Lie symmetry analysis and reduction of a new integrable coupled KdV system

In this paper, Lie symmetry is investigated for a new integrable coupled Korteweg--de Vries (KdV) equation system. Using some symmetry subalgebra of the equation system, we obtain five types of the significant similarity reductions. Abundant solutions of the coupled KdV equation system, such as the solitary wave solution, exponential solution, rational solution and polynomial solution, etc. are obtained from the reduced equations. Especially, one type of group-invariant solution of reduced equations can be acquired by means of the Painlev\'e I transcendent function.     

NEW SYMMETRIES AND LIE ALGEBRAS OF THE GENERAL KdV EQUATION

The strong symmetry of the general KdV equation is factorized to a simple form and then the inverse strong symmetry is obtained explicitly. Acting a strong symmetry of the general KdV equation on the trivial symmetry and the known re symmetry, we obtain four new sets of symmetries of the general KdV equation. All these sets of symmetries constitute an infinite dimensional Lie algebra.     

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.     

A note on the Lie symmetries of complex partial differential equations and their split real systems

Folklore suggests that the split Lie-like operators of a complex partial differential equation are symmetries of the split system of real partial differential equations. However, this is not the case generally. We illustrate this by using the complex heat equation, wave equation with dissipation, the nonlinear Burgers equation and nonlinear KdV equations. We split the Lie symmetries of a complex partial differential equation in the real domain and obtain real Lie-like operators. Further, the complex partial differential equation is split into two coupled or uncoupled real partial differential equations which constitute a system of two equations for two real functions of two real variables. The Lie symmetries of this system are constructed by the classical Lie approach. We compare these Lie symmetries with the split Lie-like operators of the given complex partial differential equation for the examples considered. We conclude that the split Lie-like operators of complex partial differential equations are not in general symmetries of the split system of real partial differential equations. We prove a proposition that gives the criteria when the Lie-like operators are symmetries of the split system.     

Symmetry and general symmetry groups of the coupled Kadomtsev--Petviashvili equation

In this paper, the Lie symmetry algebra of the coupled Kadomtsev--Petviashvili (cKP) equation is obtained by the classical Lie group method and this algebra is shown to have a Kac--Moody--Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et al。 From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.     

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.     

Twistor correspondences for the soliton hierarchies

In this article we propose a new overview on the theory of integrable systems based on symmetry reduction of the anti-self-dual Yang—Mills equations and its twistor correspondence. First, the non-linear Schrödinger (NS) equations and the Korteweg de Vries (KdV) equations are shown to be symmetry reductions of the anti-self-dual Yang—Mills (ASDYM) equation with real forms of SL (2, ) as gauge groups.

We obtain a twistor correspondence between solutions of the NS and KdV equations and certain holomorphic vector bundles with a symmetry on the total space of the complex line bundle of Chern class two on the Riemann sphere. Remarkably, when the Chern class is increased, the correspondence extends to the NS and KdV hierarchies. If the symmetry condition is dropped we obtain a twistor correspondence for a hierarchy for the Bogomolny equations, which yields the KdV and NS hierarchies when the symmetry is imposed.

The inverse scattering transform is shown to be a coordinate realization of the twistor correspondence. Both the pure solitons and the solitonless cases are treated. The k-soliton solutions arise from the kth “Ward ansatze” in an analogous fashion to the monopole solutions.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

Similarity Reduction Analysis for a Coupled KdV System

Using the direct method for a coupled KdV system, six types of the similarity reductions are obtained. The group explanation of the results is also given. It is pointed out that, in order to find all the results by nonclassical Lie approach, two additional condition equations should be satisfied at the same time together with two original equations.     

Similarity Reduction Analysis for a Coupled KdV System

Using the direct method for a coupled KdV system, six types of the similarity reductions are obtained. The group explanation of the results is also given. It is pointed out that, in order to find all the results by nonclassical Lie approach, two additional condition equations should be satisfied at the same time together with two original equations.     

Double-wave solutions and Lie symmetry analysis to the (2 + 1)-dimensional coupled Burgers equations

This paper investigates the (2 + 1)-dimensional coupled Burgers equations (CBEs) which is an important nonlinear physical model. In this respect, by making use of the generalized unified method (GUM), a series of double-wave solutions of the (2 + 1)-dimensional coupled Burgers equations are derived. The Lie symmetry technique (LST) is also utilized for the symmetry reductions of the (2 + 1)-dimensional coupled Burgers equations and extracting a non-traveling wave solution. Through some figures, we discussed the wave structures of the double-wave solutions of the CBEs for different values of parameters in these solutions.     

Higher-Dimensional Lie Algebra and New Integrable Coupling of Discrete KdV Equation

Based on semi-direct sums of Lie subalgebra \tilde{G}, a higher-dimensional 6 x 6 matrix Lie algebra sμ(6) is constructed. A hierarchy of integrable coupling KdV equation with three potentials is proposed, which is derivedfrom a new discrete six-by-six matrix spectral problem. Moreover, the Hamiltonian forms is deduced for lattice equation in the resulting hierarchy by means of the discrete variational identity --- a generalized trace identity. A strong symmetry operator of the resulting hierarchy is given. Finally, we provethat the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian systems.     

Invariant analysis with conservation law of time fractional coupled Ablowitz–Kaup–Newell–Segur equations in water waves

ABSTRACT

In this paper, the symmetry analysis and conservation laws of time fractional coupled Ablowitz–Kaup–Newell–Segur (AKNS) equations have been presented. Initially, the Lie symmetry method has been used for similarity reduction of time fractional coupled AKNS equations. Here, the Erdélyi–Kober differential operators have been used for symmetry reduction. Also, the new conservation vectors for time fractional coupled AKNS equations have been derived by using the new conservation theorem with formal Lagrangian.     

耦合KdV方程的约化及求解

利用Clarkson和Kruskal(CK)直接方法,对耦合KdV方程进行相似约化,同时从李群出发对该约化方程作了群论解释.进一步地,借助Ablowitz-Ramani-Segur(ARS)算法对耦合方程展开Painlevé测试,找到了3个Painlevé可积模型.最后通过形变映射法,求得耦合KdV方程的准确解析解. 关键词： 耦合KdV方程 CK直接法 Painlevé分析法 准确解析解     

修正cKdV方程组的孤立波结构及其稳定性

利用函数展开法求解修正耦合KdV(Coupled KdV,cKdV)方程组,得到几类孤立波解,包括扭结型-钟型、双扭结型、双钟型以及双扭结-双钟型结构的单孤子解.在不同的极限情况下,这些解分别退化为修正cKdV方程的扭结状或钟状孤波解.对孤立波的稳定性进行了数值研究,结果表明:修正cKdV方程既存在稳定的孤立波解,也存在不稳定的孤立波解.     

On the biHamiltonian structure of the supersymmetric KdV hierarchies. A Lie superalgebraic approach

We give a Lie superalgebraic interpretation of the biHamiltonian structure of known supersymmetric KdV equations. We show that the loop algebra of a Lie superalgebra carries a natural Poisson pencil, and we subsequently deduce the biHamiltonian structure of the supersymmetric KdV hierarchies by applying to loop superalgebras an appropriate reduction technique. This construction can be regarded as a superextension of the Drinfeld-Sokolov method for building a KdV-type hierarchy from a simple Lie algebra.     

Comparative study of travelling-wave and numerical solutions for the coupled short pulse （CSP） equation

The Lie symmetry analysis is performed for the coupled short plus (CSP) equation. We derive the infinitesimals that admit the classical symmetry group. Five types arise depending on the nature of the Lie symmetry generator. In all types, we find reductions in terms of system of ordinary differential equations, and exact solutions of the CSP equation are derived, which are compared with numerical solutions using the classical fourth-order Runge-Kutta scheme.     

Eigenvectors of the recursion operator and a symmetry structure for the coupled KdV hierarchy

Abstract

It is shown that eigenvectors of the recursion operator L with the eigenvalue λ _i and the inverse of the recursion operator L _i ≡ L?λ_i for the coupled KdV hierarchy (CKdVH) can be obtained in terms of squared eigenfunctions of the associated linear problem. The symmetry structure and corresponding infinite dimensional Lie algebras of CKdVH are also given. Using both the local and nonlocal symmetries of CKdVH, one can obtain some exact group invariant solutions and various new infinite-dimensional and finite-dimensional integrable models.     

New periodic solutions to a generalized Hirota－Satsuma coupled KdV system

Using expansions in terms of the Jacobi elliptic cosine function and third Jacobi elliptic function, some new periodic solutions to the generalized Hirota－Satsuma coupled KdV system are obtained with the help of the algorithm Mathematica. These periodic solutions are also reduced to the bell-shaped solitary wave solutions and kink-shape solitary solutions. As special cases, we obtain new periodic solution, bell-shaped and kink-shaped solitary solutions to the well-known Hirota－Satsuma equations.     

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.