Contact Geometry and Non-linear Differential Equations

Title page
Preface
Part 1 Symmetries and Integrals
1 Distributions
1.1 Distributions and integral manifolds
1.1.1 Distributions
1.1.2 Morphisms of distributions
1.1.3 Integral manifolds
1.2 Symmetries of distributions
1.3 Characteristic and shuffting symmetries
1.4 Curvature of a distribution
1.5 Flat distributions and the Frobenius theorem
1.6 Complex distributions on real manifolds
1.7 The Lie-Bianchi theorem
1.7.1 The Maurer-Cartan equations
1.7.2 Distributions with a commutative symmetry algebra
1.7.3 Lie-Bianchi theorem
2 Ordinary differential equations
2.1 Symmetries of ODEs
2.1.1 Generating functions
2.1.2 Lie algebra structure on generating functions
2.1.3 Commutative symmetry algebra
2.2 Non-linear second-order ODEs
2.2.1 Equation y" = y'+F(y)
2.2.2 Integration
2.2.3 Non-linear third-order equations
2.3 Linear differential equations and linear symmetries
2.3.1 The variation of constants method
2.3.2 Linear symmetries
2.4 Linear symmetries of self-adjoint operators
2.5 Schrödinger operators
2.5.1 Integrable potentials
2.5.2 Spectral problems for KdV potentials
2.5.3 Lagrange integrals
3 Model differential equations and the Lie superposition principle
3.1 Symmetry reduction
3.1.1 Reductions by symmetry ideals
3.1.2 Reductions by symmetry subalgebras
3.2 Model differential equations
3.2.1 One-dimensional model equations
3.2.2 Riccati equations
3.3 Model equations: the series A_k, D_k, C_k
3.3.1 Series A_k
3.3.2 Series D_k
3.3.3 Series C_k
3.4 The Lie superposition principle
3.4.1 Bianchi equations
3.5 AP-structures and their invariants
3.5.1 Decomposition of the de Rham complex
3.5.2 Classical almost product structures
3.5.3 Almost complex structures
3.5.4 AP-structures on five-dimensional manifolds
Part II Symplectic Algebra
4 Linear algebra of symplectic vector spaces
4.1 Symplectic vector spaces
4.1.1 Bilinear skew-symmetric forms on vector spaces
4.1.2 Symplectic structures on vector spaces
4.1.3 Canonical bases and coordinates
4.2 Symplectic transformations
4.2.1 Matrix representation of symplectic transformations
4.3 Lagrangian subspaces
4.3.1 Symplectic and Kähler spaces
5 Exterior algebra on symplectic vector spaces
5.1 Operators ? and ?
5.2 Effective forms and the Hodge-Lepage theorem
5.2.1 sl₂-method
6 A symplectic classification of exterior 2-forms in dimension 4
6.1 Pfaffian
6.2 Normal forms
6.3 Jacobi planes
6.3.1 Classification of Jacobi planes
6.3.2 Operators associated with Jacobi planes
7 Symplectic classification of exterior 2-forms
7.1 Pfaffians and linear operators associated with 2-forms
7.2 Symplectic classification of 2-forms with distinct real characteristic numbers
7.3 Symplectic classification of 2-forms with distinct complex characteristic numbers
7.4 Symplectic classification of 2-forms with multiple characteristic numbers
7.5 Symplectic classification of effective 2-forms in dimension 6
8 Classification of exterior 3-forms on a six-dimensional symplectic space
8.1 A symplectic invariant of effective 3-forms
8.1.l The case of trivial invariants
8.1.2 The case of non-trivial invariants
8.1.3 Hitchin's results on the geometry of 3-forms
8.2 The stabilizers of orbits and their prolongations
8.2.1 Stabilizers
8.2.2 Prolongations
Part III Monge-Ampère Equations
9 Symplectic manifolds
9.1 Symplectic structures
9.1.1 The cotangent bundle and the standard symplectic structure
9.1.2 Kähler manifolds
9.1.3 Orbits and homogeneous symplectic spaces
9.2 Vector fields on symplectic manifolds
9.2.1 Poisson bracket and Hamiltonian vector fields
9.2.2 Canonical coordinates
9.3 Submanifolds of symplectic manifolds
9.3.1 Presymplectic manifolds
9.3.2 Lagrangian submanifolds
9.3.3 Involutive submanifolds
9.3.4 Lagrangian polarizations
10 Contact manifolds
10.1 Contact structures
10.1.1 Examples
10.2 Contact transformations and contact vector fields
10.2.1 Examples
10.2.2 Contact vector fields
10.3 Darboux theorem
10.4 A local description of contact transformations
10.4.1 Generating functions of Lagrangian submanifolds
10.4.2 A description of contact transformations in R3
11 Monge-Ampère equations
11.1 Monge-Ampère operators
11.2 Effective differential forms
11.3 Calculus on Ω*(C*)
Il.4 The Euler operator
11.5 Solutions
11.6 Monge-Ampère equations of divergent type
12 Symmetries and contact transformations of Monge-Ampère equations
12.1 Contact transformations
12.2 Lie equations for contact symmetries
12.3 Reduction
12.4 Examp1es
12.4.1 The boundary layer equation
12.4.2 The thermal conductivity equation
12.4.3 The Petrovsky-Kolmogorov-Piskunov equation
12.4.4 The Von Karman equation
12.5 Symmetries of the reduction
12.6 Monge-Ampère equations in symplectic geometry
13 Conservation laws
13.1 Definition and examp1es
13.2 Calculus for conservation laws
13.3 Symmetries and conservations laws
13.4 Shock waves and the Hugoniot-Rankine condition
13.4.1 Shock Waves for ODEs
13.4.2 Discontinuous solutions
13.4.3 Shock waves
13.5 Calculus of variations and the Monge-Ampère equation
13.5.1 The Euler operator
13.5.2 Symmetries and conservation laws in variational problems
13.5.3 Classical variational problems
13.6 Effective cohomology and the Euler operator
14 Monge-Ampère equations on two-dimensional manifolds and geometric structures
14.1 Non-holonomic geometric structures associated with Monge-Ampère equations
14.1.1 Non-holonomic structures on contact manifolds
14.1.2 Non-holonomic fields of endomorphisms on generated by Monge-Ampère equations
14.1.3 Non-degenerate equations
14.1.4 Parabolic equations
14.2 Intermediate integrals
14.2.1 Classical and non-holonomic intermediate integrals
14.2.2 Cauchy problem and non-holonomic intermediate integrals
14.3 Symplectic Monge-Ampère equations
14.3.1 A field of endomorphisms A_ω on T*M
14.3.2 Non-degenerate symplectic equations
14.3.3 Symplectic parabolic equations
14.3.4 Intermediate integrals
14.4 Cauchy problem for hyperbolic Monge-Ampère equations
14.4.1 Constructive methods for integration of Cauchy problem
15 Systems of first-order partial differential equations on two-dimensional manifolds
15.1 Non-linear differential operators of first order on two-dimensional manifolds
15.2 Jacobi equations
15.3 Symmetries of Jacobi equations
15.4 Geometric structures associated with Jacobi's equations
15.5 Conservation laws of Jacobi equations
15.6 Cauchy problem for hyperbolic Jacobi equations
Part IV Applications
16 Non-linear acoustics
16.1 Symmetries and conservation laws of the KZ equation
16.1.1 KZ equation and its contact symmetries
16.1.2 The structure of the symmetry algebra
16.1.3 Classification of one-dimensional subalgebras of sl(2,R)
16.1.4 Classification of symmetries
16.1.5 Conservation laws
16.2 Singu1arities of solutions of the KZ equation
16.2.1 Caustics
16.2.2 Contact shock waves
17 Non-linear thermal conductivity
17.1 Symmetries of the TC equation
17.1.l TC equation
17.1.2 Group classification of TC equation
17.2 Invariant solutions
18 Meteorology applications
18.1 Shallow water theory and balanced dynamics
18.2 A geometric approach to semi-geostrophic theory
18.3 Hyper-Kähler structure and Monge-Ampère operators
18.4 Monge-Ampère operators with constant coefficients and plane balanced models
Part V Classification of Monge-Ampère equations
19 Classification of symplectic MAOs on two-dimensional manifolds
19.1 e-Structures
19.2 Classification of non-degenerate Monge-Ampère operators
19.2.1 Differentiai invariants of non-degenerate operators
19.2.2 Hyperbolic operators
19.2.3 Elliptic operators
19.3 Classification of degenerate Monge-Ampère operators
19.3.1 Non-linear mixed-type operators
19.3.2 Linear mixed-type operators
20 Classification of symplectic MAEs on two-dimensional manifolds
20.1 Monge-Ampère equations with constant coefficients
20.1.1 Hyperbolic equations
20.1.2 Elliptic equations
20.1.3 Parabolic equations
20.2 Non-degenerate quasilinear equations
20.3 Intermediate integrals and classification
20.4 Classification of generic Monge-Ampère equations
20.4.1 Monge-Ampère equations and e-structures
20.4.2 Normal forms of mixed-type equations
20.5 Applications
20.5.1 The Born-Infeld equation
20.5.2 Gas-dynamic equations
20.5.3 Two-dimensional stationary irrotational isentropic flow of a gas
21 Contact classification of MAEs on two-dimensional manifolds
21.1 Classes H_{k,j}
21.2 Invariants of non-degenerate Monge-Ampère equations
21.2.1 Tensor invariants
21.2.2 Absolute and relative invariants
21.3 The problem of contact linearization
21.4 The problem of equivalence for non-degenerate equations
21.4.1 e-Structure for non-degenerate equations
21.4.2 Functional invariants
22 Symplectic classification of MAEs on three-dimensional manifolds
22.1 Jets of submanifolds and differential equations on submanifolds
22.2 Prolongations of contact and symplectic manifolds and overdetermined Monge-Ampère equations
22.2.1 Prolongations of symplectic manifolds
22.2.2 Prolongations of contact manifolds
22.3 Differential equations for symplectic equivalence
References
Index