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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 | Lump wave and hybrid solutions of a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles显示文摘We investigate a generalized (3 + 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors. | Hui WANG Shoufu TIAN Tiantian ZHANG Yi CHEN | 2019 | Frontiers of Mathematics in China2019，14，3： | 0 |
| 3 | Inverse Scattering Transform and Soliton Solutions for the Hirota Equation with N Distinct Arbitrary Order Poles显示文摘We employ the Riemann-Hilbert(RH)method to study the Hirota equation with arbitrary order zero poles under zero boundary conditions.Through the spectral analysis,the asymptoticity,symmetry,and analysis of the Jost functions are obtained,which play a key role in constructing the RH problem.Then we successfully established the exact solution of the equation without reflection potential by solving the RH problem.Choosing some appropriate parameters of the resulting solutions,we further derive the soliton solutions with different order poles,including four cases of a fourthorder pole,two second-order poles,a third-order pole and a first-order pole,and four first-order points.Finally,the dynamical behavior of these solutions are analyzed via graphic analysis. | Xiaofan Zhang Shoufu Tian Jinjie Yang | 2022 | Advances in Applied Mathematics and Mechanics2022，14，4： | 0 |
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