Symmetry properties,conservation laws and exact solutions of time-fractional irrigation equation

In this paper, we deal with the complete algebra of Lie point symmetries for the generalized model of an irrigation system of fractional order. By means of Lie symmetry method, the vector fields has been investigated which are utilized for obtaining the conservation laws of equation. In addition, through the sub-equation method, we construct some exact solutions for the considered equation by reducing the fractional partial differential equation to a ordinary fractional differential equation.     

Symmetries,Symmetry Reductions and Exact Solutions to the Generalized Nonlinear Fractional Wave Equations

In this paper, the Lie group classification method is performed on the fractional partial differential equation (FPDE), all of the point symmetries of the FPDEs are obtained. Then, the symmetry reductions and exact solutions to the fractional equations are presented, the compatibility of the symmetry analysis for the fractional and integer-order cases is verified. Especially, we reduce the FPDEs to the fractional ordinary differential equations (FODEs) in terms of the Erdélyi-Kober (E-K) fractional operator method, and extend the power series method for investigating exact solutions to the FPDEs.     

Lie Symmetry Analysis and Conservation Laws of a Generalized Time Fractional Foam Drainage Equation

In this paper, a generalized time fractional nonlinear foam drainage equation is investigated by means of the Lie group analysis method. Based on the Riemann-Liouville derivative, the Lie point symmetries and symmetry reductions of the equation are derived, respectively. Furthermore, conservation laws with two kinds of independent variables of the equation are performed by making use of the nonlinear self-adjointness method.     

Symmetry Reduction of Two-Dimensional Damped Kuramoto-Sivashinsky Equation

In this paper, the problem of determining the largest possible set of symmetries for an important nonlinear dynamical system: the two-dimensional damped Kuramoto-Sivashinsky ((2D) DKS ) equation is studied. By applying the basic Lie symmetry method for the (2D) DKS equation, the classical Lie point symmetry operators are obtained. Also, the optimal system of one-dimensional subalgebras of the equation is constructed. The Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. The nonclassical symmetries of the (2D) DKS equation are also investigated.     

The symmetries and conservation laws of some Gordon-type equations in Milne space-time

In this letter, the Lie point symmetries of a class of Gordon-type wave equations that arise in the Milne space-time are presented and analysed. Using the Lie point symmetries, it is showed how to reduce Gordon-type wave equations using the method of invariants, and to obtain exact solutions corresponding to some boundary values. The Noether point symmetries and conservation laws are obtained for the Klein–Gordon equation in one case. Finally, the existence of higherorder variational symmetries of a projection of the Klein–Gordon equation is investigated using the multiplier approach.     

Nonclassical Lie symmetry and conservation laws of the nonlinear time-fractional Korteweg–de Vries equation

In this paper, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept. In this study, first, we employ the classical and nonclassical Lie symmetries(LS) to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries(KdV)equation, and second, we find the related exact solutions for the derived generators. Finally,according to the LS generators acquired, we construct conservation laws for related classical and nonclassical vector fields of the fractional far field Kd V equation.     

Symmetry groups and spiral wave solution of a wave propagation equation

We study a third-order nonlinear evolution equation, which can be transformed to the modified KdV equation, using the Lie symmetry method. The Lie point symmetries and the one-dimensional optimal system of the symmetry algebras are determined. Those symmetries are some types of nonlocal symmetries or hidden symmetries of the modified KdV equation. The group-invariant solutions, particularly the travelling wave and spiral wave solutions, are discussed in detail, and a type of spiral wave solution which is smooth in the origin is obtained.     

Symmetries and conservation laws of one Blaszak–Marciniak four-field lattice equation

In this paper, by using the classical Lie symmetry approach, Lie point symmetries and reductions of one Blaszak– Marciniak(BM) four-field lattice equation are obtained. Two kinds of exact solutions of a rational form and an exponential form are given. Moreover, we show that the equation has a sequence of generalized symmetries and conservation laws of polynomial form, which further confirms the integrability of the BM system.     

Symmetries and symmetry reductions of the combined KP3 and KP4 equation

To find symmetries,symmetry groups and group invariant solutions are fundamental and significant in nonlinear physics.In this paper,the finite point symmetry group of the combined KP3 and KP4(CKP34) equation is found by means of a direct method.The related point symmetries can be obtained simply by taking the infinitesimal form of the finite point symmetry group.The point symmetries of the CKP34 equation constitute an infinite dimensional KacMoody-Virasoro algebra.The point symmetry invariant so...     

Kac-Moody-Virasoro symmetry algebra of a （2＋1）-dimensional bilinear system

Based on some known facts of integrable models, this paper proposes a new （2＋1）-dimensional bilinear model equation. By virtue of the formal series symmetry approach, the new model is proved to be integrable because of the existence of the higher order symmetries. The Lie point symmetries of the model constitute an infinite dimensional Kac- Moody Virasoro symmetry algebra. Making use of the infinite Lie point symmetries, the possible symmetry reductions of the model are also studied     

Symmetries and Exact Solutions of the Breaking Soliton Equation

With the aid of the classical Lie group method and nonclassical Lie group method, we derive the classical Lie point symmetry and the nonclassical Lie point symmetry of (2+1)-dimensional breaking soliton (BS) equation. Using the symmetries, we find six classical similarity reductions and two nonclassical similarity reductions of the BS equation. Varieties of exact solutions of the BS equation are obtained by solving the reduced equations.     

Symmetry Analysis and Conservation Laws for the Hunter-Saxton Equation

In this paper,the problem of determining the most general Lie point symmetries group and conservation laws of a well known nonlinear hyperbolic PDE in mathematical physics called the Hunter-Saxton equation(HSE) is analyzed.By applying the basic Lie symmetry method for the HSE,the classical Lie point symmetry operators are obtained.Also,the algebraic structure of the Lie algebra of symmetries is discussed and an optimal system of onedimensional subalgebras of the HSE symmetry algebra which creates the preliminary classification of group invariant solutions is constructed.Particularly,the Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained.Mainly,the conservation laws of the HSE are computed via three different methods including Boyer's generalization of Noether's theorem,first homotopy method and second homotopy method.     

耦合Burgers系统的Painlevé性质分析，对称和对称约化

本文给出了耦合Burgers系统的Painlevé性质，逆强对称算子，无穷多对称和李对称约化。通过把强对称和逆强对称算子重复多次作用到耦合Burgers模型的一些平庸对称，如恒等变换，空间平移变换和标度变换上，我们得到了三族无穷多对称。这些对称构成了无穷维李代数。用其中的有限维子代数——点李代数对模型进行对称约化，得到了模型的群不变解。     

Symmetries and casimir of an extended classical long wave system

In this paper, we derive Lie point, generalized, master and time-dependent symmetries of a dispersionless equation, which is an extension of a classical long wave system. This equation also admits an infinite-dimensional Lie algebraic structure of Virasoro-type, as in the dispersive integrable systems. We discuss the construction of a sequence of negative ranking symmetries through the property of uniformity in rank. More interestingly, we obtain the conserved quantities directly from the casimir of Poisson pencil.     

微扰Kepler系统轨道微分方程的近似Lie对称性与近似不变量

把极角θ视为独立变量,得到Kepler系统的轨道微分方程.首先讨论Kepler系统轨道微分方程的Lie对称性和不变量,微扰Kepler系统轨道微分方程的精确Lie对称性和精确不变量,其次讨论微扰Kepler系统轨道微分方程的近似Lie对称性和近似不变量,并得到了微扰Kepler系统的9个一阶近似Lie对称性和6个一阶近似不变量,其中1个实为精确不变量,而其余5个分别等于微扰系数ε乘以Kepler系统相应的5个不变量。     

Exact Solutions of (2+1)-Dimensional HNLS Equation

In this paper, we use the classical Lie group symmetry method to get the Lie point symmetries of the (2+1)-dimensional hyperbolic nonlinear Schrödinger (HNLS) equation and reduce the (2+1)-dimensional HNLS equation to some (1+1)-dimensional partial differential systems. Finally, many exact travelling solutions of the (2+1)-dimensional HNLS equation are obtained by the classical Lie symmetry reduced method.     

Nonlocal Symmetry Reductions,CTE Method and Exact Solutions for Higher-Order KdV Equation

The nonlocal symmetries for the higher-order KdV equation are obtained with the truncated Painlev′e method. The nonlocal symmetries can be localized to the Lie point symmetries by introducing suitable prolonged systems.The finite symmetry transformations and similarity reductions for the prolonged systems are computed. Moreover, the consistent tanh expansion(CTE) method is applied to the higher-order KdV equation. These methods lead to some novel exact solutions of the higher-order KdV system.     

Loop Algebra of Lie symmetries for a short-wave equation

It is demonstrated that Lie point symmetries associated with a nonlinear equation for short waves in three dimensions generate an infinite-dimensional Lie algebra—a loop Algebra. Classification of the independent sets of the subalgebra is done through the adjoint action of the corresponding generators. Different forms of similarity solutions are discussed.     

New Exact Solutions and Conservation Laws to (3+1)-Dimensional Potential-YTSF Equation

Using the modified CK's direct method, we build the relationship between new solutions and old ones and find some new exact solutions to the (3+1)-dimensional potential-YTSF equation. Based on the invariant group theory, Lie point symmetry groups and Lie symmetries of the (3+1)-dimensional potential-YTSF equation are obtained. We also get conservation laws of the equation with the given Lie symmetry.     

Symmetry Reductions of a Hamilton-Jacobi-Bellman Equation Arising in Financial Mathematics

Abstract

We determine the solutions of a nonlinear Hamilton-Jacobi-Bellman equation which arises in the modelling of mean-variance hedging subject to a terminal condition. Firstly we establish those forms of the equation which admit the maximal number of Lie point symmetries and then examine each in turn. We show that the Lie method is only suitable for an equation of maximal symmetry. We indicate the applicability of the method to cases in which the parametric function depends also upon the time.