Lie symmetries and differential Galois groups of linear equations

For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In connection with this a new algorithm for computing the Lie symmetries of a linear ordinary differential equation is presented.     

Master partial differential equations for a Type II hidden symmetry

An approach for determining a class of master partial differential equations from which Type II hidden point symmetries are inherited is presented. As an example a model nonlinear partial differential equation (PDE) reduced to a target PDE by a Lie symmetry gains a Lie point symmetry that is not inherited (hidden) from the original PDE. On the other hand this Type II hidden symmetry is inherited from one or more of the class of master PDEs. The class of master PDEs is determined by the hidden symmetry reverse method. The reverse method is extended to determine symmetries of the master PDEs that are not inherited. We indicate why such methods are necessary to determine the genesis of Type II symmetries of PDEs as opposed to those that arise in ordinary differential equations (ODEs).     

一类微分-差分方程组的内禀对称和等价群变换

研究一类微分-差分方程组的对称和等价群变换.采取内禀的无穷小算子方法,给出了方程组的内禀对称和等价群变换.为结合抽象Lie代数结构,给方程完全分类提供了理论基础.     

Symmetry analysis and conservation laws of the geodesic equations for the Reissner-Nordström de Sitter black hole with a global monopole

This paper is devoted to the comprehensive analysis of the problem of symmetries and conservation laws for the geodesic equations of the Reissner-Nordström de Sitter (RNdS) black hole with a global monopole. For this purpose, the system of geodesic equations is determined and the corresponding classical Lie point symmetry operators are obtained. An optimal system of one dimensional subalgebras is constructed and a brief discussion about the algebraic structure of the Lie algebra of symmetries is presented. Also, the Noether symmetries of the geodesic Lagrangian is calculated. Finally, by applying two methods including Noether’s theorem and direct method the conservation laws associated to the system of geodesic equations are obtained.     

Classification of Discrete Symmetries of Ordinary Differential Equations

A simple method for determining all discrete point symmetries of a given differential equation has been developed recently. The method uses constant matrices that represent inequivalent automorphisms of the Lie algebra spanned by the Lie point symmetry generators. It may be difficult to obtain these matrices if there are three or more independent generators, because the matrix elements are determined by a large system of algebraic equations. This paper contains a classification of the automorphisms that can occur in the calculation of discrete symmetries of scalar ordinary differential equations, up to equivalence under real point transformations. (The results are also applicable to many partial differential equations.) Where these automorphisms can be realized as point transformations, we list all inequivalent realizations. By using this classification as a look-up table, readers can calculate the discrete point symmetries of a given ordinary differential equation with very little effort.     

Hidden symmetries of the two-particle dirac equation with linear interaction

We investigate Lie and non-Lie symmetries of the two-particle Dirac equation with linear interaction in the one-dimensional case. The integrals of motion and hidden parasupersymmetries are found. By using algebraic method and non-Lie symmetries, we obtain the energy spectra of the considered system Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1295–1296, September, 1996.     

Symmetries for the Ablowitz–Ladik Hierarchy: Part II. Integrable Discrete Nonlinear Schrödinger Equations and Discrete AKNS Hierarchy

In the paper, we continue to consider symmetries related to the Ablowitz–Ladik hierarchy. We derive symmetries for the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. The integrable discrete nonlinear Schrödinger hierarchy is in scalar form and its two sets of symmetries are shown to form a Lie algebra. We also present discrete AKNS isospectral flows, non‐isospectral flows and their recursion operator. In continuous limit these flows go to the continuous AKNS flows and the recursion operator goes to the square of the AKNS recursion operator. These discrete AKNS flows form a Lie algebra that plays a key role in constructing symmetries and their algebraic structures for both the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. Structures of the obtained algebras are different structures from those in continuous cases which usually are centerless Kac–Moody–Virasoro type. These algebra deformations are explained through continuous limit and degree in terms of lattice spacing parameter h.     

KP系统Lax算子及主对称的换位公式

本文讨论的是ＫＰ系统Ｌａｘ算子及主对称的换位公式．通过拓广速降函数空间及对 ＫＰ方程 Ｌａｘ算子的讨论,找到了 Ｌａｘ算子的表示向量;并通过对 Ｌａｘ算子、 Ｌａｘ流、 Ｌａｘ算子表示向量之间联系的讨论,得出了计算 Ｌａｘ算子李括号的表示向量的方法,从而解决了 ＫＰ方程主对称的换位公式问题．最后本文还利用伴随算子给出了从ＫＰ方程任一主对称得到其一个对称的公式．     

二阶非完整力学系统的Lie对称性与守恒量

研究二阶非完整力学系统的Lie对称与守恒量.首先利用系统运动微分方程在无限小变换下的不变性建立Lie对称的确定方程和限制方程,得到Lie对称的结构方程和守恒量;其次研究上述问题的逆问题;最后举例说明结果的应用.     

On the geometry of the Clairin theory of conditional symmetries for higher-order systems of PDEs with applications

This work presents a geometrical formulation of the Clairin theory of conditional symmetries for higher-order systems of partial differential equations (PDEs). We devise methods for obtaining Lie algebras of conditional symmetries from known conditional symmetries, and unnecessary previous assumptions of the theory are removed. As a consequence, new insights into other types of conditional symmetries arise. We then apply the so-called PDE Lie systems to the derivation and analysis of Lie algebras of conditional symmetries. In particular, we develop a method for obtaining solutions of a higher-order system of PDEs via the solutions and geometric properties of a PDE Lie system, whose form gives a Lie algebra of conditional symmetries of the Clairin type. Our methods are illustrated with physically relevant examples such as nonlinear wave equations, the Gauss–Codazzi equations for minimal soliton surfaces, and generalised Liouville equations.     

Nonlocal symmetries,exact solutions and conservation laws of the coupled Hirota equations

Using the Lax pair, nonlocal symmetries of the coupled Hirota equations are obtained. By introducing an appropriate auxiliary dependent variable, the nonlocal symmetries are successfully localized to Lie point symmetries. With the help of Lie symmetries of the closed prolongation, exact solutions and nonlocal conservation laws of the coupled Hirota equations are studied.     

非Chetaev型非完整系统的Lie对称性与Noether对称性

研究非Ｃｈｅｔａｅｖ型非完整系统的Ｌｉｅ对称性与Ｎｏｅｔｈｅｒ对称性,具体研究了非Ｃｈｅｔａｅｖ型常 质量非完整系统和非Ｃｈｅｔａｅｖ型变质量非完整系统的Ｌｉｅ对称性与Ｎｏｅｔｈｅｒ对称性．给出Ｌｉｅ对称 性导致Ｎｏｅｔｈｅｒ对称性以及Ｎｏｅｔｈｅｒ对称性导致Ｌｉｅ对称性的条件．     

偏微分方程组的Lie群与高阶对称群的Taylor多项式逐步精化算法

本文基于生成函数的Ｔａｙｌｏｒ展开式及逐步简化步骤，提出了计算偏微分方程组的Ｌｉｅ群与高阶对称群的Ｔａｙｌｏｒ多项式算法，把标准算法中的求解超定偏微分方程组的问题转化为求解代数方程组的问题，降低了求解的难度，提高了计算效率，并且易用计算机代数系统在计算机上全过程实现，并得到重要的对称群     

The σ-isotypic decomposition and the σ-index of reversible-equivariant systems

This work is concerned with dynamical systems in presence of symmetries and reversing symmetries. We describe a construction process of subspaces that are invariant by linear Γ-reversible-equivariant mappings, where Γ is the compact Lie group of all the symmetries and reversing symmetries of such systems. These subspaces are the σ-isotypic components, first introduced by Lamb and Roberts in (1999) [10] and that correspond to the isotypic components for purely equivariant systems. In addition, by representation theory methods derived from the topological structure of the group Γ, two algebraic formulae are established for the computation of the σ-index of a closed subgroup of Γ. The results obtained here are to be applied to general reversible-equivariant systems, but are of particular interest for the more subtle of the two possible cases, namely the non-self-dual case. Some examples are presented.     

非Chetaev型变质量非完整系统的Lie对称性与Noether对称性

研究非Chetaev型变质量非完整系统的Lie对称性与Noether对称性以及其间的 关系,给出Lie对称性导致Noether对称性以及Noether对称性导致Lie对称性的条件.     

相空间中单面非Chataev型非完整系统的Lie对称性与守恒量

研究相空间中单面非Ｃｈｅｔａｅｖ型非完整系统的Ｌｉｅ对称性与守恒量．首先根据微分方程在无限小变换下的不变性建立Ｌｉｅ对称性所满足的确定方程和限制方程,给出结构方程和守恒量;其次讨论系统的Ｌｉｅ对称性逆问题;最后举一实例说明结果的应用．     

一个新的非等谱可积族和一些相关对称

利用李群$M_nC$的一个子群我们引入一个线性非等谱问题,该问题的相容性条件可导出演化方程的一个非等谱可积族,该可积族可约化成一个广义非等谱可积族.这个广义非等谱可积族可进一步约化成在物理学中具有重要应用的标准非线性薛定谔方程和KdV方程.基于此,我们讨论在广义非等谱可积族等谱条件下的一个广义AKNS族$u_t=K_m(u)$的$K$对称和$\tau$对称.此外,我们还考虑非等谱AKNS族$u_t=\tau_{N+1}^l$的$K$对称和$\tau$对称.最后,我们得到这两个可积族的对称李代数,并给出这些对称和李代数的一些应用,即生成了一些变换李群和约化方程的无穷小算子.     

Type-II hidden symmetries through weak symmetries for nonlinear partial differential equations

The Type-II hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie point symmetry. In [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612-622] Abraham-Shrauner and Govinder have analyzed the provenance of this kind of symmetries and they developed two methods for determining the source of these hidden symmetries. The Lie point symmetries of a model equation and the two-dimensional Burgers' equation and their descendants were used to identify the hidden symmetries. In this paper we analyze the connection between one of their methods and the weak symmetries of the partial differential equation in order to determine the source of these hidden symmetries. We have considered the same models presented in [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612-622], as well as the WDVV equations of associativity in two-dimensional topological field theory which reduces, in the case of three fields, to a single third order equation of Monge-Ampère type. We have also studied a second order linear partial differential equation in which the number of independent variables cannot be reduced by using Lie symmetries, however when is reduced by using nonclassical symmetries the reduced partial differential equation gains Lie symmetries.     

Similarity inner solutions for the Pulsar equation

Lie symmetries are applied to classify the source of the magnetic field for the Pulsar equation near to the surface of the neutron star. We find that there are six possible different admitted Lie algebras. We apply the corresponding Lie invariants to reduce the Pulsar equation close to the surface to an ordinary differential equation. This equation is solved either with the use of Lie symmetries or the application of the ARS algorithm for singularity analysis to write the analytic solution as a Laurent expansion. These solutions are called inner solutions.     

Fundamental solutions, transition densities and the integration of Lie symmetries

In this paper we present some new applications of Lie symmetry analysis to problems in stochastic calculus. The major focus is on using Lie symmetries of parabolic PDEs to obtain fundamental solutions and transition densities. The method we use relies upon the fact that Lie symmetries can be integrated with respect to the group parameter. We obtain new results which show that for PDEs with nontrivial Lie symmetry algebras, the Lie symmetries naturally yield Fourier and Laplace transforms of fundamental solutions, and we derive explicit formulas for such transforms in terms of the coefficients of the PDE.