Dirac Structures and Integrability of Nonlinear Evolution Equations

Contents ......Page 2
Preface ......Page 5
Acknowledgements ......Page 6
1 Introduction ......Page 7
2.1 Graded spaces, complexes, cohomologies ......Page 14
2.2 Complexes over Lie algebras ......Page 17
2.3 Lie derivative; symmetries and invariants; reduction procedure ......Page 21
2.4 Dirac structures ......Page 22
2.5 Symplectic operators ......Page 26
2.6 Hamiltonian operators ......Page 28
2.7 Lie algebra structures in the space of 1-forms ......Page 31
2.8 The Schouten bracket ......Page 33
2.9 The classical Yang-Baxter equation and its Belavin-Drinfeld solutions ......Page 35
2.10 Notes ......Page 37
3.1 Nijenhuis operators and deformations of Lie algebras ......Page 39
3.2 Properties of Nijenhuis operators; symmetry generation ......Page 42
3.3 Conjugate to a Nijenhuis operator; Nijenhuis torsion ......Page 43
3.4 Hierarchies of 2-forms generated by a Nijenhuis operator; regular structures ......Page 45
3.5 Hamiltonian pairs and associated Nijenhuis operators ......Page 48
3.6 Nijenhuis relations; pairs of Dirac structures ......Page 51
3.7 Lenard scheme of integrability for Dirac structures ......Page 54
3.8 Applications; Lenard scheme for Hamiltonian and symplectic pairs ......Page 57
3.9 Conditions guaranteeing symplecticity of a pair ......Page 60
3.10 Notes ......Page 61
4.1 Construction of the complex ......Page 63
4.2 Invariant operations expressed in terms of Frechet derivatives ......Page 67
4.3 Exactness problem; dependence on the choice of the basic ring ......Page 68
4.4 Cohomology group H^0(\Omega, d) ......Page 71
4.5 The equation \delta f / \delta u = g in the rings of polynomials and smooth functions ......Page 74
4.6 General procedure of solving the equation \delta f / \delta u = g ......Page 75
4.7 Notes ......Page 79
5.1 Matrix differential operators ......Page 81
5.2 Hamiltonian conditions in an explicit form ......Page 83
5.3 First-order Hamiltonian operators in the one-variable case ......Page 86
5.4 Third-order Hamiltonian operators ......Page 88
5.5 The family of equations of Korteweg-de Vries and Harry Dym type ......Page 91
5.6 Infinite-dimensional Kirillov-Kostant structures, KdV equation and coupled nonlinear wave equation ......Page 95
5.7 Structure functions; shift of the argument and deformations of Kirillov-Kostant structures ......Page 102
5.8 The Virasoro algebra and two Hamiltonian structures of the KdV equation; generalizations to multi-variable case ......Page 106
5.9 Kirillov-Kostant-type hydrodynamic structures; Benney's moment equations ......Page 108
5.10 Dubrovin-Novikov-type Hamiltonian structures ......Page 110
5.11 The Adler-Gelfand-Dikii method of constructing Hamiltonian pairs ......Page 112
5.12 Some other local Hamiltonian operators ......Page 117
5.13 Notes ......Page 119
6.1 Symplecticity conditions in an explicit form ......Page 121
6.2 One-variable case: first- and third-order symplectic operators ......Page 123
6.3 A pair of local symplectic operators for the Krichever-Novikov equation ......Page 126
6.4 Lenard scheme for the Krichever-Novikov equation ......Page 129
6.5 Two distinct Lenard schemes for the potential KdV equation ......Page 131
6.6 Dirac structures related to Liouville, sine-Gordon, modified KdV and nonlinear Schrodinger equations ......Page 137
6.7 Many-variable case with linear dependence on u_j^{(i)} ......Page 143
6.8 Upper bounds for the level of symplectic and Hamiltonian operators ......Page 148
6.9 Symplectic operators of differential-geometric type ......Page 152
6.10 Notes ......Page 153
7.1 An alternative scheme to generate commuting symmetries ......Page 154
7.2 T-Scheme for the KdV, Benjamin-Ono and Kadomtsev-Petviashvili equations ......Page 157
7.3 T-Scheme in Hamiltonian framework; symmetries of Dirac structures ......Page 160
7.4 Examples of T-Schemes with conserved Dirac structures ......Page 163
7.5 Lie derivatives in constructing Hamiltonian pairs ......Page 165
7.6 Simultaneous action of the Lenard scheme and the T-scheme ......Page 167
7.7 Notes ......Page 171
References ......Page 172
Index ......Page 179