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