Integrating Scalar Ordinary Differential Equations with Symmetry Revisited

The process of integrating an nth-order scalar ordinary differential equation with symmetry is revisited in terms of Pfaffian systems. This formulation immediately provides a completely algebraic method to determine the initial conditions and the corresponding solutions which are invariant under a one parameter subgroup of a symmetry group. To determine the noninvariant solutions the problem splits into three cases. If the dimension of the symmetry groups is less than the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions can be found by integrating a quotient Pfaffian system on a quotient space, and integrating an equation of fundamental Lie type associated with the symmetry group. If the dimension of the symmetry group is equal to the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions are obtained either by solving an equation of fundamental Lie type associated with the symmetry group, or the solutions are invariant under a one-parameter subgroup. If the dimension of the symmetry group is greater than the order of the equation, then there exists an open dense set of initial conditions where the solutions can either be determined by solving an equation of fundamental Lie type for a solvable Lie group, or are invariant. In each case the initial conditions, the quotient Pfaffian system, and the equation of Lie type are all determined algebraically. Examples of scalar ordinary differential equations and a Pfaffian system are given.     

汽车车体振动系统的对称性与守恒量研究

用Lie群方法研究汽车车体振动系统的对称性,寻找其存在的守恒量.以汽车车体做上下垂直振动和绕其质心的前后俯仰振动,采用Lagrange函数的方法,构建汽车车体振动系统.以此系统为对象,引入Lie群方法,给出该振动系统的Noether对称性理论与Lie对称性理论;由此推导该汽车系统存在的Noether对称性与Lie对称性,并得到系统相应的的守恒量.该方法对车体振动问题提出了新的对称性解法,同时扩大了Lie群方法的应用范围.     

Lie point symmetries and some group invariant solutions of the quasilinear equation involving the infinity Laplacian

Using the classical Lie method we obtain the full Lie point symmetry group of the Aronsson equation in two independent variables. Some group invariant solutions of this equation are found and a conjecture on the Lie point symmetry group of the Aronsson equation in Rⁿ is presented.     

Lie symmetries and conservation laws of the Hirota–Ramani equation

In this paper, Lie symmetry method is performed for the Hirota–Ramani (H–R) equation. We will find the symmetry group and optimal systems of Lie subalgebras. Furthermore, preliminary classification of its group invariant solutions, symmetry reduction and nonclassical symmetries are investigated. Finally conservation laws of the H–R equation are presented.     

Classification of Similarity Solutions for Inviscid Burgers’ Equation

Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the inviscid Burgers’ equation. Looking at the adjoint representation of the obtained symmetry group on its Lie algebra, we find the preliminary classification of its group-invariant solutions. The latter provides new exact solutions for the inviscid Burgers’ equation.     

Classification of four-dimensional real Lie bialgebras of symplectic type and their Poisson–Lie groups

We classify all four-dimensional real Lie bialgebras of symplectic type and obtain the classical r-matrices for these Lie bialgebras and Poisson structures on all the associated four-dimensional Poisson–Lie groups. We obtain some new integrable models where a Poisson–Lie group plays the role of the phase space and its dual Lie group plays the role of the symmetry group of the system.     

Computation of local symmetries of second-order ordinary differential equations by the Cartan equivalence method

The Cartan equivalence method is used to find out if a given equation has a nontrivial Lie group of point symmetries. In particular, we compute invariants that permit one to recognize equations with a three-dimensional symmetry group. An effective method to transform the Lie system (the system of partial differential equations to be satisfied by the infinitesimal point symmetries) into a formally integrable form is given. For equations with a three-dimensional symmetry group, the formally integrable form of the Lie system is found explicitly. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 75–91, July, 1996.     

Decompositions of analytic 1-manifolds

In an author’s previous work, analytic 1-submanifolds had been classified w.r.t. their symmetry under a given regular and separately analytic Lie group action on an analytic manifold. It was shown that such an analytic 1-submanifold is either free or (via the exponential map) analytically diffeomorphic to the unit circle or an interval. In this paper, we show that each free analytic 1-submanifold is discretely generated by the symmetry group, i.e., naturally decomposes into countably many symmetry free segments that are mutually and uniquely related by the Lie group action. This is shown under the same assumptions that were used in the author’s previous work to prove analogous decomposition results for analytic immersive curves. Together with the results obtained there, this completely classifies 1-dimensional analytic objects (analytic curves and analytic 1-submanifolds) w.r.t. their symmetry under a given regular and separately analytic Lie group action.     

Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients

This paper studies the modified Korteweg–de Vries equation with time variable coefficients of the damping and dispersion using Lie symmetry methods. We carry out Lie group classification with respect to the time-dependent coefficients. Lie point symmetries admitted by the mKdV equation for various forms for the time variable coefficients are obtained. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are determined. These are then used to determine exact group-invariant solutions, including soliton solutions, and symmetry reductions for some special forms of the equations.     

Group Classifications, Symmetry Reductions and Exact Solutions to the Nonlinear Elastic Rod Equations

In this paper, the Lie symmetry analysis are performed on the three nonlinear elastic rod (NER) equations. The complete group classifications of the generalized nonlinear elastic rod equations are obtained. The symmetry reductions and exact solutions to the equations are presented. Furthermore, by means of dynamical system and power series methods, the exact explicit solutions to the equations are investigated. It is shown that the combination of Lie symmetry analysis and dynamical system method is a feasible approach to deal with symmetry reductions and exact solutions to nonlinear PDEs.     

一类推广的Virasoro-like李代数

构造了一类无限维李代数,它是Virasoro-like李代数的推广.研究了这类李代数的两类自同构,这两类自同构均关于映射的合成构成自同构群,一类同构于对称群S3,另一类同构于Klein交换群.得到了这类李代数一些特殊的自同态、中心.证明了这类李代数不是半单李代数.     

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.     

On the equivalence of Lie symmetries and group representations

We consider families of linear, parabolic PDEs in n dimensions which possess Lie symmetry groups of dimension at least four. We identify the Lie symmetry groups of these equations with the (2n+1)-dimensional Heisenberg group and SL(2,R). We then show that for PDEs of this type, the Lie symmetries may be regarded as global projective representations of the symmetry group. We construct explicit intertwining operators between the symmetries and certain classical projective representations of the symmetry groups. Banach algebras of symmetries are introduced.     

一类非线性发展方程的对称分析及级数解

基于李对称理论分析了广义Burgers方程的推广方程,获得其有限维李对称.进一步,研究向量场的伴随表示构造优化系统.最终基于对称约化,获得了方程的约化系统及包含级数解在内的群不变解.     

Conservation Laws and Exact Solutions to the Modified Hyperbolic Geometric Flow

In this paper, we investigate Lie symmetry group, optimal system, exact solutions and conservation laws of modified hyperbolic geometric flow via Lie symmetry method. Then, conservation laws of modified hyperbolic geometric flow are obtained by applying Ibragimov method.     

A Lie symmetry analysis and explicit solutions of the two‐dimensional ∞‐Polylaplacian

In this work, we consider the Lie point symmetry analysis of a strongly nonlinear partial differential equation of third order, the ∞‐Polylaplacian, in two spatial dimensions. This equation is a higher order generalization of the ∞‐Laplacian, also known as Aronsson's equation, and arises as the analog of the Euler–Lagrange equations of a second‐order variational principle in L^∞. We obtain its full symmetry group, one‐dimensional Lie subalgebras and the corresponding symmetry reductions to ordinary differential equations. Finally, we use the Lie symmetries to construct new invariant ∞‐Polyharmonic functions.     

Invariance properties of a general bond-pricing equation

We perform the group classification of a bond-pricing partial differential equation of mathematical finance to discover the combinations of arbitrary parameters that allow the partial differential equation to admit a nontrivial symmetry Lie algebra. As a result of the group classification we propose “natural” values for the arbitrary parameters in the partial differential equation, some of which validate the choices of parameters in such classical models as that of Vasicek and Cox-Ingersoll-Ross. For each set of these natural parameter values we compute the admitted Lie point symmetries, identify the corresponding symmetry Lie algebra and solve the partial differential equation.     

The similarity forms and invariant solutions of two-layer shallow-water equations

In this study symmetry group properties and general similarity forms of the two-layer shallow-water equations are discussed by Lie group theory. We represent that Lie group theory can be used as an effective approach for investigation of the self-similar solutions for the shallow-water equations with variable inflow as the generalization of dimensional analysis that was used so far for a regular approach in the literature. We also represent that the results obtained by dimensional analysis are just a special case of the results obtained by Lie group theory and it is possible to obtain the new similarity forms and the new variable inflow functions for the study of gravity currents in two-layer flow under shallow-water approximations based on Lie group theory. The symmetry groups of the system of nonlinear partial differential equations are found and the corresponding similarity and reduced forms are obtained. Some similarity solutions of the reduced equations are investigated. It is shown that reduced equations and similarity forms of the system depend on the group parameters. We show that an analytic similarity solution for the system of equations can be found for some special values of them. For other values of the group parameters, the similarity solutions of the two-layer shallow-water equations representing the gravity currents with a variable inflow are found by the numeric integration.     

Symmetry analysis of the option pricing model with dividend yield from financial markets

In this work, the option pricing Black–Scholes model with dividend yield is investigated via Lie symmetry analysis. As a result, the complete Lie symmetry group and infinitesimal generators of the one-dimensional Black–Scholes equation are derived. On the basis of these infinitesimal generators, the similarity variables and newly explicit solutions of the Black–Scholes equation are obtained by solving the corresponding characteristic equations. Finally, figures for an explicit solution with different dividend yields are presented to demonstrate the novel properties.