deformations: a computational theory for recursion operators. Acta Appl. Math.,

41(1-3):167“191, 1995. Geometric and algebraic structures in di¬erential equations.

[60] I. S. Krasil shchik, V. V. Lychagin, and A. M. Vinogradov. Geometry of jet spaces

and nonlinear partial di¬erential equations. Gordon & Breach Science Publishers,

New York, 1986. Translated from the Russian by A. B. Sosinski˜ ±.

[61] I. S. Krasil shchik and A. M. Vinogradov. Nonlocal symmetries and the theory of

coverings: an addendum to Vinogradov™s “Local symmetries and conservation laws”

[Acta Appl. Math. 2 (1984), no. 1, 21“78. Acta Appl. Math., 2(1):79“96, 1984.

[62] I. S. Krasil shchik and A. M. Vinogradov. Nonlocal trends in the geometry of dif-

ferential equations: symmetries, conservation laws, and B¨cklund transformations.

a

Acta Appl. Math., 15(1-2):161“209, 1989. Symmetries of partial di¬erential equa-

tions, Part I.

[63] S. Krivonos, A. Pashnev, and Z. Popowicz. Lax pairs for N = 2, 3 supersymmetric

KdV equations and their extensions. Modern Phys. Lett. A, 13(18):1435“1443, 1998.

[64] S. Krivonos and A. Sorin. Extended N = 2 supersymmetric matrix (1, s)-KdV hier-

archies. Phys. Lett. A, 251(2):109“114, 1999.

[65] S. Krivonos, A. Sorin, and F. Toppan. On the super-NLS equation and its relation

with the N = 2 super-KdV equation within the coset approach. Phys. Lett. A,

206(3-4):146“152, 1995.

[66] M. D. Kruskal, R. M. Miura, C. S. Gardner, and N. J. Zabusky. Korteweg-de Vries

equation and generalizations. V. Uniqueness and nonexistence of polynomial con-

servation laws. J. Mathematical Phys., 11:952“960, 1970.

[67] B. A. Kupershmidt. Elements of superintegrable systems. D. Reidel Publishing Co.,

Dordrecht, 1987. Basic techniques and results.

[68] P. Labelle and P. Mathieu. A new N = 2 supersymmetric Korteweg-de Vries equa-

tion. J. Math. Phys., 32(4):923“927, 1991.

[69] V. V. Lychagin, V. N. Rubtsov, and I. V. Chekalov. A classi¬cation of Monge-

´

Amp`re equations. Ann. Sci. Ecole Norm. Sup. (4), 26(3):281“308, 1993.

e

[70] S. Mac Lane. Homology. Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition.

[71] B. Malgrange. Ideals of di¬erentiable functions. Tata Institute of Fundamental Re-

search, Bombay, 1967. Tata Institute of Fundamental Research Studies in Mathe-

matics, No. 3.

[72] Yu. I. Manin and A. O. Radul. A supersymmetric extension of the Kadomtsev-

Petviashvili hierarchy. Comm. Math. Phys., 98(1):65“77, 1985.

[73] M. Marvan. Another look on recursion operators. In Di¬erential geometry and ap-

plications (Brno, 1995), pages 393“402. Masaryk Univ., Brno, 1996.

[74] P. Mathieu. Supersymmetric extension of the Korteweg-de Vries equation. J. Math.

Phys., 29(11):2499“2506, 1988.

[75] R. M. Miura. Korteweg-de Vries equation and generalizations. I. A remarkable ex-

plicit nonlinear transformation. J. Mathematical Phys., 9:1202“1204, 1968.

BIBLIOGRAPHY 377

[76] R. M. Miura. The Korteweg-de Vries equation: a survey of results. SIAM Rev.,

18(3):412“459, 1976.

[77] R. M. Miura, C. S. Gardner, and M. D. Kruskal. Korteweg-de Vries equation and

generalizations. II. Existence of conservation laws and constants of motion. J. Math-

ematical Phys., 9:1204“1209, 1968.

[78] A. Nijenhuis and R. W. Richardson. Deformations of Lie algebra structures. J. Math.

Mech., 17:89“105, 1967.

[79] W. Oevel and Z. Popowicz. The bi-Hamiltonian structure of fully supersymmetric

Korteweg-de Vries systems. Comm. Math. Phys., 139(3):441“460, 1991.

[80] P. J. Olver. Applications of Lie groups to di¬erential equations. Springer-Verlag,

New York, 1986.

[81] L. V. Ovsiannikov. Group analysis of di¬erential equations. Academic Press Inc.

[Harcourt Brace Jovanovich Publishers], New York, 1982. Translated from the Rus-

sian by Y. Chapovsky, Translation edited by William F. Ames.

[82] Z. Popowicz. The Lax formulation of the “new” N = 2 SUSY KdV equation. Phys.

Lett. A, 174(5-6):411“415, 1993.

[83] M. K. Prasad. Instantons and monopoles in Yang-Mills gauge ¬eld theories. Phys.

D, 1(2):167“191, 1980.

[84] M. K. Prasad and C. M. Sommer¬eld. Exact classical solutions for the ™t Hooft

Monopole and the Julia-Zee Dyon. Phys. Rev. Lett., 35:760“762, 1975.

[85] G. H. M. Roelofs. The LIESUPER package for Reduce, Memorandum 943. Depart-

ment of Applied Mathematics, University of Twente, Enschede, The Netherlands,

1991.

[86] G. H. M. Roelofs. The INTEGRATOR package for Reduce, Memorandum 1100.

Department of Applied Mathematics, University of Twente, Enschede, The Nether-

lands, 1992.

[87] G. H. M. Roelofs. Prolongation Structures of Supersymmetric Systems, PhD Thesis.

Department of Applied Mathematics, University of Twente, Enschede, The Nether-

lands, 1993.

[88] G. H. M. Roelofs and P. H. M. Kersten. Supersymmetric extensions of the nonlinear

Schr¨dinger equation: symmetries and coverings. J. Math. Phys., 33(6):2185“2206,

o

1992.

[89] A. Roy Chowdhury and M. Naskar. On the complete integrability of the supersym-

metric nonlinear Schr¨dinger equation. J. Math. Phys., 28(8):1809“1812, 1987.

o

[90] E. Schr¨fer, F. W. Hehl, and J. D. McCrea. Exterior calculus on the computer: the

u

REDUCE-package EXCALC applied to general relativity and to the Poincar´ gauge e

theory. Gen. Relativity Gravitation, 19(2):197“218, 1987.

[91] F. Schwarz. Automatically determining symmetries of partial di¬erential equations.

Computing, 34(2):91“106, 1985.

[92] W. M. Sluis and P. H. M. Kersten. Nonlocal higher-order symmetries for the Feder-

bush model. J. Phys. A, 23(11):2195“2204, 1990.

[93] D. C. Spencer. Overdetermined systems of linear partial di¬erential equations. Bull.

Amer. Math. Soc., 75:179“239, 1969.

[94] W.-H. Steeb, W. Erig, and W. Strampp. Symmetries and the Dirac equation. J.

Math. Phys., 22(5):970“973, 1981.

[95] W.-H. Steeb, W. Erig, and W. Strampp. Similarity solutions of nonlinear Dirac

equations and conserved currents. J. Math. Phys., 23(1):145“153, 1982.

[96] C. H. Su and C. S. Gardner. Korteweg-de Vries equation and generalizations. III.

Derivation of the Korteweg-de Vries equation and Burgers equation. J. Mathematical

Phys., 10:536“539, 1969.

[97] T. Tsujishita. Homological method of computing invariants of systems of di¬erential

equations. Di¬erential Geom. Appl., 1(1):3“34, 1991.

378 BIBLIOGRAPHY

[98] A. Verbovetsky. Notes on the horizontal cohomology. In Secondary calculus and co-

homological physics (Moscow, 1997), pages 211“231. Amer. Math. Soc., Providence,

RI, 1998.

[99] A. M. Vinogradov. The algebra of logic of the theory of linear di¬erential operators.

Dokl. Akad. Nauk SSSR, 205:1025“1028, 1972.

[100] A. M. Vinogradov. Category of nonlinear di¬erential equations. In Equations on

manifolds, pages 26“51. Voronezh. Gos. Univ., Voronezh, 1982.

[101] A. M. Vinogradov. Local symmetries and conservation laws. Acta Appl. Math.,

2(1):21“78, 1984.

[102] A. M. Vinogradov. The C-spectral sequence, Lagrangian formalism, and conservation

laws. I. The linear theory. J. Math. Anal. Appl., 100(1):1“40, 1984.

[103] A. M. Vinogradov, editor. Symmetries of partial di¬erential equations. Kluwer Aca-

demic Publishers, Dordrecht, 1989. Conservation laws”applications”algorithms,

Reprint of Acta Appl. Math. 15 (1989), no. 1-2, and 16 (1989), no. 1 and no. 2.

[104] A. M. Vinogradov and I. S. Krasil shchik. What is Hamiltonian formalism? Uspehi

Mat. Nauk, 30(1(181)):173“198, 1975.

[105] A. M. Vinogradov and I. S. Krasil shchik. A method of calculating higher symmetries

of nonlinear evolutionary equations, and nonlocal symmetries. Dokl. Akad. Nauk

SSSR, 253(6):1289“1293, 1980.

[106] M. M. Vinogradov. Basic functors of di¬erential calculus in graded algebras. Uspekhi

Mat. Nauk, 44(3(267)):151“152, 1989.

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