伊布拉基莫夫、伊布拉基莫夫编著的《李群分析在地球物理流体动力学中的应用（英文版）》精炼并自成体系地介绍李群分析的基本概念和方法，是应用群分析方面的第一本专著，是地球物理流体动力学理论快速入门参考书。

伊布拉基莫夫、伊布拉基莫夫编著的《李群分析在地球物理流体动力学中的应用（英文版）》是第一本将李群分析应用于深海内波的传播，并提出了一种新的方法来描述深海非线性波的相互作用的著作。《李群分析在地球物理流体动力学中的应用（英文版）》的主题思想是通过李群分析来探究深海波动问题，书中提供了非常灵活易懂的内容，涵盖多个研究方向，其目的是吸引更多的物理学家和数学家利用李群的对称性分析研究非线性物理问题。

《李群分析在地球物理流体动力学中的应用（英文版）》可供对利用李群分析研究物理、工程和自然科学感兴趣的专家及教授参考，也可作为应用数学、物理及工程学专业的研究生关于非线性微分方程的对称性应用课程的教材。

part i internal waves in stratified fluid

1 introduction

2 governing equations

 2.1 stratification

 2.2 linear model for small disturbances

 2.2.1 linearization of the boundary conditions

 2.2.2 linear boundary value problem

 2.3 the boussinesq approximation for nonlinear internal waves in continuously stratified ocean

 2.3.1 two-dimensional nonlinear boussinesq equations

 2.3.2 dispersion relation and anisotropic property of internal waves

3 two model examples

 3.1 generation of internal waves

 3.1.1 harmonic tidal flow over a corrugated slope

 3.1.2 discussion about the radiation condition

 3.2 reflection of internal waves from sloping topography

 3.2.1 the problem of internal waves impinging on a sloping bottom

 3.2.2 direct answer to the question

 3.2.3 latitude anomaly as an alternative answer

part ii introduction to lie group analysis

4 calculus of differential algebra

 4.1 definitions

 4.1.1 main variables

 4.1.2 total differentiations

 4.1.3 differential functions

 4.1.4 euler-lagrange operator

 4.2 properties

 4.2.1 divergence test

 4.2.2 one-dimensional case

 4.3 exact equations

 4.3.1 definition

 4.3.2 first-order equations

 4.3.3 second-order equations

 4.3.4 linear second-order equations

 4.4c hange of variables in the space

 4.4.1 one independent variable

 4.4.2 several independent variables

5 transformation groups

 5.1 preliminaries

 5.1.1 examples from elementary mathematics

 5.1.2 examples from physics

 5.1.3 examples from fluid mechanics

 5.2 one-parameter groups

 5.2.1 introduction of transformation groups

 5.2.2 local one-parameter groups

 5.2.3 local groups in canonical parameter

 5.3 infinitesimal description of one-parameter groups

 5.3.1 infinitesimal transformation

 5.3.2 lie equations

 5.3.3 exponential map

 5.4 invariants and invariant equations

 5.4.1 invariants

 5.4.2 invariant equations

 5.4.3 canonical variables

 5.4.4 construction of groups using canonical variables

 5.4.5 frequently used groups in the plane

6 symmetry of differential equations

 6.1 notation

 6.1.1 differential equations

 6.1.2 transformation groups

 6.2 prolongation of group generators

 6.2.1 prolongation with one independent variable

 6.2.2 several independent variables

 6.3 definition of symmetry groups

 6.3.1 definition and determining equations

 6.3.2 construction of equations with given symmetry

 6.3.3 calculation of infinitesimal symmetry

 6.4 lie algebra

 6.4.1 definition of lie algebra

 6.4.2 examples of lie algebra

 6.4.3 invariants of multizparameter groups

 6.4.4 lie algebra l2 in the plalae: canonical variables

 6.4.5 calculation of invariants in canonical variables

7 applications of symmetry

 7.1 ordinary differential equations

 7.1.1 integration of first-order equations

 7.1.2 integration of second-order equations

 7.2 partial differential equations

 7.2.1 symmetry of the burgers equation

 7.2.2 invariant solutions

 7.2.3 group transformations of solutions

 7.3 from symmetry to conservation laws

 7.3.1 introduction

 7.3.2 noether's theorem

 7.3.3 theorem of nonlocal conservation laws

8 part hi group analysis of internal waves

8 generalities

 8.1 introduction

 8.1.1 basic equations

 8.1.2 adjoint system

 8.1.3 formal lagrangian

 8.2 self-adjointness of basic equations

 8.2.1 adjoint system to basic equations

 8.2.2 self-adjointness

 8.3 symmetry

 8.3.1 obvious symmetry

 8.3.2 general admitted lie algebra

 8.3.3 admitted lie algebra in the case f = 0

9 conservation laws

 9.1 introduction

 9.1.1 general discussion of conservation equations

 9.1.2 variational derivatives of expressions with jacobians

 9.1.3 nonlocal conserved vectors

 9.1.4 computation of nonlocal conserved vectors

 9.1.5 local conserved vectors

 9.2 utilization of obvious symmetry

 9.2.1 translation of v

 9.2.2 translation of p

 9.2.3 translation of ψ

 9.2.4 derivation of the flux of conserved vectors with known densities

 9.2.5 translation ofx

 9.2.6 time translation

 9.2.7 conservation of energy

 9.3 use of semi-dilation

 9.3.1 computation of the conserved density

 9.3.2 conserved vector

 9.4 conservation law due to rotation

 9.5 summary of conservation laws

 9.5.1 conservation laws in integral form

 9.5.2 conservation 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 invariant solution.

 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 resonant triads

 a.4.3 model example: resonant interactions between 20 000 internal waves

 a.5 stability of the gm spectrum and open question on dissipation modelling

references

index