Lie Group Analysis of Differential Equations: Invariant Solutions of Nonlinear Physical Phenomena

Preface
Acknowledgments
1 Lie group analysis of differential equations
	1.1 Basic concepts and examples from elementary mathematics
	1.2 One-parameter group
	1.3 Lie equations
		1.3.1 Lie theorem
		1.3.2 Inverse Lie theorem
	1.4 Canonical variables
	1.5 Invariants and invariant equations
		1.5.1 Invariant equations
		1.5.2 Universal invariant
	1.6 Transformations in terms of canonical variables
	1.7 Exponential map
	1.8 The invariance principle for differential equations
		1.8.1 The prolongation formulae
		1.8.2 Generalization
		1.8.3 Illustrative examples: prolonged generators
		1.8.4 Invariant solutions (G/H factor system)
	1.9 Group transformations admitted by nonlinear partial differential equations
		1.9.1 How to find a group admitted by a nonlinear PDE?
		1.9.2 Lie algebras and multi-parameter groups
		1.9.3 Commutator of the group
	1.10 Summary
	1.11 Exercises
2 Integration of ordinary differential equations
	2.1 First-order equations
		2.1.1 Integrating factor
		2.1.2 Integration using canonical variables
		2.1.3 Integration of ordinary differential equations admitting groups
	2.2 Second-order equations
		2.2.1 Algorithm
		2.2.2 Exact second-order equations
	2.3 Exercises
3 Illustration: invariant solutions as internal singularities of nonlinear differential equations
	3.1 Invariant solutions
		3.1.1 Derivation of invariant solutions
		3.1.2 Behavior of invariant solutions
	3.2 Perturbation of singular solutions
		3.2.1 Application to the first case
		3.2.2 Application to the second case
		3.2.3 Analysis of numerical or implicit solutions
	3.3 Concluding remarks
	3.4 Exercises
4 Modeling scenario 1: blood flow of variable density
	4.1 Mathematical modeling
	4.2 First approach: approximate analysis
		4.2.1 Failure of the direct approach
		4.2.2 Multi-scale approach
		4.2.3 Stability of perturbed steady flows
	4.3 Second approach: group theoretical point of view
		4.3.1 Traveling waves
		4.3.2 Similarity solution
	4.4 Concluding remarks
	4.5 Exercises
5 Modeling scenario 2: invariant solutions as dispersion relation
	5.1 Preliminaries
	5.2 Group theoretical derivation of dispersion relations
		5.2.1 Homogeneous linear equations
		5.2.2 Nonhomogeneous polynomial linear equations
	5.3 Concluding remarks
	5.4 Exercises
6 Modeling scenario 3: invariant solutions of nonlinear surface waves in the ocean and atmosphere
	6.1 Preliminaries
	6.2 Cauchy–Poisson free boundary problem
	6.3 Linear theory
		6.3.1 Two-dimensional case
		6.3.2 Three particular cases
	6.4 The Lagrange method for the long wave theory
		6.4.1 Nondimensionalization
	6.5 Shallow water equations
	6.6 Nonlinear approximations
		6.6.1 The second approximation
		6.6.2 Linear analysis
		6.6.3 Boussinesq approximation for surface gravity waves
		6.6.4 KdV equation
		6.6.5 Periodic and solitary solutions
	6.7 Modeling equatorial planetary atmospheric waves
		6.7.1 Shallow water approximation
	6.8 Invariant solution and Fibonacci spiral
		6.8.1 Case studies
		6.8.2 Asymptotic analysis of invariant solutions
		6.8.3 Approximation of the similarity solution for large and small k
	6.9 Nonlinear analysis
	6.10 Fibonacci spirals
	6.11 Hodograph method
		6.11.1 Mapping the nonlinear shallow water model to a linear system
		6.11.2 Reduction to a second-order linear equation
		6.11.3 Characteristics
	6.12 Riemann’s integration method
		6.12.1 Preliminaries
		6.12.2 Toward Riemann’s function via the invariance principle
	6.13 Shock waves
	6.14 Perturbed system
		6.14.1 Hodograph transform and simplified perturbation
		6.14.2 Hodograph transformation of symmetries
		6.14.3 Approximate symmetry for the perturbed system
		6.14.4 Approximately invariant solution
	6.15 Concluding remarks
	6.16 Exercises
7 Modeling scenario 4: Rossby nonlinear atmospheric waves along a spherical planet
	7.1 Euler and Navier–Stokes equations
	7.2 Nonlinear nonviscous flows in rotating reference frame
	7.3 Invariant solutions of Euler equations
		7.3.1 Invariant solution based on X2 and X4
		7.3.2 Invariant solution based on X4 and X5
	7.4 Invariant solutions of Navier–Stokes equations
	7.5 Discussion of invariant solutions
	7.6 Effects of Rossby waves on the energy balance of zonal flows
	7.7 Concluding remarks
	7.8 Exercises
8 Modeling scenario 5: invariant solutions as ocean whirlpools
	8.1 Preliminaries
	8.2 Model equations of internal waves
		8.2.1 Linear model and utilization of the Kelvin hypothesis for ur≪1
		8.2.2 Effects of rotation on oscillatory modes
	8.3 Invariant solutions for the case ur=0
		8.3.1 Well-known invariant internal oscillation solution
		8.3.2 Invariant nonstationary solution
		8.3.3 Invariant stationary solution
		8.3.4 Invariant solutions as nonlinear whirlpools
	8.4 Approximately invariant solutions for the case ur=εvr
		8.4.1 Well-known approximately invariant internal oscillation solution
		8.4.2 Approximately invariant nonstationary solution
		8.4.3 Approximately invariant stationary solution
		8.4.4 Approximately invariant nonlinear whirlpools
	8.5 Discussions of invariant and approximately invariant solutions
	8.6 Weakly nonlinear model
		8.6.1 Evanescent modes
		8.6.2 Neutral stability
	8.7 Existence of nonlinear whirlpools
		8.7.1 Two examples
		8.7.2 Oscillatory solution
	8.8 Exercises
9 Modeling scenario 6: invariant solutions of internal waves in the ocean
	9.1 Mathematical modeling
	9.2 Rotationally invariant solutions and comparison with linear theory
	9.3 Lagrangian, conservation laws, and exact solutions of the nonlinear internal waves
		9.3.1 Adjoint system to the Boussinesq model (9.52)–(9.54)
		9.3.2 Self-adjointness of the Boussinesq model (9.52)–(9.54)
		9.3.3 Conservation laws
		9.3.4 Variational derivatives of expressions with Jacobians
		9.3.5 Nonlocal conserved vectors
		9.3.6 Computation of nonlocal conserved vectors
		9.3.7 Local conserved vectors
	9.4 Concluding remarks
Bibliography
Subject Index