In this paper, we investigate the Lie point symmetries of Klein-Gordon equation and Schr?dinger equation by applying the geometric concept of Noether point symmetries for the below defined Lagrangian. Moreover, we organize a strong relationship among the Lie symmetries related to Klein-Gordon as well as Schr?dinger equations. Finally, we utilize the consequences of Lie point symmetries of Poisson and heat equations within Riemannian space to conclude the Lie point symmetries of Klein-Gordon equation and Schr?dinger equation within universal Riemannian space.
We first propose a way for generating Lie algebras from which we get a few kinds of reduced 6 6 Lie algebras, denoted by R6, R8 and R1,R6/2, respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra R6 whose compatibility gives rise to an integrable hierarchy with 4- potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra R6 to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again,via using the Lie algebra R62, we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.
By considering the one-dimensional model for describing long, small amplitude waves in shallow water, a generalized fifth-order evolution equation named the Olver water wave （OWW） equation is investigated by virtue of some new pseudo-potential systems. By introducing the corresponding pseudo-potential systems, the authors systematically construct some generalized symmetries that consider some new smooth functions {Xiβ}β=1,2…,N^i=1,2…,n depending on a finite number of partial derivatives of the nonlocal variables vβ and a restriction i,α,β∑（ ξi/ vβ）^2＋（ ηα/ vβ）^2≠0,ie.,i,α,β∑（ ξi/ vβ）^2≠0. Furthermore, i,a,B i,a,~ the authors investigate some structures associated with the Olver water wave （AOWW） equations including Lie algebra and Darboux transformation. The results are also extended to AOWW equations such as Lax, Sawada-Kotera, Kaup-Kupershmidt, It6 and Caudrey-Dodd-Cibbon-Sawada-Kotera equations, et al. Finally, the symmetries are ap- plied to investigate the initial value problems and Darboux transformations.
Two isomorphic groups R2 and M are firstly constructed. Then we extend them into the differential manifold R2n and n products of the group M for which four kinds of Lie algebras are obtained. By using these Lie algebras and the Tu scheme, integrable hierarchies of evolution equations along with multi-component potential functions can be generated, whose Hamiltonian structures can be worked out by the variational identity. As application illustrations, two integrable Hamiltonian hierarchies with 4 component potential functions are obtained, respectively, some new reduced equations are followed to present. Specially remark that the integrable hierarchies obtained by taking use of the approach presented in the paper are not integrable couplings. Finally, we generalize an equation obtained in the paper to introduce a general nonlinear integrable equation with variable coefficients whose bilinear form, BEcklund transformation, Lax pair and infinite conserved laws are worked out, respectively, by taking use of the Bell polynomials.
This paper investigates a real version of a (2 + 1) dimensional nonlinear Schr?dinger equation through adoption of Painlevé test by means of which the (2 + 1) dimensional nonlinear Schr?dinger equation is studied according to the Weiss et al. method and Kruskal’s simplification algorithms. According to Painlevé test, it is found that the number of arbitrary functions required for explaining the Cauchy-Kovalevskaya theorem exist. Finally, the associated B?cklund transformation and bilinear form is directly obtained from the Painlevé test.
