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Index

Abelian covering, 106 Cartan plane, 17, 59
adapted coordinate system, see special Cartan submodule, 18
coordinate system category FC(A), 178
category M∞ , 9
adjoint operator, 73
annihilating operators in the Federbush category DM∞ , 99
model, 137, 143 category GDE(M ), 262
C-cohomology, 187
B¨cklund auto-transformations, 149
a
of a graded extension, 255
B¨cklund transformations, 149
a
of an equation, 192
in the category DM∞ , 149
C-di¬erential operator, 63
Belavin“Polyakov“Schwartz“Tyupkin
classical symmetries, 22, 25
instanton, 50
¬nite, 22
bosonic symmetries, 281
in¬nitesimal, 25
Boussinesq equation, 93, 233
of the Burgers equation, 33
deformations, 233
of the Federbush model, 129
graded extensions, 322
of the Hilbert“Cartan equation, 87
conservation laws, 323
of the nonlinear di¬usion equation,
coverings, 323
35“37
deformations, 325
higher symmetries, 324 of the nonlinear Dirac equation, 39,
nonlocal symmetries, 324 42, 43
recursion operators, 325 of the self-dual Yang“Mills equations,
higher symmetries, 96 46
recursion operators, 233 of the static Yang“Mills equations, 51
recursion symmetries, 96 C-natural extension, 253
bundle of k-jets, 4 coefficient check switch, 360
Burgers equation, 30, 80, 109, 215, 221, cogluing transformation, 160
362 Cole“Hopf transformation, 110
classical symmetries, 33 compatibility complex, 75
Cole“Hopf transformation, 110 compatibility conditions, 28
coverings, 109
connection, 16, 176, 252
deformations, 215, 221
connection form, 177, 187
higher symmetries, 84
conservation laws, 72
nonlocal symmetries, 109
of supersymmetric extensions of the
recursion operators, 221
Boussinesq equation, 323
of supersymmetric extensions of the
Cartan connection, 61
KdV equation, 328, 333, 339
Cartan covering
of supersymmetric extensions of the
even, 100
NLS equation, 318, 320
odd, 268
of the Dirac equations, 77
Cartan di¬erential, 66, 198
of the Federbush model, 130
Cartan distribution, 17, 59
Cartan forms, 18 of the KdV equation, 111, 227
379
380 INDEX

of the Kupershmidt super KdV equa- of the heat equation, 214
tion, 274 of the supersymmetric KdV equation,
314
of the Kupershmidt super mKdV
depl!* list, 363
equation, 279
de Rham complex
of the massive Thirring model, 121
graded, 248
of the supersymmetric KdV equation,
of an algebra, 166
283, 311
on E ∞ , 58
of the supersymmetric mKdV equa-
on J ∞ (π), 11
tion, 291
de Rham di¬erential, 6, 11, 14, 164, 166,
of the supersymmetric NLS equation,
355, 358
297, 304
graded, 248
of the Sym equation, 238
derivation, 160, 358
conserved densities, 72; see also conser-
bigraded, 356
vation laws
graded, 353
trivial, 72
di¬erential forms, 358
contact transformations, 22
graded, 244, 354
contraction, 172, 246, 356, 358
of an algebra, 164, 165
coverings, 263
on E ∞ , 58
Abelian, 106
on J ∞ (π), 10
Cartan even covering, 100
di¬erential operators of in¬nite order, 12
Cartan odd covering, 268
differentiation switch, 361
dimension, 101
Diff-prolongation, 158
equivalent, 100
dimension of a covering, 101
in the category DM∞ , 99
dimension of a graded manifold, 354
irreducible, 101
discrete symmetries of the Federbush
linear, 100
model, 138
over E ∞ , 99
distribution on J ∞ (π), 12
over supersymmetric extensions of the
KdV equation, 328, 333
equation associated to an operator, 13
over supersymmetric extensions of the
equivalent coverings, 100
NLS equation, 318
equ operator, 359
over the Burgers equation, 109
Euler“Lagrange equation, 76
over the supersymmetric KdV equa-
Euler“Lagrange operator, 74, 141
tion, 311
evolutionary equation, 16
reducible, 101
evolutionary vector ¬eld, 70
trivial, 101
exterior derivative, see de Rham di¬er-
universal Abelian, 106
ential
creating operators in the Federbush
model, 137, 143 Federbush model, 129
C-spectral sequence, 65, 202 annihilating operators, 137
curvature form, 177, 188, 252 classical symmetries, 129
conservation laws, 130
deformations Hamiltonian structures, 140
of a graded extension, 257 higher symmetries, 130, 138, 144
of an equation structure, 192 nonlocal symmetries, 146
of supersymmetric extensions of the recursion symmetries, 135
Boussinesq equation, 325 fermionic symmetries, 281
of supersymmetric extensions of the ¬nitely smooth algebra, 176
KdV equation, 332, 337, 348 ¬‚at connection, 16, 178, 252
of supersymmetric extensions of the formally integrable equation, 30
NLS equation, 318, 320 Fr´chet derivative, see Euler“Lagrange
e
of the Boussinesq equation, 233 operator
of the Burgers equation, 215 free di¬erential extension, 253
INDEX 381

Fr¨licher“Nijenhuis bracket, 175
o of the Hilbert“Cartan equation, 91“93
graded, 249 of the KdV equation, 111
of the Kupershmidt super KdV equa-
gauge coupling constant, 43 tion, 272
gauge potential, 43 of the Kupershmidt super mKdV
gauge symmetries, 24 equation, 277
gauge transformations, 52 of the massive Thirring model, 116
of the Yang“Mills equations, 49 of the supersymmetric KdV equation,
generating form, see generating function 282, 312
generating function of the supersymmetric mKdV equa-
of a conservation law, 75 tion, 291
of a contact ¬eld, 26 of the supersymmetric NLS equation,
of a graded evolutionary derivation, 297, 304
206 of the Sym equation, 235
of a Lie ¬eld, 27 Hilbert“Cartan equation, 84
of an evolutionary vector ¬eld, 70 classical symmetries, 87
generating section, see generating func- higher symmetries, 91“93
tion hodograph transformation, 23
generic point of maximal integral mani- horizontal de Rham cohomology, 65
fold of the Cartan distribution, 20 horizontal de Rham complex, 65, 198
geometrical module, 167 with coe¬cients in Cartan forms, 66
geometrization functor, 167 horizontal de Rham di¬erential, 65, 256
g-invariant solution, 28 horizontal distribution of a connection,
gluing homomorphism, 158 187
gluing transformation, 158 horizontal forms, 65, 197
graded algebra, 353 horizontal plane, 15
graded commutative algebra, 353 H-spectral sequence, 199
graded evolutionary derivation, 206
ideal of an equation, 58
graded extensions of a di¬erential equa-
in¬nite prolongation of an equation, 57
tion, 253; see also supersymmetric
in¬nitesimal deformation of a graded ex-
extensions
tension, 257
graded Jacobi identity, 353
in¬nitesimal Stokes formula, 11
graded manifold, 354
in¬nitesimal symmetries, 25
graded module, 353
inner di¬erentiation, see contraction
graded polyderivations, 244
inner product, see contraction
graded vector space, 352
instanton solutions of the Yang“Mills
Green™s formula, 73
equations, 45, 49
Hamiltonian structures of the Federbush integrable distribution on J ∞ (π), 12
model, 140 integrable element, 250
heat equation, 110, 214 integral manifold of a distribution on
deformations, 214 J ∞ (π), 12
higher Jacobi bracket, 70 INTEGRATION package, 362
graded, 207 interior symmetry, 22
higher symmetries, 68 internal coordinates, 59
of supersymmetric extensions of the invariant recursion operators, 183
Boussinesq equation, 324 invariant solutions, 27
of supersymmetric extensions of the of the Yang“Mills equations, 49
KdV equation, 330, 336, 343 invariant submanifold of a covering, 103
of supersymmetric extensions of the inversion of a recursion operator, 151
NLS equation, 318, 320 involutive subspace, 19
of the Boussinesq equation, 96 irreducible coverings, 101
of the Burgers equation, 84
of the Federbush model, 130, 144 jet of a section, 4
382 INDEX

jet of a section at a point, 3 local equivalence of di¬erential equa-
tions, 22
jet operator, 159
Jet-prolongation, 160
manifold of k-jets, 4
massive Thirring model, 115
KdV equation, 111, 150, 227, 373
conservation laws, 121
conservation laws, 111, 227
higher symmetries, 116
deformations, 227
nonlocal symmetries, 120, 121, 124
graded extensions, 271, 281, 311, 326,
recursion symmetries, 128
333, 339
maximal integral manifolds of the Car-
conservation laws, 274, 283, 311,
tan distribution
328, 333, 339
on E ∞ , 60
coverings, 311, 328, 333, 339
on J ∞ (π), 60
deformations, 314, 332, 337, 348
on J k (π), 20
higher symmetries, 272, 282, 312,
330, 336, 343 maximal involutive subspace, 19
nonlocal symmetries, 274, 283, 312, mKdV equation, 152
330, 336, 343 graded extensions, 276, 291
recursion operators, 315, 332, 337 conservation laws, 279, 291
recursion symmetries, 348 higher symmetries, 277, 291
higher symmetries, 111 nonlocal symmetries, 279, 291
nonlocal symmetries, 111 recursion operators, 152
recursion operators, 113, 227 modi¬ed Korteweg de Vries equation, see
killing functor, 268 mKdV equation
Korteweg de Vries equation, see KdV module of k-jets, 159
equation module of in¬nite jets, 159
Kupershmidt super KdV equation, 271, module of symbols, 170
271; see also graded extensions of Monge“Ampere equations, 14, 67
the KdV equation monopole solutions of the Yang“Mills
Kupershmidt super mKdV equation, equations, 45, 52, 55
276; see also graded extensions of morphism of coverings, 100
the mKdV equation
N¨ther symmetry, 76
o
Leibniz rule N¨ther theorem, 76
o
bigraded, 356 -cohomology, 179
graded, 353 -complex, 179
Lenard recursion operator, 113, 150 Nijenhuis torsion, 182
Lie algebra graded, 254
bigraded, 356 NLS equation, 231
graded, 353 deformations, 231
Lie derivative, 172, 174, 357, 358 graded extensions, 294, 317
graded, 248 conservation laws, 297, 304, 318,
Lie ¬eld, 25 320
Lie transformation, 21 coverings, 318, 320
lifting deformations, 318, 320

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