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| 1 | On the Lie Algebras, Generalized Symmetries and Darboux Transformations of the Fifth-Order Evolution Equations in Shallow Water显示文摘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}i=1,2,···,nβ =1,2,···,N depending on a finite number of partial derivatives of the nonlocal variables vβand a restriction ∑iα,β(? ξi/?vβ)2+(?ηα/?vβ)2≠0, i.e.,∑i,α,β(?Gα/?vβ)2≠0. Furthermore,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, It and Caudrey-Dodd-Gibbon-Sawada-Kotera equations, et al. Finally, the symmetries are applied to investigate the initial value problems and Darboux transformations. | Shoufu TIAN Yufeng ZHANG Binlu FENG Hongqing ZHANG | 2015 | Chinese Annals of Mathematics,Series B2015，36，4： | 2 |
| 2 | The uniform boundedness of solution {or nonlinear di{ferential equations显示文摘 | Feng Binlu Yu Yuanhong | 1997 | Modern Mathemat- ics and Mechanics-VII1997，11，： | 1 |
| 3 | A New Non-Isospectral Integrable Hierarchy and Some Associated Symmetries显示文摘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. | Yufeng ZHANG Yiyi LIU Jian-gen LIU Binlu FENG | 2023 | Journal of Mathematical Research with Applications2023，43，5： | 0 |
| 4 | Expanding Integrable Models and Their Some Reductions as Well as Darboux Transformations显示文摘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. | Binlu FENG Y.C.HON | 2016 | Journal of Mathematical Research with Applications2016，36，3： | 0 |
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