.

0 (hodnocen0 x )

BK

1st pub.

Cambridge : Cambridge University Press, 2007

xxi, 496 s. : il. ; 24 cm

ISBN 0-521-82476-1 (váz.)

ISBN 978-0-521-82476-7 (váz.)

Encyclopedia of mathematics and its applications ; 101

Obsahuje bibliografii na s. 487-492 a rejstřík

000235835

Preface XIII // Part I Symmetries and Integrals 1 // 1 Distributions 3 // 1.1 Distributions and integral manifolds 3 // 1.1.1 Distributions 3 // 1.1.2 Morphisms of distributions 4 // 1.1.3 Integral manifolds 3 // 1.2 Symmetries of distributions 11 // 1.3 Characteristic and shuffling symmetries 15 // 1.4 Curvature of a distribution 18 // 1.5 Flat distributions and the Frobenius theorem 20 // 1.6 Complex distributions on real manifolds 23 // 1.7 The Lie-Bianchi theorem 24 // 1.7.1 The Maurer-Cartan equations 24 // 1.7.2 Distributions with a commutative symmetry // algebra 27 // 1.7.3 Lie-Bianchi theorem 30 // r // 2 Ordinary differential equations 32 // 2.1 Symmetries of ODEs 32 // 2.1.1 Generating functions 32 // 2.1.2 Lie algebra structure on generating functions 37 // 2.1.3 Commutative symmetry algebra 38 // 2.2 Non-linear second-order ODEs 40 // 2.2.1 Equation y" = / + F(y) 43 // 2.2.2 Integration 46 // 2.2.3 Non-linear third-order equations 48 // 2.3 Linear differential equations and linear symmetries 50 // 2.3.1 The variation of constants method 50 // 2.3.2 Linear symmetries 51 // 2.4 Linear symmetries of self-adjoint operators 54 // 2.5 Schrödinger operators 56 // 2.5.1 Integrable potentials 58 // 2.5.2 Spectral problems for KdV potentials 65 // 2.5.3 Lagrange integrals 73 // 3 Model differential equations and the // Lie superposition principle 76 // 3.1 Symmetry reduction 76 // 3.1.1 Reductions by symmetry ideals 76 // 3.1.2 Reductions by symmetry subalgebras 77 // 3.2 Model differential equations 78 // 3.2.1 One-dimensional model equations 80 // 3.2.2 Riccati equations 82 // 3.3 Model equations: the series Ak,Dk, Q 83 // 3.3.1 Series 83 // 3.3.2 Series 86 // 3.3.3 Series Ck 87 // 3.4 The Lie superposition principle 89 // 3.4.1 Bianchi equations 92 // 3.5 AP-structures and their invariants 94 // 3.5.1 Decomposition of the de Rham complex 94 //

3.5.2 Classical almost product structures 96 // 3.5.3 Almost complex structures 98 // 3.5.4 ??-structures on five-dimensional manifolds 98 // Part II Symplectic Algebra 101 // 4 Linear algebra of symplectic vector spaces 103 // 4.1 Symplectic vector spaces 103 // 4.1.1 Bilinear skew-symmetric forms on vector spaces 103 // 4.1.2 Symplectic structures on vector spaces 104 // 4.1.3 Canonical bases and coordinates 107 // 4.2 Symplectic transformations 108 // 4.2.1 Matrix representation of symplectic // transformations 110 // 4.3 Lagrangian subspaces 113 // 4.3.1 Symplectic and Kähler spaces 117 // 5 Exterior algebra on symplectic vector spaces 119 // 5.1 Operators _L and T 119 // 5.2 Effective forms and the Hodge-Lepage theorem 125 // 5.2.1 s I2-method 132 // 6 A symplectic classification of exterior 2-forms in dimension 4 135 // 6.1 Pfaffian 135 // 6.2 Normal forms 137 // 6.3 Jacobi planes 142 // 6.3.1 Classification of Jacobi planes 143 // 6.3.2 Operators associated with Jacobi planes 145 // 7 Symplectic classification of exterior 2-forms 147 // 7.1 Pfaffians and linear operators associated with 2-forms 147 // 7.2 Symplectic classification of 2-forms with distinct real characteristic numbers 149 // 7.3 Symplectic classification of 2-forms with distinct // complex characteristic numbers 152 // 7.4 Symplectic classification of 2-forms with multiple // characteristic numbers 154 // 7.5 Symplectic classification of effective 2-forms in // dimension 6 160 // 8 Classification of exterior 3-forms on a six-dimensional symplectic space 162 // 8.1 A symplectic invariant of effective 3-forms 162 // 8.1.1 The case of trivial invariants 165 // 8.1.2 The case of non-trivial invariants 167 // 8.1.3 Hitchin’s results on the geometry of // 3-forms 173 // 8.2 The stabilizers of orbits and their prolongations 175 // 8.2.1 Stabilizers 175 // 8.2.2 Prolongations 178 //

Part III Monge-Ampere Equations 181 // 9 Symplectic manifolds 1 $3 // 9.1 Symplectic structures 183 // 9.1.1 The cotangent bundle and the standard symplectic structure 184 // 9.1.2 Kähler manifolds 186 // 9.1.3 Orbits and homogeneous symplectic spaces 187 // 9.2 Vector fields on symplectic manifolds 189 // 9.2.1 Poisson bracket and Hamiltonian vector fields 189 // 9.2.2 Canonical coordinates 191 // 9.3 Submanifolds of symplectic manifolds 192 // 9.3.1 Presymplectic manifolds 192 // 9.3.2 Lagrangian submanifolds 194 // 9.3.3 Involutive submanifolds 197 // 9.3.4 Lagrangian polarizations 198 // 10 Contact manifolds 201 // 10.1 Contact structures 201 // 10.1.1 Examples 202 // 10.2 Contact transformations and contact vector fields 208 // 10.2.1 Examples 209 // 10.2.2 Contact vector fields 215 // 10.3 Darboux theorem 219 // 10.4 A local description of contact transformations 221 // 10.4.1 Generating functions of Lagrangian submanifolds 221 // 10.4.2 A description of contact transformations in M3 222 // 11 Monge-Ampčre equations 224 // 11.1 Monge-Ampere operators 224 // 11.2 Effective differential forms 226 // 11.3 Calculus on Ł2*(C*) 230 // 11.4 The Euler operator 233 // 11.5 Solutions 236 // 11.6 Monge-Ampere equations of divergent type 241 // 12 Symmetries and contact transformations of Monge-Ampčre // equations 243 // 12.1 Contact transformations 243 // 12.2 Lie equations for contact symmetries 251 // 12.3 Reduction 256 // 12.4 Examples , 259 // 12.4.1 The boundary layer equation 259 // 12.4.2 The thermal conductivity equation 261 // 12.4.3 The Petrovsky-Kolmogorov-Piskunov // equation 262 // 12.4.4 The Von Karman equation 264 // 12.5 Symmetries of the reduction 267 // 12.6 Monge-Ampčre equations in symplectic geometry 270 // 13 Conservation laws 273 // 13.1 Definition and examples 273 // 13.2 Calculus for conservation laws 274 //

13.3 Symmetries and conservations laws 279 // 13.4 Shock waves and the Hugoniot-Rankine // condition 280 // 13.4.1 Shock Waves for ODEs 280 // 13.4.2 Discontinuous solutions 281 // 13.4.3 Shockwaves 283 // 13.5 Calculus of variations and the Monge-Ampere equation 285 // 13.5.1 The Euler operator 285 // 13.5.2 Symmetries and conservation laws in variational problems 286 // 13.5.3 Classical variational problems 287 // 13.6 Effective cohomology and the Euler operator 288 // 14 Monge-Ampere equations on two-dimensional // manifolds and geometric structures 294 // 14.1 Non-holonomic geometric structures associated with // Monge-Ampčre equations 295 // 14.1.1 Non-holonomic structures on contact // manifolds 295 // 14.1.2 Non-holonomic fields of endomorphisms on generated by Monge-Ampčre // equations 295 // 14.1.3 Non-degenerate equations 298 // 14.1.4 Parabolic equations 302 // 14.2 Intermediate integrals 304 // 14.2.1 Classical and non-holonomic intermediate // integrals 304 // 14.2.2 Cauchy problem and non-holonomic // intermediate integrals 307 // 14.3 Symplectic Monge-Ampčre equations 308 // 14.3.1 A field of endomorphisms on ?*? 308 // 14.3.2 Non-degenerate symplectic equations 310 // 14.3.3 Symplectic parabolic equations 312 // 14.3.4 Intermediate integrals 313 // 14.4 Cauchy problem for hyperbolic Monge-Ampčre // equations 313 // 14.4.1 Constructive methods for integration of // Cauchy problem 314 // 15 Systems of first-order partial differential equations on // two-dimensional manifolds 318 // 15.1 Non-linear differential operators of first order on // two-dimensional manifolds 319 // 15.2 Jacobi equations 321 // 15.3 Symmetries of Jacobi equations 328 // 15.4 Geometric structures associated with // Jacobi’s equations 330 // 15.5 Conservation laws of Jacobi equations 332 // 15.6 Cauchy problem for hyperbolic Jacobi equations 334 //

Part IV Applications 337 // 16 Non-linear acoustics 339 // 16.1 Symmetries and conservation laws of the KZ equation 340 // 16.1.1 KZ equation and its contact symmetries 340 // 16.1.2 The structure of the symmetry algebra 342 // 16.1.3 Classification of one-dimensional subalgebras of // sl(2,M) 345 // 16.1.4 Classification of symmetries 347 // 16.1.5 Conservation laws 348 // 16.2 Singularities of solutions of the KZ equation 349 // 16.2.1 Caustics 349 // 16.2.2 Contact shock waves 351 // 17 Non-linear thermal conductivity 356 // 17.1 Symmetries of the TC equation 356 // 17.1.1 TC equation . 356 // 17.1.2 Group classification of TC equation 357 // 17.2 Invariant solutions 363 // 18 Meteorology applications 371 // 18.1 Shallow water theory and balanced dynamics 372 // 18.2 A geometric approach to semi-geostrophic theory 374 // 18.3 Hyper-Kähler structure and Monge-Ampere operators 376 // 18.4 Monge-Ampčre operators with constant // coefficients and plane balanced models 380 // Part V Classification of Monge-Ampčre // equations 383 // 19 Classification of symplectic MAOs on two-dimensional manifolds 385 // 19.1  -Structures 386 // 19.2 Classification of non-degenerate Monge-Ampere operators 388 // 19.2.1 Differential invariants of non-degenerate operators 388 // 19.2.2 Hyperbolic operators 392 // 19.2.3 Elliptic operators 401 // 19.3 Classification of degenerate Monge-Ampere operators 406 // 19.3.1 Non-linear mixed-type operators 406 // 19.3.2 Linear mixed-type operators 416 // 20 Classification of symplectic MAEs on two-dimensional manifolds 422 // 20.1 Monge-Ampere equations with constant coefficients 422 // 20.1.1 Hyperbolic equations 423 // 20.1.2 Elliptic equations 425 // 20.1.3 Parabolic equations 426 // 20.2 Non-degenerate quasilinear equations 428 // 20.3 Intermediate integrals and classification 429 // 20.4 Classification of generic Monge-Ampere equatiopn 430 //

20.4.1 Monge-Ampere equations and  -structures 430 // 20.4.2 Normal forms of mixed-type equations 436 // 20.5 Applications 440 // 20.5.1 The Born-Infeld equation 440 // 20.5.2 Gas-dynamic equations 442 // 20.5.3 Two-dimensional stationary irrotational // isentropic flow of a gas 445 // 21 Contact classification of MAEs on two-dimensional manifolds 447 // 21.1 Classes ... 447 // 21.2 Invariants of non-degenerate Monge-Ampčre equations 454 // 21.2.1 Tensor invariants 454 // 21.2.2 Absolute and relative invariants 456 // 21.3 The problem of contact linearization 459 // 21.4 The problem of equivalence for non-degenerate // equations 464 // 21.4.1  -Structure for non-degenerate equations 464 // 21.4.2 Functional invariants 470 // 22 Symplectic classification of MAEs on three-dimensional manifolds 472 // 22.1 Jets of submanifolds and differential equations on submanifolds 473 // 22.2 Prolongations of contact and symplectic manifolds and // overdetermined Monge-Ampere equations 476 // 22.2.1 Prolongations of symplectic manifolds 476 // 22.2.2 Prolongations of contact manifolds 479 // 22.3 Differential equations for symplectic equivalence 482 // References 487 // Index 493