Analytical Properties of Nonlinear Partial Differential Equations: with Applications to Shallow Water Models (CMS/CAIMS Books in Mathematics, 10)

Preface
Contents
Named Equation Reference
1 Equations of Fluid Dynamics and the Shallow Water Approximation
	1.1 Notation
	1.2 Navier–Stokes and Euler Equations
	1.3 The Shallow Water Approximation
	1.4 Boundary Value Problems for Irrotational Flows with a Free Surface
		1.4.1 Boundary Value Problem with a Free Surface
		1.4.2 Irrotational Flows
		1.4.3 The Dimensionless Boundary Value Problem
2 Integrability and Other Analytical Properties of Nonlinear PDE Systems
	2.1 Introductory Remarks
	2.2 C-integrability
	2.3 S-integrability
	2.4 Lax Pairs and Zero-Curvature Representation
	2.5 Symmetries
	2.6 Equivalence Transformations
	2.7 Conservation Laws
	2.8 Exact Solutions
	2.9 Variational Principles and Lagrangian Structure
	2.10 Hamiltonian Structure of PDE Models
	2.11 Recursion Operators
	2.12 Bi-Hamiltonian Systems
	2.13 Painlevé Property
	2.14 Dispersion Relations
	2.15 Computations on and off of the Solution Set of a PDE System
	2.16 Symbolic Software for the Computation of Analytical Properties
	2.17 Other Notions of Integrability, Related Concepts, Methods, and Further Remarks
3 Shallow Water Models and Their Analytical Properties
	3.1 The PDE Models
	3.2 Euler Equations in Two Dimensions
		3.2.1 Derivation
		3.2.2 Integrability
		3.2.3 Hamiltonian Structure
		3.2.4 Lagrangian Structure
		3.2.5 Symmetries
		3.2.6 Conservation Laws
		3.2.7 Exact Solutions
		3.2.8 Painlevé Property
		3.2.9 Dispersion Relation
		3.2.10 Remarks
	3.3 Su-Gardner Equations
		3.3.1 Derivation
		3.3.2 Integrability
		3.3.3 Hamiltonian Structure
		3.3.4 Lagrangian Structure
		3.3.5 Symmetries
		3.3.6 Conservation Laws
		3.3.7 Exact Solutions
		3.3.8 Painlevé Property
		3.3.9 Dispersion Relation
		3.3.10 Remarks
	3.4 Korteweg-de Vries Equation
		3.4.1 Derivation
		3.4.2 Integrability
		3.4.3 Hamiltonian Structure
		3.4.4 Lagrangian Structure
		3.4.5 Symmetries
		3.4.6 Conservation Laws
		3.4.7 Exact Solutions
		3.4.8 Painlevé Property
		3.4.9 Dispersion Relation
		3.4.10 Remarks
	3.5 Burgers Equation
		3.5.1 Derivation
		3.5.2 Integrability
		3.5.3 Hamiltonian Structure
		3.5.4 Lagrangian Structure
		3.5.5 Symmetries
		3.5.6 Conservation Laws
		3.5.7 Exact Solutions
		3.5.8 Painlevé Property
		3.5.9 Dispersion Relation
		3.5.10 Remarks
	3.6 Nonlinear Schrödinger Equation
		3.6.1 Derivation
		3.6.2 Integrability
		3.6.3 Hamiltonian Structure
		3.6.4 Lagrangian Structure
		3.6.5 Symmetries
		3.6.6 Conservation Laws
		3.6.7 Exact Solutions
		3.6.8 Painlevé Property
		3.6.9 Dispersion Relation
		3.6.10 Remarks
	3.7 Benjamin-Bona-Mahony Equation
		3.7.1 Derivation
		3.7.2 Integrability
		3.7.3 Hamiltonian Structure
		3.7.4 Lagrangian Structure
		3.7.5 Symmetries
		3.7.6 Conservation Laws
		3.7.7 Exact Solutions
		3.7.8 Painlevé Property
		3.7.9 Dispersion Relation
		3.7.10 Remarks
	3.8 Classical Boussinesq Equations
		3.8.1 Derivation
		3.8.2 Integrability
		3.8.3 Hamiltonian Structure
		3.8.4 Lagrangian Structure
		3.8.5 Symmetries
		3.8.6 Conservation Laws
		3.8.7 Exact Solutions
		3.8.8 Painlevé Property
		3.8.9 Dispersion Relation
		3.8.10 Remarks
	3.9 Kaup-Boussinesq Equations
		3.9.1 Derivation
		3.9.2 Integrability
		3.9.3 Hamiltonian Structure
		3.9.4 Lagrangian Structure
		3.9.5 Symmetries
		3.9.6 Conservation Laws
		3.9.7 Exact Solutions
		3.9.8 Painlevé Property
		3.9.9 Dispersion Relation
		3.9.10 Remarks
	3.10 Boussinesq Equation
		3.10.1 Derivation
		3.10.2 Integrability
		3.10.3 Hamiltonian Structure
		3.10.4 Lagrangian Structure
		3.10.5 Symmetries
		3.10.6 Conservation Laws
		3.10.7 Exact Solutions
		3.10.8 Painlevé Property
		3.10.9 Dispersion Relation
		3.10.10 Remarks
	3.11 Shallow Water Equations
		3.11.1 Derivation
		3.11.2 Integrability
		3.11.3 Hamiltonian Structure
		3.11.4 Lagrangian Structure
		3.11.5 Symmetries
		3.11.6 Conservation Laws
		3.11.7 Exact Solutions
		3.11.8 Painlevé Property
		3.11.9 Dispersion Relation
		3.11.10 Remarks
	3.12 Generalized Shallow Water Equation
		3.12.1 Derivation
		3.12.2 Integrability
		3.12.3 Hamiltonian Structure
		3.12.4 Lagrangian Structure
		3.12.5 Symmetries
		3.12.6 Conservation Laws
		3.12.7 Exact Solutions
		3.12.8 Painlevé Property
		3.12.9 Dispersion Relation
		3.12.10 Remarks
	3.13 Camassa-Holm Equation
		3.13.1 Derivation
		3.13.2 Integrability
		3.13.3 Hamiltonian Structure
		3.13.4 Lagrangian Structure
		3.13.5 Symmetries
		3.13.6 Conservation Laws
		3.13.7 Exact Solutions
		3.13.8 Painlevé Property
		3.13.9 Dispersion Relation
		3.13.10 Remarks
	3.14 Modified Camassa-Holm Equation
		3.14.1 Derivation
		3.14.2 Integrability
		3.14.3 Hamiltonian Structure
		3.14.4 Lagrangian Structure
		3.14.5 Symmetries
		3.14.6 Conservation Laws
		3.14.7 Exact Solutions
		3.14.8 Painlevé Property
		3.14.9 Dispersion Relation
		3.14.10 Remarks
	3.15 Two-Component Camassa-Holm Equations
		3.15.1 Derivation
		3.15.2 Integrability
		3.15.3 Hamiltonian Structure
		3.15.4 Lagrangian Structure
		3.15.5 Symmetries
		3.15.6 Conservation Laws
		3.15.7 Exact Solutions
		3.15.8 Painlevé Property
		3.15.9 Dispersion Relation
		3.15.10 Remarks
	3.16 Kadomtsev-Petviashvili Equation
		3.16.1 Derivation
		3.16.2 Integrability
		3.16.3 Hamiltonian Structure
		3.16.4 Lagrangian Structure
		3.16.5 Symmetries
		3.16.6 Conservation Laws
		3.16.7 Exact Solutions
		3.16.8 Painlevé Property
		3.16.9 Dispersion Relation
		3.16.10 Remarks
A Higher-Order and Multi-parameter Shallow Water Models
References
Index