The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long...The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long derivatives V_(x),V_(y),V_(t).Two different constraints on the kernel function K are introduced under the reality of the solution u of the 2DG equation.Then,two classes of exact solutions with constant asymptotic values at infinity u|x^(2)+y^(2)→∞→0 are constructed by means of the∂¯-dressing method for the casesσ=1 andσ=i.The rational and multiple pole solutions of the 2DG equation are obtained with the kernel functions of zero-order and higher-order Dirac delta functions,respectively.展开更多
By applying a subgroup of the Lie group Mn C we introduce a linear nonisospectral problem whose compatibility condition gives rise to a nonisospectral integrable hierarchy of evolution equations, which reduces to a ge...By applying a subgroup of the Lie group Mn C we introduce a linear nonisospectral problem whose compatibility condition gives rise to a nonisospectral integrable hierarchy of evolution equations, which reduces to a generalized nonisospectral integrable hierarchy(GNIH).The GNIH further reduces to the standard nonlinear Schr¨odinger equation and the Kd V equation which have important applications in physics science. Based on this, we discuss the K symmetries and the τ symmetries of the generalized AKNS hierarchy u_t = K_m(u) with isospectral condition coming from the GNIH. Furthermore, we also consider the K symmetries and the τ symmetries of the nonisospectral AKNS hierarchy ut = τ_N~l +1. Finally, we obtain the symmetry Lie algebras for the both integrable hierarchies, and present some applications for the symmetries and the Lie algebras, which means that some Lie groups of transformations and the infinitesimal operators of reduced equations are generated.展开更多
In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, ...In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra？T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra？H to obtain an integrable hierarchy with variable coefficients.展开更多
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 ...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.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.1237125611971475)。
文摘The(2+1)-dimensional integrable generalization of the Gardner(2DG)equation is solved via the inverse scattering transform method in this paper.A kind of general solution of the equation is obtained by introducing long derivatives V_(x),V_(y),V_(t).Two different constraints on the kernel function K are introduced under the reality of the solution u of the 2DG equation.Then,two classes of exact solutions with constant asymptotic values at infinity u|x^(2)+y^(2)→∞→0 are constructed by means of the∂¯-dressing method for the casesσ=1 andσ=i.The rational and multiple pole solutions of the 2DG equation are obtained with the kernel functions of zero-order and higher-order Dirac delta functions,respectively.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 1197147512371256)。
文摘By applying a subgroup of the Lie group Mn C we introduce a linear nonisospectral problem whose compatibility condition gives rise to a nonisospectral integrable hierarchy of evolution equations, which reduces to a generalized nonisospectral integrable hierarchy(GNIH).The GNIH further reduces to the standard nonlinear Schr¨odinger equation and the Kd V equation which have important applications in physics science. Based on this, we discuss the K symmetries and the τ symmetries of the generalized AKNS hierarchy u_t = K_m(u) with isospectral condition coming from the GNIH. Furthermore, we also consider the K symmetries and the τ symmetries of the nonisospectral AKNS hierarchy ut = τ_N~l +1. Finally, we obtain the symmetry Lie algebras for the both integrable hierarchies, and present some applications for the symmetries and the Lie algebras, which means that some Lie groups of transformations and the infinitesimal operators of reduced equations are generated.
基金Supported by the Research Grant Council of the Hong Kong Special Administrative Region(Grant No.City U101211)the National Natural Science Foundation of China(Grant No.11371361)the Natural Science Foundation of Shandong Province(Grant No.ZR2013AL016)
文摘In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra？T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra？H to obtain an integrable hierarchy with variable coefficients.
基金supported by the National Natural Science Foundation of China(Nos.11301527,11371361)the Fundamental Research Funds for the Central Universities(No.2013QNA41)the Construction Project of the Key Discipline of Universities in Jiangsu Province During the 12th FiveYear Plans(No.SX2013008)
文摘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.
