Group analysis of KdV equation with time dependent coefficients

We study the generalized KdV equation having time dependent variable coefficients of the damping and dispersion from the Lie group-theoretic point of view. Lie group classification with respect to the time dependent coefficients is performed. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are obtained. These subalgebras are then used to construct a number of similarity reductions and exact group-invariant solutions, including soliton solutions, for some special forms of the equations.     

Exact solutions of KdV equation with time-dependent coefficients

In this paper, we study the Korteweg-de Vries (KdV) equation having time dependent coefficients from the Lie symmetry point of view. We obtain Lie point symmetries admitted by the equation for various forms for the time-dependent coefficients. We use the symmetries to construct the group-invariant solutions for each of the cases of the arbitrary coefficients. Subsequently, the 1-soliton solution is obtained by the aid of solitary wave ansatz method. It is observed that the soliton solution will exist provided that these time-dependent coefficients are all Riemann integrable.     

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 breaking of systems of linear second-order ordinary differential equations with constant coefficients

We show that the structure of the Lie symmetry algebra of a system of n linear second-order ordinary differential equations with constant coefficients depends on at most n-1 parameters. The tools used are Jordan canonical forms and appropriate scaling transformations. We put our approach to test by presenting a simple proof of the fact that the dimension of the symmetry Lie algebra of a system of two linear second-order ordinary differential with constant coefficients is either 7, 8 or 15. Also, we establish for the first time that the dimension of the symmetry Lie algebra of a system of three linear second-order ordinary differential equations with constant coefficients is 10, 12, 13 or 24.     

Explicit solutions of the (2 + 1)-dimensional AKNS shallow water wave equation with variable coefficients

Using an improved direct reduction method, we find the equivalence transformations of (2 + 1)-dimensional AKNS shallow water wave equation with variable coefficients, and obtain the corresponding relationship between explicit solutions of AKNS equation and those of the corresponding reduced equation. In addition, we get some new explicit solutions of AKNS equation by applying Lie symmetry method.     

Symmetry analysis, conservation laws of a time fractional fifth-order Sawada-Kotera equation

In this paper, we intend to study the symmetry properties and conservation laws of a time fractional fifth-order Sawada-Kotera (S-K) equation with Riemann-Liouville derivative. Applying the well-known Lie symmetry method, we analysis the symmetry properties of the equation. Based on this, we find that the S-K equation can be reduced to a fractional ordinary differential equation with Erdelyi-Kober derivative by the similarity variable and transformation. Furthermore, we construct some conservation laws for the S-K equation using the idea in the Ibragimov theorem on conservation laws and the fractional generalization of the Noether operators.     

Self-adjointness and conservation laws of two variable coefficient nonlinear equations of Schrödinger type

In this paper, some recent concepts and results on self-adjointness and conservation laws are applied to two variable coefficient nonlinear equations of Schrödinger type: the generalized variable coefficient nonlinear Schrödinger (GVCNLS) equation and the cubic-quintic nonlinear Schrödinger (CQNLS) equation with variable coefficients. The two equations are changed to two real systems by a proper transformation. To obtain the formal Lagrangians of the two systems, we discuss their self-adjointness and find that the GVCNLS system is weak self-adjoint and the CQNLS system is quasi self-adjoint. Having performed Lie symmetry analysis for the two systems, we find five nontrivial conservation laws for the GVCNLS system and four nontrivial conservation laws for the CQNLS system by using a general theorem on conservation laws given by Ibragimov.     

Benjamin-Bona-Mahony (BBM) equation with variable coefficients: Similarity reductions and Painlevé analysis

The Lie-group formalism is applied to investigate the symmetries of the Benjamin-Bona-Mahony (BBM) equation with variable coefficients. We derive the infinitesimals and the admissible forms of the coefficients that admit the classical symmetry group. The ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.     

变系数KdV-Burgers-Kuramoto方程的群论分析——容许变换及对称群

提出变系数KdV-Burgers-Kuramoto(VCKdVBK)方程$u_t+f(x,t)uu_x+g(x,t)u_{xx}+h(x,t)u_{xxx}+k(x,t)u_{xxxx}=0$的容许变换, 它不改变方程的形式但可能改变方程的系数$f, g, h$及$k$.然后, 基于这个容许变换, 给出了VCKdVBK方程的7种不同类型的等价类. 最后, 也给出了VCKdVBK方程的Lie点对称群.     

Conservation Laws For the (1+2)-Dimensional Wave Equation in Biological Environment

The derivation of conservation laws for the wave equation on sphere, cone and flat space is considered. The partial Noether approach is applied for wave equation on curved surfaces in terms of the coefficients of the first fundamental form (FFF) and the partial Noether operator's determining equations are derived. These determining equations are then used to construct the partial Noether operators and conserved vectors for the wave equation on different surfaces. The conserved vectors for the wave equation on the sphere, cone and flat space are simplified using the Lie point symmetry generators of the equation and conserved vectors with the help of the symmetry conservation laws relation.     

On a generalized Fisher equation

A generalized non-linear Fisher equation in cylindrical coordinates with radial symmetry is studied from Lie symmetry point of view. In the classical Fisher equation the reaction diffusion term is replaced with a general function to accommodate more equations of this type. Moreover, the diffusivity is assumed to be a function of the dependent variable to account for many real situations. An attempt is made to classify the diffusivity function and exact solutions are obtained in some cases.     

一些数学物理问题中的Hamilton方程

讨论了新的一系列在数学物理方程中微分方程的Hamilton正则表示,其中包括变系数2阶对称方程的Hamilton系统,关于常系数的4阶对称方程新的非齐次Hamilton表示,MKdV方程以及KP方程的正则表示.     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

Complex Lie symmetries for scalar second-order ordinary differential equations

A scalar complex ordinary differential equation can be considered as two coupled real partial differential equations, along with the constraint of the Cauchy–Riemann equations, which constitute a system of four equations for two unknown real functions of two real variables. It is shown that the resulting system possesses those real Lie symmetries that are obtained by splitting each complex Lie symmetry of the given complex ordinary differential equation. Further, if we restrict the complex function to be of a single real variable, then the complex ordinary differential equation yields a coupled system of two ordinary differential equations and their invariance can be obtained in a non-trivial way from the invariance of the restricted complex differential equation. Also, the use of a complex Lie symmetry reduces the order of the complex ordinary differential equation (restricted complex ordinary differential equation) by one, which in turn yields a reduction in the order by one of the system of partial differential equations (system of ordinary differential equations). In this paper, for simplicity, we investigate the case of scalar second-order ordinary differential equations. As a consequence, we obtain an extension of the Lie table for second-order equations with two symmetries.     

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.     

Symmetry reductions and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power

With the aid of Maple symbolic computation and Lie group method, PKPp equation is reduced to some (1+1)-dimensional partial differential equations, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Using the separation of variables, homoclinic test technique and auxiliary equation methods, we obtain new abundant exact non-traveling solutions with arbitrary functions for the PKPp.     

Group properties and conservation laws for nonlocal shallow water wave equation

Symmetry groups, symmetry reductions, optimal system, conservation laws and invariant solutions of the shallow water wave equation with nonlocal term are studied. First, Lie symmetries based on the invariance criterion for nonlocal equations and the solution approach for nonlocal determining equations are found and then the reduced equations and optimal system are obtained. Finally, new conservation laws are generated and some similarity solutions for symmetry reduction forms are discussed.     

变质量非完整系统的形式不变性与Lie对称性

研究变质量非完整系统的形式不变性和Lie对称性．给出变质量非完整系统在无限小变换下形式不变性和Lie对称性的定义、判据及存在守恒量的定理,得到形式不变性和Lie对称性的关系,并举例说明结果的应用．     

Symmetry Analysis of Abel's Equation

A solution algorithm for Abel's equation and some generalizations based on a nontrivial Lie symmetry of a particular kind, i.e., so-called structure-preserving symmetry, is described. For the existence of such a symmetry a criterion in terms of the coefficients of the so-called rational normal form of the given equation is derived. If it is affirmative, solving Abel's equation is reduced to a well-defined integration problem. It is shown that almost all known ad hoc methods for obtaining closed form solutions are consequences of this type of symmetry. Possible extensions of this scheme to more general classes of first-order ordinary differential equations are pointed out.     

On Lie symmetry analysis,conservation laws and solitary waves to a longitudinal wave motion equation

Under investigation in this work is a longitudinal wave motion equation, which describes the solitary waves propagation with dispersion caused by transverse Poisson’s effect in a magneto-electro-elastic circular rod. The Lie symmetry method is employed to study its vector fields and optimal systems, respectively. Furthermore, the symmetry reductions and eight families of soliton wave solutions of the equation are obtained on the basis of the optimal systems, including hyperbolic-type and trigonometric-type solutions. Two of reduced equations are Painlevé-like equations. Finally, by virtue of conservation law multiplier, the complete set of local conservation laws of the equation for the arbitrary constant coefficients is well constructed with a detailed derivation.