Optimal system and approximate solutions of nonlinear dissipative media

In this paper, the problem of approximate symmetries of a class of nonlinear wave equations with a small nonlinear dissipation has been investigated. In order to compute the first-order approximate symmetry, we have applied the method that was proposed by Valenti basically based on the expansion of the dependent variables in perturbation series but removing the drawback of the impossibility to work in hierarchy in calculating symmetries. The algebraic structure of the approximate symmetries is discussed, an optimal system of one-dimensional subalgebras is defined and constructed, and, finally, some invariant solutions corresponding to the resulted symmetries are obtained.     

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.     

一类变系数非线性Schrǒdinger方程的对称群分类和容许变换(英文)

从对称群和容许变换的角度讨论一类变系数非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程,给出所考察方程的非平凡点对称群     

一类组合方程的势对称分类

首先给出一类含有任意函数的变系数波动方程uxx=H(x)utt的古典对称及其势对称的完全分类,然后借助于这个波动方程的对称分类,系统讨论了含有两个任意函数的一类组合方程的势对称分类,所得结果确实扩充了原方程的对称.在计算过程中,采用微分形式的吴方法,微分特征列的程序包起到了重要作用.     

解扰动非线性发展方程的Lie-B(a)cklund方法

A new method based on Lie-B(a)cklund symmetry method to solve the perturbed nonlinear evolution equations is presented. New approximate solutions of perturbed nonlinear evolution equations stemming from the exact solutions of unperturbed equations are obtained.This method is a generalization of Burde's Lie point symmetry technique.     

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.     

KdV-Burgers方程的对称与孤子解

考虑KdV-Burgers方程的一些简单对称及其构成的李代数,并利用对称约化方法将KdV-Burgers方程化为常微分方程,从而得到该方程的群不变解.此外,利用多项式展开式的方法去获得KdV-Burgers方程的新的孤子波解.     

A NEW APPROACH TO SOLVE PERTURBED NONLINEAR EVOLUTION EQUATIONS THROUGH LIE-BACKLUND SYMMETRY METHOD

A new method based on Lie-Backlund symmetry method to solve the perturbed nonlinear evolution equations is presented. New approximate solutions of perturbed nonlinear evolution equations stemming from the exact solutions of unperturbed equations are obtained. This method is a generalization of Burde＇s Lie point symmetry technique.     

准坐标下非完整力学系统的Lie对称性和守恒量

研究准坐标下非完整系统的Ｌｉｅ对称性,首先,对准坐标下非完整力学系统定义无限小变换生成元,由微分方程在无限小变换下的不变性,建立Ｌｉｅ对称性的确定方程,得到结构方程并求出守恒量;其次,研究上述问题的逆问题;根据已知积分求相应的Ｌｉｅ对称性,举例说明结果的应用。     

Nonlocal symmetries and exact solutions of a variable coefficient AKNS system

In this paper, nonlocal symmetries of variable coefficient Ablowitz-Kaup-Newell-Segur(AKNS) system are studied for the first time. In order to construct some new analytic solutions, a new variable is introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of closed system, we give out two types of symmetry reductions and some analytic solutions. For some interesting solutions, such as interaction solutions among solitons and other complicated waves, we give corresponding images to describe their dynamic behavior.     

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations is performed. First, complete infinite-order approximate symmetry classification of the equation is obtained by means of the method originated by Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is derived and used to construct general formulas of approximate symmetry reductions and similarity solutions. Second, we study approximate homotopy symmetry of the equation and construct connections between the two symmetry methods for the first-order and higher-order cases, respectively. The series solutions derived by the two methods are compared.     

Optimal portfolio for a defined-contribution pension plan under a constant elasticity of variance model with exponential utility

Based on the Lie symmetry method, we derive the explicit optimal invest strategy for an investor who seeks to maximize the expected exponential (CARA) utility of the terminal wealth in a defined-contribution pension plan under a constant elasticity of variance model. We examine the point symmetries of the Hamilton-Jacobi-Bellman (HJB) equation associated with the portfolio optimization problem. The symmetries compatible with the terminal condition enable us to transform the (2+ 1)-dimensional HJB equation into a (1+ 1)-dimensional nonlinear equation which is linearized by its infinite-parameter Lie group of point transformations. Finally, the ansatz technique based on variables separation is applied to solve the linear equation and the optimal strategy is obtained. The algorithmic procedure of the Lie symmetry analysis method adopted here is quite general compared with conjectures used in the literature.     

Investigation of new solutions for an extended (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schif equation

We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schif (eCBS) equation using the commutator table which results in a system of nonlinear ordinary differential equations (ODEs) which can be manually solved. Through two stages of Lie symmetry reductions, the eCBS equation is reduced to non-solvable nonlinear ODEs using different combinations of optimal Lie vectors. Using the integration method and the Riccati and Bernoulli equation methods, we investigate new analytical solutions to those ODEs. Back substituting to the original variables generates new solutions to the eCBS equation. These results are simulated through three- and two-dimensional plots.     

一类非线性波动方程的势对称分类

先给出了含有一个任意函数的线性波动方程的古典和势对称的完全分类.然后,在此基础上给出了含有两个任意函数的一类非线性波动方程的两种情形势对称分类,得到了该方程的新势对称.在处理对称群分类问题的难点-求解确定方程组时我们提出了微分形式吴方法算法,克服了以往难于处理的困难.在整个计算过程中反复使用了吴方法,吴方法起到了关键的作用.     

Second-order approximate symmetry classification and optimal system of a class of perturbed nonlinear wave equations

A complete second-order approximate symmetry classification of a class of perturbed nonlinear wave equations with a arbitrary function is performed by means of the method originated from Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is given. Based on the Lie symmetry reduction method, large classes of approximate reductions and invariant solutions of the equations are constructed.     

Lie group symmetries and Riemann function of Klein–Gordon–Fock equation with central symmetry

In the present paper Lie symmetry group method is applied to find new exact invariant solutions for Klein–Gordon–Fock equation with central symmetry. The found invariant solutions are important for testing finite-difference computational schemes of various boundary value problems of Klein–Gordon–Fock equation with central symmetry. The classical admitted symmetries of the equation are found. The infinitesimal symmetries of the equation are used to find the Riemann function constructively.     

Lie symmetry analysis,optimal system,and generalized group invariant solutions of the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, Lie group theoretic method is used to carry out the similarity reduction and solitary wave solutions of (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation. The equation describes the propagation of nonlinear dispersive waves in inhomogeneous media. Under the invariance property of Lie groups, the infinitesimal generators for the governing equation have been obtained. Thereafter, commutator table, adjoint table, invariant functions, and one-dimensional optimal system of subalgebras are derived by using Lie point symmetries. The symmetry reductions and some group invariant solutions of the DJKM equation are obtained based on some subalgebras. The obtained solutions are new and more general than the rest while known results reported in the literature. In order to show the physical affirmation of the results, the obtained solutions are supplemented through numerical simulation. Thus, the solitary wave, doubly soliton, multi soliton, and dark soliton profiles of the solutions are traced to make this research physically meaningful.     

Lie symmetries of nonlinear boundary value problems

Nonlinear boundary value problems (BVPs) by means of the classical Lie symmetry method are studied. A new definition of Lie invariance for BVPs is proposed by the generalization of existing those on much wider class of BVPs. A class of two-dimensional nonlinear boundary value problems, modeling the process of melting and evaporation of metals, is studied in details. Using the definition proposed, all possible Lie symmetries and the relevant reductions (with physical meaning) to BVPs for ordinary differential equations are constructed. An example how to construct exact solution of the problem with correctly-specified coefficients is presented and compared with the results of numerical simulations published earlier.     

EXACT SOLUTIONS TO NONLINEAR WAVE EQUATION

In this paper,we use an invariant set to construct exact solutions to a nonlinear wave equation with a variable wave speed. Moreover,we obtain conditions under which the equation admits a nonclassical symmetry. Several different nonclassical symmetries for equations with different diffusion terms are presented.     

Residual symmetry,nth Bäcklund transformation,and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV equation

This paper is concerned with the nth Bäcklund transformation (BT) related to multiple residual symmetries and soliton-cnoidal wave interaction solution for the combined modified KdV–negative-order modified KdV (mKdV-nmKdV) equation. The residual symmetry derived from the truncated Painlevé expansion can be extended to the multiple residual symmetries, which can be localized to Lie point symmetries by prolonging the combined mKdV-nmKdV equation to a larger system. The corresponding finite symmetry transformation, ie, nth BT, is presented in determinant form. As a result, new multiple singular soliton solutions can be obtained from known ones. We prove that the combined mKdV-nmKdV equation is integrable, possessing the second-order Lax pair and consistent Riccati expansion (CRE) property. Furthermore, we derive the exact soliton and soliton-cnoidal wave interaction solutions by applying the nonauto-BT obtained from the CRE method.