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| On the Lie Algebras,Generalized Symmetries and Darboux Transformations of the Fifth-Order Evolution Equations in Shallow Water |
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Citation： |
Shoufu TIAN,Yufeng ZHANG,Binlu FENG,Hongqing ZHANG.On the Lie Algebras,Generalized Symmetries and Darboux Transformations of the Fifth-Order Evolution Equations in Shallow Water[J].Chinese Annals of Mathematics B,2015,36(4):543~560 |
| Page view： 1257
Net amount： 1137 |
Authors： |
Shoufu TIAN; Yufeng ZHANG; Binlu FENG; Hongqing ZHANG; |
Foundation： |
supported by the National Natural Science Foundation of China (Nos. 11301527,
11371361), the Fundamental Research Funds for the Central Universities (No. 2013QNA41) and the
Construction Project of the Key Discipline of Universities in Jiangsu Province During the 12th Five-
Year Plans (No. SX2013008). |
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| Abstract： |
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
$\left\{X_{i\beta}\right\}^{i=1,2,\cdots,n}_{\beta=1,2,\cdots,N}$
depending on a finite number of partial derivatives of the nonlocal
variables $v^{\beta}$ and a restriction
$\sum\limits_{i,\alpha,\beta}\big(\frac{\partial \xi^{i}}{\partial
v^{\beta}}\big)^{2}+\big(\frac{\partial \eta^{\alpha}}{\partial
v^{\beta}}\big)^{2}\neq 0$, i.e.,
$\sum\limits_{i,\alpha,\beta}\big(\frac{\partial
G^{\alpha}}{\partial v^{\beta}}\big)^{2}\neq 0$. 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\^{o} and
Caudrey-Dodd-Gibbon-Sawada-Kotera equations, et al. Finally, the
symmetries are applied to investigate the initial value problems and
Darboux transformations. |
Keywords： |
Generalized symmetries, Darboux transformations, Analytical solutions |
Classification： |
35Q51, 35Q53, 35C99, 68W30, 74J35 |
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