part i internal waves in stratified fluid
1introduction
2 governing equations
2.1stratification
2.2linear model for small disturbances
2.2.1linearization of the boundary conditions
2.2.2linear boundary value problem
2.3the boussinesq approximation for nonlinear internal waves incontinuously stratified ocean
2.3.1two-dimensional nonlinear boussinesq equations
2.3.2dispersion relation and anisotropic property of internalwaves
3 two model examples
3.1generation of internal waves
3.1.1harmonic tidal flow over a corrugated slope
3.1.2discussion about the radiation condition
3.2reflection of internal waves from sloping topography

part i internal waves in stratified fluid
1introduction
2 governing equations
2.1stratification
2.2linear model for small disturbances
2.2.1linearization of the boundary conditions
2.2.2linear boundary value problem
2.3the boussinesq approximation for nonlinear internal waves incontinuously stratified ocean
2.3.1two-dimensional nonlinear boussinesq equations
2.3.2dispersion relation and anisotropic property of internalwaves
3 two model examples
3.1generation of internal waves
3.1.1harmonic tidal flow over a corrugated slope
3.1.2discussion about the radiation condition
3.2reflection of internal waves from sloping topography
3.2.1the problem of internal waves impinging on a slopingbottom
3.2.2direct answer to the question
3.2.3latitude anomaly as an alternative answer
part ii introduction to lie group analysis

4 calculus of differential algebra
4.1definitions
4.1.1main variables
4.1.2total differentiations
4.1.3differential functions
4.1.4euler-lagrange operator
4.2properties
4.2.1divergence test
4.2.2one-dimensional case
4.3exact equations
4.3.1definition
4.3.2first-order equations
4.3.3second-order equations
4.3.4linear second-order equations
4.4change of variables in the space
4.4.1one independent variable
4.4.2several independent variables

5 transformation groups
5.1preliminaries
5.1.1examples from elementary mathematics
5.1.2examples from physics
5.1.3examples from fluid mechanics
5.2one-parameter groups
5.2.1introduction of transformation groups
5.2.2local one-parameter groups
5.2.3local groups in canonical parameter
5.3infinitesimal description of one-parameter groups
5.3.1infinitesimal transformation
5.3.2lie equations
5.3.3exponential map
5.4invariants and invariant equations
5.4.1invariants
5.4.2invariant equations
5.4.3canonical variables
5.4.4construction of groups using canonical variables
5.4.5frequently used groups in the plane
6 symmetry of differential equations
6.1notation
6.1.1differential equations
6.1.2transformation groups
6.2prolongation of group generators
6.2.1prolongation with one independent variable
6.2.2several independent variables
6.3definition of symmetry groups
6.3.1definition and determining equations
6.3.2construction of equations with given symmetry
6.3.3calculation of infinitesimal symmetry
6.4lie algebra
6.4.1definition of lie algebra
6.4.2examples of lie algebra
6.4.3invariants of multizparameter groups
6.4.4lie algebra l2 in the plalae: canonical variables
6.4.5calculation of invariants in canonical variables

7 applications of symmetry
7.1ordinary differential equations
7.1.1integration of first-order equations
7.1.2integration of second-order equations
7.2partial differential equations
7.2.1symmetry of the burgers equation
7.2.2invariant solutions
7.2.3group transformations of solutions
7.3from symmetry to conservation laws
7.3.1introduction
7.3.2noether's theorem
7.3.3theorem of nonlocal conservation laws

8 part hi group analysis of internal waves
8 generalities
8.1introduction
8.1.1basic equations
8.1.2adjoint system
8.1.3formal lagrangian
8.2self-adjointness of basic equations
8.2.1adjoint system to basic equations
8.2.2self-adjointness
8.3symmetry
8.3.1obvious symmetry
8.3.2general admitted lie algebra
8.3.3admitted lie algebra in the case f = 0
9 conservation laws

9.1introduction
9.1.1general discussion of conservation equations
9.1.2variational derivatives of expressions with jacobians
9.1.3nonlocal conserved vectors
9.1.4computation of nonlocal conserved vectors
9.1.5local conserved vectors
9.2utilization of obvious symmetry
9.2.1translation of v
9.2.2translation of p
9.2.3translation of ψ
9.2.4derivation of the flux of conserved vectors with knowndensities
9.2.5translation ofx
9.2.6time translation
9.2.7conservation of energy
9.3 use of semi-dilation
9.3.1computation of the conserved density
9.3.2conserved vector
9.4conservation law due to rotation
9.5summary of conservation laws
9.5.1conservation laws in integral form
9.5.2conservation laws in differential form

10 group invariant solutions
10.1 use of translations and dilation
10.1.1 construction of the invariant solution
10.1.2 generalized invariant solution and wave beams
10.1.3 energy of the generalized invariant solution
10.1.4 conserved density p of the generalized invariantsolution.
10.2 use of rotation and dilation
10.2.1 the invariants
10.2.2 candidates for the invariant solution
10.2.3 construction of the invariant solution
10.2.4 qualitative analysis of the invariant solution
10.2.5 energy of the rotationally symmetric solution
10.2.6 comparison with linear theory
10.3 concluding remarks
a resonant triad model
a. 1 weakly nonlinear model
a.2 two questions
a.3 solutions to the resonance conditions
a.4 resonant triad model
a.4.1 utilization of the gm spectrum
a.4.2 model example: energy conservation for two resonanttriads
a.4.3 model example: resonant interactions between 20 000 internalwaves
a.5 stability of the gm spectrum and open question on dissipationmodelling
references
index