Formal Series Symmetries and Truncated Symmetries of a Bilinear Schrödinger Equation

For a type of nonlinear Schrodinger equations, we get a set of formal series symmetries. For a special integrable bilinear Schrodinger equation in (2 + 1)-dimensional spacetime, some truncated symmetries which constitute an infinite dimensional Lie algebra are obtained.     

Lie Symmetries,Infinite-Dimensional Lie Algebras and Similarity Reductions of Certain (2+1)-Dimensional Nonlinear Evolution Equations

Abstract

The Lie point symmetries associated with a number of (2 + 1)-dimensional generalizations of soliton equations are investigated. These include the Niznik – Novikov – Veselov equation and the breaking soliton equation, which are symmetric and asymmetric generalizations respectively of the KDV equation, the (2+1)-dimensional generalization of the nonlinear Schrödinger equation by Fokas as well as the (2+1)-dimensional generalized sine-Gordon equation of Konopelchenko and Rogers. We show that in all these cases the Lie symmetry algebra is infinite-dimensional; however, in the case of the breaking soliton equation they do not possess a centerless Virasorotype subalgebra as in the case of other typical integrable (2+1)-dimensional evolution equations. We work out the similarity variables and special similarity reductions and investigate them.     

Symmetries,Integrability and Exact Solutions to the (2+1)-Dimensional Benney Types of Equations

This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integrable condition of the equation is given. Then, the symmetry reductions and exact solutions to the (2+1)-dimensional nonlinear wave equations are presented. Especially, the shock wave solutions of the Benney equations are investigated by the symmetry reduction and trial function method.     

Lie Symmetries,Kac-Moody-Virasoro Algebras and Integrability of Certain (2+1)-Dimensional Nonlinear Evolution Equations

Abstract

In this paper we study Lie symmetries, Kac-Moody-Virasoro algebras, similarity reductions and particular solutions of two different recently introduced (2+1)-dimensional nonlinear evolution equations, namely (i) (2+1)-dimensional breaking soliton equation and (ii) (2+1)-dimensional nonlinear Schrüdinger type equation introduced by Zakharov and studied later by Strachan. Interestingly our studies show that not all integrable higher dimensional systems admit Kac-Moody-Virasoro type sub-algebras. Particularly the two integrable systems mentioned above do not admit Virasoro type subalgebras, eventhough the other integrable higher dimensional systems do admit such algebras which we have also reviewed in the Appendix. Further, we bring out physically interesting solutions for special choices of the symmetry parameters in both the systems.     

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     

A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications

With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding(2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation(BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing(2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the(2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the(2+1)-dimensional AKNS equation(also called the Davey-Stewartson hierarchy), a kind of(2+1)-dimensional Schr¨odinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new(2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the(2+1)-dimensional integrable coupling, which is further reduced to the standard(2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known(1+1)-dimensional AKNS hierarchy, the(1+1)-dimensional nonlinear Schr¨odinger equation are all special cases of the(2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the(2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated.     

A Maple Package to Compute Lie Symmetry Groups and Symmetry Reductions of （1＋1）-Dimensional Nonlinear Systems

Armed with the computer algebra system Maple, using a direct algebraic substitution method, we obtain Lie point symmetries, Lie symmetry groups and the corresponding symmetry reductions of one component nonlinear integrable and nonintegrable equations only by clicking the ‘Enter＇ key. Abundant （1＋1）-dimensional nonlinear mathematical physical systems are analysed effectively by using a Maple package LieSYMGRP proposed by us.     

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.     

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.     

On geometric approach to Lie symmetries of differential-difference equations

Based upon Cartan's geometric formulation of differential equations, Harrison and Estabrook proposed a geometric approach for the symmetries of differential equations. In this Letter, we extend Harrison and Estabrook's approach to analyze the symmetries of differential-difference equations. The discrete exterior differential technique is applied in our approach. The Lie symmetry of (2+1)-dimensional Toda equation is investigated by means of our approach.     

（2＋2）-Dimensional Discrete Soliton Equations and Integrable Coupling System

In this paper, we extend a （2＋2）-dimensional continuous zero curvature equation to （2＋2）-dimensional discrete zero curvature equation, then a new （2＋2）-dimensional cubic Volterra lattice hierarchy is obtained. Fhrthermore, the integrable coupling systems of the （2＋2）-dimensional cubic Volterra lattice hierarchy and the generalized Toda lattice soliton equations are presented by using a Lie algebraic system sl（4）.     

Three New (2+1)-dimensional Integrable Systems and Some Related Darboux Transformations

We introduce two operator commutators by using different-degree loop algebras of the Lie algebra A₁, then under the framework of zero curvature equations we generate two (2+1)-dimensional integrable hierarchies, in-cluding the (2+1)-dimensional shallow water wave (SWW) hierarchy and the (2+1)-dimensional Kaup-Newell (KN) hierarchy. Through reduction of the (2+1)-dimensional hierarchies, we get a (2+1)-dimensional SWW equation and a (2+1)-dimensional KN equation. Furthermore, we obtain two Darboux transformations of the (2+1)-dimensional SWW equation. Similarly, the Darboux transformations of the (2+1)-dimensional KN equation could be deduced. Finally, with the help of the spatial spectral matrix of SWW hierarchy, we generate a (2+1) heat equation and a (2+1) nonlinear generalized SWW system containing inverse operators with respect to the variables x and y by using a reduction spectral problem from the self-dual Yang-Mills equations.     

A new (2+1)-dimensional supersymmetric Boussinesq equation and its Lie symmetry study

According to the conjecture based on some known facts of integrable models, a new (2+1)-dimensional supersymmetric integrable bilinear system is proposed. The model is not only the extension of the known (2+1)-dimensional negative Kadomtsev--Petviashvili equation but also the extension of the known (1+1)-dimensional supersymmetric Boussinesq equation. The infinite dimensional Kac--Moody--Virasoro symmetries and the related symmetry reductions of the model are obtained. Furthermore, the traveling wave solutions including soliton solutions are explicitly presented.     

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.     

A symmetry-preserving difference scheme for high dimensional nonlinear evolution equations

In this paper, a procedure for constructing discrete models of the high dimensional nonlinear evolution equanons is presented. In order to construct the difference model, with the aid of the potential system of the original equation and compatibility condition, the difference equations which preserve all Lie point symmetries can be obtained. As an example, invariant difference models of the （2＋1）-dimensional Burgers equation are presented.     

Symmetries and Similarity Reductions of a New （2＋1）-Dimensional Shallow Water Wave System

The symmetries of a （2＋1）-dimensional shallow water wave system, which is newly constructed through applying variation principle of analytic mechanics, are researched in this paper. The Lie symmetries and the corresponding reductions are obtained by means of classical Lie group approach. The （1＋1） dimensional displacement shallow water wave equation can be derived from the reductions when special symmetry parameters are chosen.     

Generation of Nonlinear Evolution Equations by Reductions of the Self-Dual Yang–Mills Equations

With the help of some reductions of the self-dual Yang Mills （briefly written as sdYM） equations, we introduce a Lax pair whose compatibility condition leads to a set of （2 ＋ 1）-dimensional equations. Its first reduction gives rise to a generalized variable-coefficient Burgers equation with a forced term. Furthermore, the Burgers equation again reduces to a forced Burgers equation with constant coefficients, the standard Burgers equation, the heat equation, the Fisher equation, and the Huxley equation, respectively. The second reduction generates a few new （2 ＋ 1）-dimensional nonlinear integrable systems, in particular, obtains a kind of （2 ＋ 1）-dimensional integrable couplings of a new （2 ＋ 1）- dimensional integrable nonlinear equation.     

Symmetry Reductions of a 1.5-Layer Ocean Circulation Model

The （2＋1）-dimensional nonlinear 1.5-layer ocean circulation model without external wind stress forcing is analyzed by using the classical Lie group approach. Some Lie point symmetries and their corresponding two-dimensional reduction equations are obtained.     

Group Classification and Exact Solutions of a Class of Variable Coefficient Nonlinear Wave Equations

Complete group classification of a class of variable coefficient （1＋1）-dimensional wave equations is performed. The possible additional equivalence transformations between equations from the class under consideration and the conditional equivalence groups are also investigated. These allow simplification of the results of the classification and further applications of them. The derived Lie symmetries are used to construct exact solutions of special forms of these equations via the classical Lie method. Nonclassical symmetries of the wave equations are discussed.     

Kaup-Kupershmidt方程的对称代数

利用Ｍｉｕｒａ变换和Ｃｏｌｅ－Ｈｏｐｆ变换，证明了Ｋａｕｐ－Ｋｕｐｅｒｓｈｍｉｄｔ方程的１０族时间无关对称构成一个非Ａｂｅｌ李代数。与１０族对称相应的１０族方程具有共同的强对称，１０族可积模型的时间无关对称及其构成的李代数亦被给定。