| 致谢 | 第3-4页 |
| Acknowledgements | 第4-5页 |
| 摘要 | 第5-6页 |
| Abstract | 第6页 |
| 详细的总结（Detailed summary） | 第7-15页 |
| Chapter 1 Introduction | 第15-20页 |
| 1.1 Research background | 第15页 |
| 1.2 Other Research in the area | 第15-16页 |
| 1.3 Research significance | 第16-17页 |
| 1.4 Research objectives | 第17-18页 |
| 1.5 Outline of Thesis | 第18-20页 |
| Chapter 2 Lie symmetries and Point transformations for Invariant functions | 第20-40页 |
| 2.1 Introduction | 第20页 |
| 2.2 Point Transformations | 第20-22页 |
| 2.3 Invariant functions | 第22页 |
| 2.4 Infinitesimal Point Transformations | 第22-24页 |
| 2.5 Lie Algebra | 第24-25页 |
| 2.6 Prolongation of point transformations | 第25-27页 |
| 2.7 Lie symmetries of ordinary differential equations (ODEs) | 第27-32页 |
| 2.8 Lie symmetries of partial differential equations (PDEs) | 第32-35页 |
| 2.9 Noether point symmetries of ODEs | 第35-38页 |
| 2.10 Noether symmetries of PDEs | 第38-39页 |
| Conclusion | 第39-40页 |
| Chapter 3 Lie point symmetries of an ordinary group of Partial Differential Equations(PDEs) | 第40-52页 |
| 3.1 Introduction | 第40页 |
| 3.2 Lie symmetries of a 2nd order PDEs | 第40-43页 |
| 3.3 Linear function G ( , p, ) and Lie symmetries conditions | 第43-45页 |
| 3.4 Lie symmetries of Poisson equation | 第45-46页 |
| 3.5 Lie symmetries of Laplace equation | 第46页 |
| 3.6 Lie symmetries of the conformal Poisson equation | 第46-47页 |
| 3.7 Lie symmetries of conformal Laplace equation within Riemannian spaces | 第47-48页 |
| 3.8 Heat equation by a flux in a Riemannian space | 第48-51页 |
| 3.9 Conclusion | 第51-52页 |
| Chapter 4 Lie point symmetries of Klein Gordon and Schr?dinger equations | 第52-63页 |
| 4.1 Introduction | 第52-53页 |
| 4.2 Noether point symmetries of conformal Lagrangian in Riemannian space | 第53-55页 |
| 4.3 Lie symmetries of Klein-Gordon equation | 第55-56页 |
| 4.4 Lie point symmetries of conformal Klein Gordon equation | 第56-58页 |
| 4.5 Lie point symmetries of Schr?dinger equation | 第58-59页 |
| 4.6 Klein Gordon equation and sl(2, R) algebra | 第59页 |
| 4.7 Oscillator system | 第59-60页 |
| 4.8 Kepler Ermakov potential by means of an oscillator term | 第60-62页 |
| 4.9 Conclusion | 第62-63页 |
| Chapter 5 Generalization of Kepler-Ermakov Dynamical System in a Riemannian space | 第63-76页 |
| 5.1 Introduction | 第63-64页 |
| 5.2 Lie symmetries of Kepler Ermakov system | 第64-65页 |
| 5.3 Generalization of the two-dimensional KEDS | 第65-69页 |
| 5.4 Generalization of the three-dimensional KEDS | 第69页 |
| 5.5 The three-dimensional KEDS of class I | 第69-70页 |
| 5.6 The three-dimensional KEDS of class II | 第70-71页 |
| 5.7 The -dimensional Riemannian KEDS | 第71-72页 |
| 5.8 The non-Hamiltonian Riemannian KEDS of n-dimensional | 第72-73页 |
| 5.9 The Conservative Riemannian KEDS of n-dimensional space | 第73-75页 |
| 5.10 Conclusion | 第75-76页 |
| Chapter 6 Reduction of Type-II Hidden symmetries of Laplace equation and Homogeneous heatequation | 第76-90页 |
| 6.1 Introduction | 第76-77页 |
| 6.2 In general Riemannian spaces Lie point Symmetries of the Laplace equation | 第77-78页 |
| 6.3 In general Riemannian spaces reduction of the Laplace equation | 第78页 |
| 6.4 A gradient HV introducing by Riemannian space | 第78-79页 |
| 6.5 A gradient KV introducing by Riemannian spaces | 第79-80页 |
| 6.6 A gradient sp.CKVs introducing by Riemannian space | 第80-83页 |
| 6.7 In definite Riemannian space reduction of Homogeneous Heat equation | 第83-84页 |
| 6.8 In a space the heat equation which introduce a gradient KV | 第84-86页 |
| 6.9 In a space the heat equation which introduce a gradient HV | 第86-88页 |
| Conclusion | 第88-90页 |
| Chapter 7 Lie symmetries and Painlevé analysis for a (2+1)-dimensional nonlinear Schr?dingerequation | 第90-101页 |
| 7.1 Introduction | 第90-91页 |
| 7.2 Painlevé analysis for a (2+1) dimensional NLSE | 第91-93页 |
| 7.3 Lie symmetry algebra for a (2+1) dimensional NLSE | 第93-98页 |
| 7.4 Hirota’s Bilinear form and some new special solutions of Eq.(7.1) | 第98-100页 |
| Conclusions | 第100-101页 |
| Chapter 8 Summary and Conclusions | 第101-105页 |
| References | 第105-115页 |
| Author’s resume | 第115-118页 |
| 学位论文数据集 | 第118页 |
