| Internal Waves in Stratified Fluid | p. 1 |
| Introduction | p. 3 |
| Governing Equations | p. 5 |
| Stratification | p. 6 |
| Linear model for small disturbances | p. 7 |
| Linearization of the boundary conditions | p. 10 |
| Linear boundary value problem | p. 11 |
| The Boussinesq approximation for nonlinear internal waves in continuously stratified ocean | p. 13 |
| Two-dimensional nonlinear Boussinesq equations | p. 15 |
| Dispersion relation and anisotropic property of internal waves | p. 17 |
| Two Model Examples | p. 25 |
| Generation of internal waves | p. 25 |
| Harmonic tidal flow over a corrugated slope | p. 26 |
| Discussion about the radiation condition | p. 29 |
| Reflection of internal waves from sloping topography | p. 33 |
| The problem of internal waves impinging on a sloping bottom | p. 34 |
| Direct answer to the question | p. 36 |
| p. 39 |
| Introduction to Lie Group Analysis | p. 43 |
| Calculus of Differential Algebra | p. 47 |
| Definitions | p. 47 |
| Main variables | p. 7 |
| Total differentiations | p. 48 |
| Differential functions | p. 48 |
| Euler-Lagrange operator | p. 49 |
| Properties | p. 49 |
| Divergence test | p. 49 |
| One-dimensional case | p. 51 |
| Exact equations | p. 53 |
| Definition | p. 53 |
| First-order equations | p. 53 |
| Second-order equations | p. 54 |
| Linear second-order equations | p. 56 |
| Change of variables in the space A | p. 57 |
| One independent variable | p. 57 |
| Several independent variables | p. 59 |
| Transformation Groups | p. 61 |
| Preliminaries | p. 61 |
| Examples from elementary mathematics | p. 61 |
| Examples from physics | p. 64 |
| Examples from fluid mechanics | p. 66 |
| One-parameter groups | p. 69 |
| Introduction of transformation groups | p. 69 |
| Local one-parameter groups | p. 71 |
| Local groups in canonical parameter | p. 74 |
| Infinitesimal description of one-parameter groups | p. 75 |
| Infinitesimal transformation | p. 75 |
| Lie equations | p. 76 |
| Exponential map | p. 79 |
| Invariants and invariant equations | p. 82 |
| Invariants | p. 82 |
| Invariant equations | p. 83 |
| Canonical variables | p. 85 |
| Construction of groups using canonical variables | p. 89 |
| Frequently used groups in the plane | p. 90 |
| Symmetry of Differential Equations | p. 91 |
| Notation | p. 91 |
| Differential equations | p. 91 |
| Transformation groups | p. 92 |
| Prolongation of group generators | p. 92 |
| Prolongation with one independent variable | p. 92 |
| Several independent variables | p. 94 |
| Definition of symmetry groups | p. 95 |
| Definition and determining equations | p. 95 |
| Construction of equations with given symmetry | p. 96 |
| Calculation of infinitesimal symmetry | p. 98 |
| Lie algebra | p. 99 |
| Definition of Lie algebra | p. 99 |
| Examples of Lie algebra | p. 100 |
| Invariants of multi-parameter groups | p. 103 |
| Lie algebra Lq_ in the plane: Canonical variables | p. 106 |
| Calculation of invariants in canonical variables | p. 107 |
| Applications of Symmetry | p. 111 |
| Ordinary differential equations | p. 111 |
| Integration of first-order equations | p. 111 |
| Integration of second-order equations | p. 114 |
| Partial differential equations | p. 116 |
| Symmetry of the Burgers equation | p. 116 |
| Invariant solutions | p. 117 |
| Group transformations of solutions | p. 120 |
| From symmetry to conservation laws | p. 121 |
| Introduction | p. 121 |
| Noether's theorem | p. 123 |
| Theorem of nonlocal conservation laws | p. 125 |
| Group Analysis of Internal Waves | p. 131 |
| Generalities | p. 135 |
| Introduction | p. 135 |
| Basic equations | p. 135 |
| Adjoint system | p. 136 |
| Formal Lagrangian | p. 136 |
| Self-adjointness of basic equations | p. 137 |
| Adjoint system to basic equations | p. 138 |
| Self-adjointness | p. 139 |
| Symmetry | p. 139 |
| Obvious symmetry | p. 139 |
| General admitted Lie algebra | p. 141 |
| Admitted Lie algebra in the case f = 0 | p. 142 |
| Conservation Laws | p. 143 |
| Introduction | p. 143 |
| General discussion of conservation equations | p. 143 |
| Variational derivatives of expressions with Jacobians | p. 145 |
| Nonlocal conserved vectors | p. 146 |
| Computation of nonlocal conserved vectors | p. 147 |
| Local conserved vectors | p. 149 |
| Utilization of obvious symmetry | p. 150 |
| Translation of v | p. 150 |
| Translation of Á | p. 151 |
| Translation of ¿ | p. 151 |
| Derivation of the flux of conserved vectors with known densities | p. 152 |
| Translation of x | p. 153 |
| Time translation | p. 153 |
| Conservation of energy | p. 154 |
| Use of semi-dilation | p. 156 |
| Computation of the conserved density | p. 157 |
| Conserved vector | p. 158 |
| Conservation law due to rotation | p. 159 |
| Summary of conservation laws | p. 159 |
| Conservation laws in integral form | p. 160 |
| Conservation laws in differential form | p. 160 |
| Group Invariant Solutions | p. 163 |
| Use of translations and dilation | p. 163 |
| Construction of the invariant solution | p. 163 |
| Generalized invariant solution and wave beams | p. 166 |
| Energy of the generalized invariant solution | p. 167 |
| Conserved density P of the generalized invariant solution | p. 168 |
| Use of rotation and dilation | p. 172 |
| The invariants | p. 172 |
| Candidates for the invariant solution | p. 173 |
| Construction of the invariant solution | p. 174 |
| Qualitative analysis of the invariant solution | p. 175 |
| Energy of the rotationally symmetric solution | p. 176 |
| Comparison with linear theory | p. 177 |
| Concluding remarks | p. 180 |
| Resonant Triad Model | p. 183 |
| Weakly nonlinear model | p. 184 |
| Two questions | p. 188 |
| Solutions to the resonance conditions | p. 189 |
| Resonant triad model | p. 192 |
| Utilization of the GM spectrum | p. 196 |
| Model example: Energy conservation for two resonant triads | p. 198 |
| Model example: Resonant interactions between 20 000 internal waves | p. 202 |
| Stability of the GM spectrum and open question on dissipation modelling | p. 205 |
| References | p. 209 |
| Index | p. 213 |
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