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