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of a Lie ¬eld, 25 higher symmetries, 297, 304, 318,
of a Lie transformation, 24 320
of a linear di¬erential operator, 63 nonlocal symmetries, 297, 304, 318
linear coverings, 100 recursion operators, 318, 320
linear di¬erential equation, 12 recursion operators, 231
linear di¬erential operator, 5, 156 nonlinear di¬erential equation, 12
graded, 245 formally integrable, 30
over J ∞ (π), 11 -normal, 75
linear recursion operators, 150 local equivalence, 22
-normal equation, 75, 211 regular, 17
INDEX 383

nonlinear di¬erential operator, 5 recursion operators, 149, 150, 181, 260
for supersymmetric extensions of the
over a mapping, 10
Boussinesq equation, 325
nonlinear di¬usion equation, 34
for supersymmetric extensions of the
classical symmetries, 35“37
KdV equation, 332, 337
nonlinear Dirac equation
for supersymmetric extensions of the
classical symmetries, 39, 42, 43
NLS equation, 318, 320
nonlinear Dirac equations
for the Boussinesq equation, 233
conservation laws, 77
for the Burgers equation, 221
nonlinear Schr¨dinger equation, see NLS
o
for the KdV equation, 113, 227
equation
for the mKdV equation, 152
nonlocal coordinates, 102
for the NLS equation, 231
nonlocal symmetries, 104
for the supersymmetric KdV equation,
•-symmetry, 104
315
of supersymmetric extensions of the
for the Sym equation, 241
Boussinesq equation, 324
invariant, 183
of supersymmetric extensions of the
inversion, 151
KdV equation, 330, 336, 343
linear, 150
of supersymmetric extensions of the
recursion symmetries
NLS equation, 318, 320
for supersymmetric extensions of the
of the Burgers equation, 109
KdV equation, 348
of the Federbush model, 146
of the Boussinesq equation, 96
of the KdV equation, 111
of the Federbush model, 135
of the Kupershmidt super KdV equa-
of the massive Thirring model, 128
tion, 274
reducible coverings, 101
of the Kupershmidt super mKdV
regular equation, 17
equation, 279
regular point, 17
of the massive Thirring model, 120,
121, 124 relation equations, 218, 267
of the supersymmetric KdV equation, representative morphism, 6
283, 312 Richardson“Nijenhuis bracket, 174
of the supersymmetric mKdV equa- graded, 247
tion, 291 R-plane, 4
of the supersymmetric NLS equation,
self-dual gauge ¬eld, 45
297, 304
self-dual Yang“Mills equations, 45
of the Sym equation, 238
shadow, 151, 267
nonlocal variables, 264
of a nonlocal symmetry, 105
nontrivial symmetry
of recursion operators, 219
of a graded extension, 256
of the Cartan distribution on J ∞ (pi), shadow equations, 218, 267
smooth algebra, 168
68
smooth bundle over J ∞ (π), 8
one-line theorem, 74 smooth functions
on E ∞ , 58
operator of k-jet, 4
on J ∞ (π), 10
point of J ∞ (π), 8 smooth mapping of J ∞ (π), 8
point symmetries, 34 solution of a di¬erential equation, 13
Poisson bracket, 170 solve equation procedure, 358
prolongation space of in¬nite jets, 7
of a di¬erential equation, 28, 29 special coordinate system, 4
in J ∞ (π), 8
in¬nite, 57
of a di¬erential operator, 6 spectral sequence associated to a connec-
Diff-prolongation, 158 tion, 180
Jet-prolongation, 160 structural element, 192, 253
384 INDEX

of a covering, 217 universal cocompositon operation, 160
universal composition transformation,
of an equation, 62
structure of sym(E ∞ ), 72 158
universal linearization operator, 71, 210
structure of sym(π), 69
unshu¬„e, 156
structure of Lie ¬elds, 26
structure of Lie transformations, 24
variational bicomplex, 65, 74
structure of maximal integral manifolds
associated to a connection, 180
of Cartan distribution, 21
vector ¬elds, 358
super evolutionary derivation, 206
graded, 354
supersymmetric extensions
on E ∞ , 58
of the Boussinesq equation, 322
on J ∞ (π), 10
of the KdV equation, 281, 311
vectors in involution, 19
(N = 2), 326, 327, 333, 339
vertical derivation, 179
of the mKdV equation, 291
V-spectral sequence, 203
of the NLS equation, 294, 317
Sym equation, 235 wedge product of polyderivations, 162
conservation laws, 238 Whitney product of coverings, 101
deformations, 241
higher symmetries, 235 Yang“Mills equations, 43, 75
nonlocal symmetries, 238 Belavin“Polyakov“Schwartz“Tyupkin
recursion operators, 241 instanton, 50
symbol of an operator, 170 classical symmetries, 46, 51
symmetries, 22, 72, 181 instanton solutions, 45, 49
bosonic, 281 invariant solutions, 49
classical, 22, 25 monopole solutions, 45, 52, 55
discrete, 138 self-dual, 45
fermionic, 281 ™t Hooft instanton, 51
gauge, 24
higher, 68
nonlocal, 104
of an object of the category DM∞ ,
103
of the Cartan distribution on J ∞ (π),
68
point, 34
recursion, 96, 128, 135

tangent vector to J ∞ (π), 8
te variable, 359
™t Hooft instanton, 51
total derivatives, 26
total di¬erential operators, see C-di¬er-
ential operators
trivial covering, 101, 267
trivial deformations, 193
trivial symmetry of the Cartan distribu-
tion on J ∞ (π), 68
two-line theorem, 75
2-trivial object, 182
type of maximal integral manifold of the
Cartan distribution, 20
type of maximal involutive subspace, 20

universal Abelian covering, 106

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