformations. Furthermore, the relationship between Gauss kernels for two equations related by an equivalence transformation is identified.

New infinite-dimensional symmetry groups for the stationary axisymmetric Einstein-Maxwell equations with multiple Abelian gauge fields

The so-called extended hyperbolic complex （EHC） function method is used to study further the stationary axisymmetric Einstein Maxwell theory with p Abelian gauge fields （EM-p theory, for short）, Two EHC structural Riemann- Hilbert （RH） transformations are constructed and are then shown to give an infinite-dimensional symmetry group of the EM-p theory. This symmetry group is verified to have the structure of semidirect product of Kac-Moody group SU（p ＋ 1, 1） and Virasoro group. Moreover, the infinitesimal forms of these two RH transformations are calculated and found to give exactly the same infinitesimal transformations as in previous author＇s paper by a different scheme, This demonstrates that the results obtained in the present paper provide some exponentiations of all the infinitesimal symmetry transformations obtained before.

Non-Lie Symmetry Groups and New Exact Solutions to the (2 + 1)-Dimensional Broer-Kaup System

For the (2 + 1)-dimensional Broer-Kaup system, we study the corresponding Lie symmetry groups, and obtain the symmetry group theorem and the Backlund transformation formula of solutions finding. At the same time, we find some new exact solutions of the (2 + 1)-dimensional Broer-Kaup system and extend the results in the papers [1-4].

Lie Algebra of Infinitesimal Generators of the Symmetry Group of the Heat Equation

Last time symmetry methods have been recognized to be of great importance for the study of the differential equations arising in mathematics and physics. The purpose of this paper is to provide some application of Lie groups to heat equation. In this example, we determine Lie algebra of infinitesimal generators of symmetry group of heat equation and construct group-invariant solutions of this equation. The some computational methods are presented so that researchers in other fields can readily learn to use them.

Some New Lie Symmetry Groups of Differential-Difference Equations Obtained from a Simple Direct Method

In this paper,based on the symbolic computing system Maple,the direct method for Lie symmetry groupspresented by Sen-Yue Lou [J.Phys.A:Math.Gen.38 (2005) L129] is extended from the continuous differential equationsto the differential-difference equations.With the extended method,we study the well-known differential-difference KPequation,KZ equation and (2+1)-dimensional ANNV system,and both the Lie point symmetry groups and the non-Liesymmetry groups are obtained.