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Dorfman I. Ч Dirac Structures and Integrability of Nonlinear Evolution Equations
Dorfman I. Ч Dirac Structures and Integrability of Nonlinear Evolution Equations

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Ќазвание: Dirac Structures and Integrability of Nonlinear Evolution Equations

јвтор: Dorfman I.

јннотаци€:

An introduction to the area for non-specialists with an original approach to the mathematical basis of one of the hottest research topics in nonlinear science. Deals with specific aspects of Hamiltonian theory of systems with finite or infinite dimensional phase spaces. Emphasizes systems which occur in soliton theory. Outlines current work in the Hamiltonian theory of evolution equations.


язык: en

–убрика: ћатематика/ћатематическа€ ‘изика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1993

 оличество страниц: 183

ƒобавлена в каталог: 24.04.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$\tau$-scheme for the KdV, Benjamin Ч Ono and Kadomtsev Ч Petviashvili equations      151Ч154
$\tau$-scheme in Hamiltonian framework, symmetries of Dirac structures      154Ч157
$\tau$-scheme of integrability      148Ч165
$\tau$-scheme of integrability, examples with conserved Dirac structures      157Ч159
$\tau$-scheme, Hamiltonian pairs      159Ч160
$\tau$-scheme, Lenard scheme      161Ч165
$\tau$-scheme, Lie derivatives in constructing      167
$\tau$-scheme, Liouville equation      164Ч165
$\tau$-scheme, sine-Gordon equation      163Ч164
2-cococycles      35 98Ч99
3-dihedral group of algebraic structures      137Ч139
Adler Ч Gelfand Ч Dikii method      106Ч111 113Ч114
Algebraic theory, Dirac structures      8Ч32
Belavin Ч Drinfeld solutions, Yang Ч Baxter equation      29Ч32
Benjamin Ч Ono (BO) equation      153 157
BenneyТs moment equations      102Ч104
Brackets      see УPoisson bracketsФ УSchouten
Characteristic foliation      19Ч20
Coboundary      10 14 33Ч35
Cochains      10
Cocycles      10
Cohomologies      10 12 14 65Ч68 99
Commutative symmetry algebra      148Ч154
Complex of formal variational calculus      57Ч74
Complex of formal variational calculus, cohomology group $H^{\circ}(\sqrt{}, d)$      65Ч68
Complex of formal variational calculus, construction      57Ч60
Complex of formal variational calculus, exactness problem      62Ч65
Complex of formal variational calculus, general procedure of solving $\delta f/\delta u = g$      69Ч73
Complex of formal variational calculus, invariant operations expressed in terms of Frechet derivatives      61Ч62
Complex of formal variational calculus, the equation $\delta f/\delta u = g$ in the ring of polynomials and smooth functions      68Ч69
Complexes, de Rham      8 12Ч14 18Ч19 22 57Ч60
Complexes, over Lie algebra      11Ч15
Condition for operator to be Hamiltonian      77Ч79
Conjugate operator      14Ч15
Coupled nonlinear wave equation (CNW)      89Ч97
de Rham complex      8
de Rham complex of a ring      12Ч14
de Rham complex, construction      57Ч60
de Rham complex, Dirac structures      18Ч19
de Rham complex, symplectic operators      22
Density of the functional      59
Differential-geometric type symplectic operators of      146Ч147
Dirac structures, $\tau$-scheme      154Ч159
Dirac structures, algebraic theory      8Ч32
Dirac structures, basic to Hamiltonian theory      4
Dirac structures, constituting a pair      127Ч130
Dirac structures, definition      16Ч17
Dirac structures, finite-dimensional class      18Ч19
Dirac structures, Hamiltonian operators      22Ч27
Dirac structures, Lenard scheme of integrability      48Ч51
Dirac structures, Nijenhuis relations      45Ч51
Dirac structures, Nijenhuis relations; pairs of Dirac structures      45Ч51
Dirac structures, pairs of, and Nijenhuis operators      33Ч56
Dirac structures, reduction to symplectic structures      5Ч6
Dirac structures, related to Liouville, sine-Gordon, modified KdV and nonlinear Schrodinger equations      131Ч136
Dirac structures, symplectic operators      20Ч22
Dubrovin Ч Novikov-type Hamiltonian structures      104Ч106
Evolution equations, integrability      34 6Ч7
Evolution equations, related to local Hamiltonian operators      75Ч114
Evolution equations, related to local symplectic operators      115Ч147
Exactness problem in the complex of formal variational calculus      62Ч65
FaddeevТs auxiliary conditions      5
Foliation, characteristic      19Ч20
Frechet derivatives, complex of formal variational calculus      60Ч62
Frechet derivatives, Hamiltonian conditions      77Ч79
Frechet derivatives, local symplectic operators      116Ч119
Frobenius theorem      19Ч20 31
Functional, density of the      59
Graded spaces      8Ч10
Hamiltonian operators, $\tau$-scheme      154Ч157
Hamiltonian operators, Adler Ч Gelfand Ч Dikii method of constructing, pairs      106Ч111 113Ч114
Hamiltonian operators, conditions in an explicit form      77Ч79
Hamiltonian operators, Dirac structure      22Ч27
Hamiltonian operators, Dubrovin Ч Novikov-type      104Ч106
Hamiltonian operators, first order, in the one-variable case      80Ч82
Hamiltonian operators, Kirillov Ч Kostant-type hydrodynamic structures      102Ч104
Hamiltonian operators, Korteweg Ч de Vries and Harry Dym equations      85Ч89
Hamiltonian operators, Lenard scheme for Hamiltonian and symplectic pairs      51Ч56
Hamiltonian operators, Lie derivatives in constructing Hamiltonian pairs      159Ч160
Hamiltonian operators, local, and evolution equations      75Ч114
Hamiltonian operators, pairs and associated Nijenhuis operators      42Ч48
Hamiltonian operators, third-order      82Ч85
Hamiltonian operators, upper bounds for the level of symplectic operators and      142Ч146
Hamiltonian operators, Virasoro algebra      100Ч102
Hamiltonian operators, Yang Ч Baxter equation      29Ч31
Hamiltonian theory, general algebraic scheme      8Ч32
Hamiltonian theory, introduction      1Ч7
Hamiltonian vector fields, $\tau$-scheme      154Ч157
Hamiltonian vector fields, Dirac structures      17
Hamiltonian vector fields, symplectic operators      21Ч22
Harry Dym(HD) equations      85Ч89
Hereditary or A-operators      56
Homotypy, algebraic      63Ч64
Integrability, x-scheme      148Ч165 (see also УParticular subjectsФ)
Invariant operations expressed in terms of Frechet derivatives      61Ч62
Invariants and symmetries      15Ч16
Inverse scattering method      34
Kadomtsev Ч Petviashvili(KP) equation      153Ч154 158
KdV (Korteweg-de Vries) equation, $\tau$-scheme      151Ч153 158Ч159 162Ч163
KdV (Korteweg-de Vries) equation, coupled nonlinear wave equation      89Ч97
KdV (Korteweg-de Vries) equation, Dirac structures      135
KdV (Korteweg-de Vries) equation, Hamiltonian dynamical system      5Ч6
KdV (Korteweg-de Vries) equation, Hamiltonian theory      3Ч7
KdV (Korteweg-de Vries) equation, Lenard schemes for the potential      125Ч130
KdV (Korteweg-de Vries) equation, local Hamiltonian operators      85Ч89
KdV (Korteweg-de Vries) equation, Virasoro algebra and two Hamiltonian structures of the      100Ч102
Kirillov Ч Kostant structures, Adler Ч Gelfand Ч Dikii method      106Ч111 113
Kirillov Ч Kostant structures, canonical example of a Poisson structure      4
Kirillov Ч Kostant structures, deformations of      96Ч100 113
Kirillov Ч Kostant structures, hydrodynamic, BenneyТs moment equations      102Ч104
Kirillov Ч Kostant structures, infinite-dimensional      89Ч97
Kirillov Ч Kostant structures, symplectic structure      24
Kirillov Ч Kostant structures, Virasoro algebra      100Ч102
KN equation      see УKrichever Ч Novikov(KN) equationФ
Korteweg-de Vries equation      see УKdV (Korteweg-de Vries) equationФ
KP equation      see УKadomtsev Ч Petriashvili(KP) equationФ
Krichever Ч Novikov(KN) equation, Hamiltonian presentation      6
Krichever Ч Novikov(KN) equation, Lenard scheme      122Ч125
Krichever Ч Novikov(KN) equation, pair of local symplectic operators      120Ч125 147
Lax type equations      109Ч111
Lenard scheme, $\tau$-scheme      161Ч165
Lenard scheme, applied to the Krichever Ч Novikov(KN) equation      122Ч125
Lenard scheme, for Hamiltonian and symplectic pairs      51Ч56
Lenard scheme, hierarchy of the Liouville equation      134Ч135
Lenard scheme, integrability for Dirac structures      33 48Ч51
Lenard scheme, KdV (Korteweg-de Vries) equation      93Ч94
Lenard scheme, Liouville equation      165
Lenard scheme, local Hamiltonian operators      85Ч89
Lenard scheme, sine-Gordon equation      133Ч134 163Ч164
Lenard scheme, two, for the potential KdV equation      125Ч130
Lie algebra, commutative symmetry      148Ч154
Lie algebra, complexes over      11Ч15
Lie algebra, de Rham complex      8
Lie algebra, deformations of, and Nijennuis operators      33Ч35
Lie algebra, framework for Dirac structures      6
Lie algebra, graded spaces      9Ч10
Lie algebra, Lie derivatives      15Ч16 159Ч160
Lie algebra, local Hamiltonian operators      90Ч97
Lie algebra, Poisson bracket      2Ч3
Lie algebra, skew-symmetry      2Ч3
Lie algebra, structures in the space of 1-forms      25Ч27
Lie algebra, Yang Ч Baxter equation      29Ч32
Liouville equation      3 41 134Ч135 164Ч165
Matrix differential operators, local Hamiltonian operators      75Ч77
Matrix differential operators, symplecticity conditions      115Ч117
Matrix differential operators, theorem      101Ч102
Nijennuis operators, deformations of Lie algebras      33Ч35
Nijennuis operators, Hamiltonian pairs      42Ч48
Nijennuis operators, hierarchies of 2-forms generated by regular structures      39Ч42
Nijennuis operators, Nijenhuis torsion, conjugate to      37Ч39
Nijennuis operators, pairs of Dirac structures      33Ч56
Nijennuis operators, pairs of Dirac structures, Nijenhuis relations      45Ч51
Nijennuis operators, properties of and two Dirac structures      129Ч130
Nijennuis operators, properties of, symmetry generation      36Ч37
Nonlinear Schrodinger(NLS) equations      135Ч136
Poisson brackets, Lenard scheme of integrability      50 52
Poisson brackets, Lie algebra structure      2Ч3 17Ч18
Poisson brackets, presymplectic manifold      4Ч5
Poisson brackets, reference      114
Presymplectic manifold      4Ч5
Reduction procedure      15Ч16
references      166Ч172
Ring of polynomials, complex of formal variational calculus      65 68Ч69
Schouten bracket, the      27Ч29 90Ч92
Schrodinger (NLS) system, nonlinear      135Ч136
Sine-Gordon equation      133Ч134 163Ч164
Skew-symmetry, Hamiltonian operators      26Ч27
Skew-symmetry, Lie algebra      2Ч3
Skew-symmetry, local Hamiltonian operators      76Ч77
Skew-symmetry, local symmetric operators      116Ч120
Skew-symmetry, mapping      14
Soliton theory      3 114
Structure function      96Ч100
Superalgebra, Lie      9Ч10 28
Symmetries and invariants      15Ч16
Symmetry generation, properties of Nijenhuis operators      36Ч37
Symplectic manifold      1Ч2 4Ч5
Symplectic operators, conditions guaranteeing symplecticity of a pair      54Ч55
Symplectic operators, constituting a pair      48
Symplectic operators, differential-geometric type      146Ч147
Symplectic operators, graphs of linear operators      20Ч22
Symplectic operators, Lenard scheme for Hamiltonian and symplectic pairs      51Ч56
Symplectic operators, local, and evolution equations related to them      115Ч147
Symplectic operators, many-variable case with linear dependence on $u_j^{(i)}$      137Ч142
Symplectic operators, one-variable case: first- and third-order      117Ч120
Symplectic operators, upper bounds for the level of, and Hamiltonian operators      142Ч146
Virasoro algebra      100Ч102
Yang Ч Baxter equation, Belavin Ч Drinfeld solutions      29Ч32
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