| 摘要 | 第1-5页 |
| Abstract | 第5-6页 |
| 1 Introduction | 第6-9页 |
| 2 The Camassa-Holm equation | 第9-34页 |
| §2.1 Multi-symplecticity of the Camassa-Holm equation | 第9-13页 |
| ·Lagrangian and multi-symplectic formulations | 第9-12页 |
| ·The local properties of the Camassa-Holm equation | 第12-13页 |
| §2.2 Energy-preserving algorithms | 第13-34页 |
| ·Introduction of the variational derivative | 第13-15页 |
| ·Fourier pseudospectral method and AVF method | 第15-19页 |
| ·Galerkin method and AVF method | 第19-27页 |
| ·Fourier pseudospectral method and DPD method | 第27-30页 |
| ·Wavelet collocation method and DPD method | 第30-34页 |
| 3 The Benjamin-type equations | 第34-46页 |
| §3.1 Multi-symplecticity of the Benjamin-type equations | 第34-37页 |
| ·A modified multi-symplectic formulation | 第35-36页 |
| ·The local properties of the Benjamin-type equations | 第36-37页 |
| §3.2 The operators H,L and their discretizations | 第37-39页 |
| ·Introduction of Hilbert transform H | 第37-38页 |
| ·Discretization of the non-local operators H and L | 第38-39页 |
| §3.3 Energy-preserving algorithms | 第39-46页 |
| ·Fourier pseudospectral method and AVF method | 第41-43页 |
| ·Galerkin method and AVF method | 第43-45页 |
| ·Wavelet collocation method and AVF method | 第45-46页 |
| 4 Numerical Examples | 第46-67页 |
| §4.1 Numerical simulation for the Camassa-Holm equation | 第46-59页 |
| §4.2 Numerical simulation for the Benjamin equation | 第59-67页 |
| 5 Concluding remarks | 第67-69页 |
| Bibliography | 第69-74页 |
| 致谢 | 第74页 |
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