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.

Extended Group Foliation Method and Functional Separation of Variables to Nonlinear Wave Equations

Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.

A Kind of Diophantine Equations in Finite Simple Groups

In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=(t,0,0,0,t,0), where t is a non negative integer. We also give all solutions of a kind of generalized Ramanujan Nagell equations by using the theories of imaginary quadratic field and Pells equation.

Besov Estimates for Sub-Elliptic Equations in the Heisenberg Group

In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.

Preliminary Group Classification for One Class of Nonlinear Wave Equations

The group classification is carried out on the nonlinear wave equation utt = f（x,u, ux）uzz ＋ g（x,u,uz） by using the preliminary group classification approach. The generators of equivalence group are determined and the corresponding reduced forms are obtained. The result of the work is shown in table form.

SOLVABILITY OF THE FOURTH ORDER NONLINEAR SUBELLIPTIC EQUATIONS ON THE HEISENBERG GROUP

In this paper,some existence results for the fourth order nonlinear subelliptic equations on the Heisenberg group are given by means of variational methods.

Preliminary group classification of quasilinear third-order evolution equations

Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six non- equivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.

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.

ON THE LOCAL SOLVABILITY OF DIFFERENTIAL EQUATIONS ASSOCIATED TO UNSOLVABLE INHOMOGENEOUS LEFT INVARIANT DIFFERENTIAL OPERATORS ON THE HEISENBERG GROUP

This paper studies the local solvability of the differental equations associated to unsolvable inhomogeneous left invariant differential operators on the Heisenberg group.It is provedthat for a class of inhomogeneous left invariant differental operators on the Heisenberg group,the local solvability of the corresponding equations is equivalent to the local sovability of the equations associated to their highest order terms.Then,under certain conditions on the highest order term,we obtain the necessary and sufficient conditions for the functon f to satisfy ill order for the differential equation Lu=f to be locally solvable

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.

Lie Group Classifications and Stability of Exact Solutions for Multidimensional Landau-Lifshitz Equations

In this paper, based on classical Lie group method, we study multi-dimensional Landau-Lifshitz equation, and get its infinitesimal generator, symmetry group and new solutions. In particular, we build the connection between new exact solutions and old exact solutions. At the same time, we also prove that the initial boundary value condition of the three-dimensional Landau-Lifshitz equation admits a unique solution and discuss the stability of the solution.

Symmetry Groups and New Exact Solutions of(2+1)-Dimensional Dispersive Long-Wave Equations

Using the modified CK＇s direct method, we derive a symmetry group theorem of （2＋1）-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of （2＋1）- dimensional dispersive long-wave equations are obtained.

Diophantine Equations and the Freeness of Mobius Groups

Let p and q be two fixed non zero integers verifying the condition gcd(p,q) = 1. We check solutions in non zero integers a1,b1,a2,b2 and a3 for the following Diophantine equations: (B1) (B2) . The equations (B1) and (B2) were considered by R.C. Lyndon and J.L. Ullman in [1] and A.F. Beardon in [2] in connection with the freeness of the M?bius group generated by two matrices of namely and where .?They proved that if one of the equations (B1) or (B2) has solutions in non zero integers then the group is not free. We give algorithms to decide if these equations admit solutions. We obtain an arithmetical criteria on p and q for which (B1) admits solutions. We show that for all p and q the equations (B1) and (B2) have only a finite number of solutions.

Functional Separable Solutions to Nonlinear Diffusion Equations by Group Foliation Method

We consider the functional separation of variables to the nonlinear diffusion equation with source and convection term： ut = （A（x）D（u）ux）x ＋ B（x）Q（u）, Ax ≠ 0. The functional separation of variables to this equation is studied by using the group foliation method. A classification is carried out for the equations which admit the function separable solutions. As a consequence, some solutions to the resulting equations are obtained.

Solution of ODE u″+p(u)(u′)2+q(u)=0 and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations

Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc., are reduced to an integrable ODE expressed by u＂ ＋p（u）（u＇）^2 ＋ q（u） = 0 whose generai solution can be given. Furthermore, combining complete discrimination system for polynomiai, the classifications of all single travelling wave solutions to these equations are obtained. The equation u＂＋p（u）（u＇）^2＋q（u） = 0 includes the equation （u＇）^2 = f（u） as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.

Conditional Similarity Reductions of Jimbo—Miwa Equation via the Classical Lie Group Approach

Recently, the Clarkson and Kruskal direct method has been modified to find new similarity reductions (conditional similarity reductions) of nonlinear systems and the results obtained by the modified direct method cannot be obtained by the current classical and/or non-classical Lie group approach. In this paper, we show that the conditional similarity reductions of the Jimbo-Miwa equation can be reobtained by adding an additional constraint equation to the original model to form a conditional equation system first and then solving the model system by means of the classical Lie group approach.

Estimation of Acentric Factor of Organic Compounds with Corresponding States Group Contribution Method

A new estimation method was proposed by combining the corresponding state principle with the group contribution method through introducing the concept of assumed-critical properties. Combining it with the Reidel equation, a new acentric factor correlation equation (CSGC-Reidel) was developed. Contribution values of 70 groups were obtained by correlating acentric factor data of 228 organic compounds of 14 type substances including saturated hydrocarbons, unsaturated hydrocarbons, cyclanes, aromatics, oxygen compounds, nitrogen compounds,halohydrocarbons, etc. The average error of acentric factor is 3.52% between the literature data and the predicated with the new estimation method.

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.

TAYLOR POLYNOMIAL STEPWISE REFINEMENTALGORITHM FOR LIE AND HIGH SYMMETRIES OF PARTIAL DIFFERENTIAL EQUATIONS

In this article, based on the Taylor expansions of generating functions and stepwise refinement procedure, authors suggest a algorithm for finding the Lie and high (generalized) symmetries of partial differential equations (PDEs). This algorithm transforms the problem having to solve over-determining PDEs commonly encountered and difficulty part in standard methods into one solving to algebraic equations to which one easy obtain solution. so, it reduces significantly the difficulties of the problem and raise computing efficiency. The whole procedure of the algorithm is carried out automatically by using any computer algebra system. In general, this algorithm can yields many more important symmetries for PDEs.

Symmetry Analysis of Nonlinear Incompressible Non-Hydrostatic Boussinesq Equations

The symmetries of the （2＋1）-dimensional nonlinear incompressible non-hydrostatic Boussinesq （INHB） equations, which describe the atmospheric gravity waves （GWs）, are researched in this paper. The Lie symmetries and the corresponding reductions are obtained by means of classical Lie group approach. Calculation shows the INHB equations are invariant under some Galilean transformations, scaling transformations, and space-time translations. The symmetry reduction equations and similar solutions of the INHB equations are proposed.