势Korteweg-de Vries方程的暗方程分类
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摘要
通过要求势Korteweg-de Vries (KdV)方程和线性扩展方程具有高阶对称,获得了势KdV方程的完整的暗方程标量分类:8类具有若干自由参数的相互独立的暗方程.然后通过一个直接假设方法,获得了势KdV方程的8类暗方程的递归算子.
In this paper, we obtain a complete scalar classification for dark potential Korteweg-de Vries(pKdV) equation system by requiring the existence of higher order differential polynomial symmetries. There exist eight independent classes of dark pKdV equation with some free parameters. The recursion operators of dark pKdV equations are then constructed using a direct assumption method.
In this paper, we obtain a complete scalar classification for dark potential Korteweg-de Vries(pKdV) equation system by requiring the existence of higher order differential polynomial symmetries. There exist eight independent classes of dark pKdV equation with some free parameters. The recursion operators of dark pKdV equations are then constructed using a direct assumption method.
引文
[1]蔡荣根,周宇峰.暗物质与暗能量研究新进展[J].中国基础科学,2010,12(3):3-9.
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[11]Huang L L,Chen Y.Nonlocal symmetry and exact solutions of the(2+1)-dimensional modified BogoyavlenskiiSchiff equation[J].Chinese Physics B,2016,25(6):63-70.
[12]Olver P J.Applications of Lie Groups to Differential Equations[M].New York:Springer,1998:75-238.
[13]Li B,Li Y Q,Chen Y.Finite symmetry transformation groups and some exact solutions to(2+1)-dimensional cubic nonlinear Schr?dinger equation[J].Communications in Theoretical Physics,2009,51:773-776.
[14]Tian S F,Zhou S W,Jiang W Y,et al.Analytic solutions,Darboux transformation operators and super symmetry for a generalized one-dimensional time-dependent Schr?dinger equation[J].Applied Mathematics and Computation,2012,218:7308-7321.
[15]Hone A N W,Kamp K P H,Quispel G R W,et al.Integrability of reductions of the discrete Korteweg-de Vries and potential Korteweg-de Vries equations[EB/OL].[2018-03-16].https://www.researchgate.net/publication/258728367.
[16]Wang G W,Xu T Z,Ebadi G,et al.Singular solitons,shock waves,and other solutions to potential Kd Vequation[J].Nonlinear Dynamics,2014,76:1059-1068.
[17]Fokas A S.Symmetries and integrability[J].Studies in Applied Mathematics,1987,77(3):253-299.
[18]Fokas A S,Santini P M.Recursion operators and bi-Hamiltonian structures in multidimensions II[J].Communications in Mathematical Physics,1988,116:449-474.
[19]Tian K L,Zhu X M,He J S.On recursion operator of the q-KP hierarchy[J].Communications in Theoretical Physics,2016,63(3):325-328.
[20]Symes W.Relations among generalized Korteweg-de Vries systems[J].Journal of Mathematical Physics,1979,20(4):721-725.
[2]Perlmutter S,Turner M S,White M.Constraining dark energy with SNe Ia and large-scale structure[J].Physical Review Letters,1999,83:670-673.
[3]Silk J.Challenges in cosmology from the big bang to dark energy,dark matter and galaxy formation[J].Nuclear Physics,1988,310:669-692.
[4]Kupershmidt B A.Dark equations[J].Journal of Nonlinear Mathematical Physics,2001,8(3):363-445.
[5]Wazwaz A M.Analytic study on the one and two spatial dimensional potential KdV equations[J].Chaos,2008,36:175-181.
[6]Tran D T,Kamp P H V D,Quispel G R W.Involutivity of integrals of sine-Gordon,modified Kd V and potential Kd V maps[J].Journal of Physics A:Mathematical and Theoretical,2010,44(29):1651-1659.
[7]Fordy A P,Gibbons J.Factorization of operators I:Miura transformations[J].Journal of Mathematical Physics,1980,21(10):2508-2510.
[8]Xiong N,Lou S Y,Li B,et al.Classification of dark modified KdV equation[J].Communications in Theoretical Physics,2017,68:13-20.
[9]陈玉福,张鸿庆.微分方程递归算子的一种推广[J].应用数学和力学,1999,20(11):1143-1148.
[10]Liu H,Li J,Zhang Q.Lie symmetry analysis and exact explicit solutions for general Burgers equation[J].Journal of Computational and Applied Mathematics,2009,228:1-9.
[11]Huang L L,Chen Y.Nonlocal symmetry and exact solutions of the(2+1)-dimensional modified BogoyavlenskiiSchiff equation[J].Chinese Physics B,2016,25(6):63-70.
[12]Olver P J.Applications of Lie Groups to Differential Equations[M].New York:Springer,1998:75-238.
[13]Li B,Li Y Q,Chen Y.Finite symmetry transformation groups and some exact solutions to(2+1)-dimensional cubic nonlinear Schr?dinger equation[J].Communications in Theoretical Physics,2009,51:773-776.
[14]Tian S F,Zhou S W,Jiang W Y,et al.Analytic solutions,Darboux transformation operators and super symmetry for a generalized one-dimensional time-dependent Schr?dinger equation[J].Applied Mathematics and Computation,2012,218:7308-7321.
[15]Hone A N W,Kamp K P H,Quispel G R W,et al.Integrability of reductions of the discrete Korteweg-de Vries and potential Korteweg-de Vries equations[EB/OL].[2018-03-16].https://www.researchgate.net/publication/258728367.
[16]Wang G W,Xu T Z,Ebadi G,et al.Singular solitons,shock waves,and other solutions to potential Kd Vequation[J].Nonlinear Dynamics,2014,76:1059-1068.
[17]Fokas A S.Symmetries and integrability[J].Studies in Applied Mathematics,1987,77(3):253-299.
[18]Fokas A S,Santini P M.Recursion operators and bi-Hamiltonian structures in multidimensions II[J].Communications in Mathematical Physics,1988,116:449-474.
[19]Tian K L,Zhu X M,He J S.On recursion operator of the q-KP hierarchy[J].Communications in Theoretical Physics,2016,63(3):325-328.
[20]Symes W.Relations among generalized Korteweg-de Vries systems[J].Journal of Mathematical Physics,1979,20(4):721-725.