<<

. 16
( 16)



exactly the same way as before, leading to an expression for F8 :
36 : solve equation(13);
equ(13) breaks into equ(18), . . . , equ(20) by u2 , u3 , u4 , u5 , u6 , . . .
366 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

37 : print equations(18, 19);
equ(18) := 2(’(F8 )u1 ,2 + 3F7 )$
Functions occurring :
F7 (t)
F8 (u1 , u, t, x)

equ(19) := 2(F12 )t ’ 4(F8 )u,u1 u1 ’ 4(F8 )u1 ,x + (F7 )t2 x + 2(F7 )t u
+ 2(F7 )t u1 x + 4F12 u1 + 6F7 uu1 $
Functions occurring :
F7 (t)
F8 (u1 , u, t, x)
F12 (t)
38 : solve equation(18);
equ(18) : Inhomogeneous integration of (F8 )u2
1


39 : print equations(19, 19);
equ(19) := ’4(F14 )u u1 ’ 4(F14 )x + 2(F12 )t + (F7 )t2 x + 2(F7 )t u
+ 2(F7 )t u1 x + 4F12 u1 + 6F7 uu1 $
Functions occurring :
F7 (t)
F12 (t)
F14 (u, t, x)

40 : solve equation(19);
equ(19) breaks into equ(21), . . . , equ(22) by u1 , u2 , u3 , u4 , u5 , . . .

41 : equ(21);
2(’2(F14 )u + (F7 )t x + 2F12 + 3F7 u)$

42 : solve equation(21);
equ(21) : Inhomogeneous integration of (F14 )u

43 : print equations(22, 22);
equ(22) := ’4(F15 )x + 2(F12 )t + (F7 )t2 x$
Functions occurring :
F7 (t)
F12 (t)
F15 (t, x)
3. OVERDETERMINED SYSTEMS OF PDE 367

44 : solve equation(22);
equ(22) : Inhomogeneous integration of (F15 )x

45 : f(8) := f(8);
F8 := (4(F12 )t u1 x + (F7 )t2 u1 x2 + 4(F7 )t uu1 x + 8F16 u1 + 8F13
+ 8F12 uu1 + 6F7 u2 u1 + 12F7 u2 )/8$ (8.82)
1

while F1 and the remaining equation, equ(20), are given as:
46 : f(1) := f(1);
F1 := (4(F12 )t u1 x + (F7 )t2 u1 x2 + 4(F7 )t uu1 x + 4(F7 )t u2 x + 8F16 u1
+ 8F13 + 8F12 uu1 + 8F12 u2 + 6F7 u2 u1 + 12F7 uu2
+ 12F7 u2 + 8F7 u3 )/8$
1

47 : print equations(1, te);
equ(20) := (8(F16 )t u1 + 8(F13 )t ’ 16(F13 )u,x u1 ’ 8(F13 )u2 u2
1
’ 8(F13 )x2 ’ 8(F13 )x u + 4(F12 )t2 u1 x + 4(F12 )t uu1 + (F7 )t3 u1 x2
+ 2(F7 )t2 uu1 x ’ 2(F7 )t2 u1 + 2(F7 )t u2 u1 + 4(F7 )t u2 ’ 8F13 u1 )/4$
1
Functions occurring :
F7 (t)
F12 (t)
F13 (u, t, x)
F16 (t) (8.83)
The remaining equation can then be solved in a straightforward way leading
to the ¬nal result:
48 : solve equation(20);
equ(20) breaks into equ(23), . . . , equ(25) by u1 , u2 , u3 , u4 , u5 , . . .

49 : equ(23);
4(’2(F13 )u2 + (F7 )t )$

50 : solve equation(23);
equ(23) : Inhomogeneous integration of (F13 )u2

51 : f(13) := f(13);
F13 := ((F7 )t u2 + 4F18 u + 4F17 )/4$

52 : print equations(24, 24);
equ(24) := ’16(F18 )x + 8(F16 )t + 4(F12 )t2 x + 4(F12 )t u + (F7 )t3 x2
+ 2(F7 )t2 ux ’ 2(F7 )t2 ’ 8F18 u ’ 8F17 $
368 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

Functions occurring :
F17 (t, x)
F7 (t)
F12 (t)
F16 (t)
F18 (t, x)

53 : solve equation(24);
equ(24) breaks into equ(26), . . . , equ(27) by u, u1 , u2 , u3 , u4 , . . .

54 : print equations(26, 27);
equ(26) := 2(2(F12 )t + (F7 )t2 x ’ 4F18 )$
Functions occurring :
F18 (t, x)
F7 (t)
F12 (t)

equ(27) := ’16(F18 )x + 8(F16 )t + 4(F12 )t2 x + (F7 )t3 x2
+ 2(F7 )t2 ’ 8F17 $
Functions occurring :
F17 (t, x)
F7 (t)
F12 (t)
F16 (t)
F18 (t, x)

55 : solve equation(26);
equ(26) : Solved for F18

56 : solve equation(27);
equ(27) : Solved forF17

57 : print equations(1, te);
equ(25) := 8(F16 )t2 + 4(F12 )t3 x
+ (F7 )t4 x2 ’ 8(F7 )t3 $
Functions occurring :
F7 (t)
F12 (t)
F16 (t) (8.84)
3. OVERDETERMINED SYSTEMS OF PDE 369

and
58 : solve equation(25);
equ(25) breaks into equ(28), . . . , equ(30) by x, u, u1 , u2 , u3 , . . .

59 : print equations(28, 30);
equ(28) := (F7 )t4 $
Functions occurring :
F7 (t)

equ(29) := 4(F12 )t3 $
Functions occurring :
F12 (t)

equ(30) := 8((F16 )t2 ’ (F7 )t3 )$
Functions occurring :
F7 (t)
F16 (t)

60 : solve equation(28);
equ(28) : Homogeneous integration of (F7 )t4

61 : f(7) := f(7);
F7 := c(4)t3 + c(3)t2 + c(2)t + c(1)$

62 : solve equation(29);
equ(29) : Homogeneous integration of (F12 )t3

63 : f(12) := f(12);
F12 := c(7)t2 + c(6)t + c(5)$

64 : equ(30) := equ(30);
equ(30) := 8(’6c(4) + (F16 )t2 )$

65 : solve equation(30);
equ(30) : Inhomogeneous integration of (F16 )t2

66 : f(16) := f(16);
F16 := c(9)t + c(8) + 3c(4)t2 $

67 : factor t, x; (8.85)
and
68 : f(1) := f(1);
370 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

F1 := (t3 c(4)(3u2 u1 + 6uu2 + 6u2 + 4u3 )
1
+ 6t2 xc(4)(uu1 + u2 )
+ t2 (4c(7)uu1 + 4c(7)u2 + 3c(4)u2 + 12c(4)u1 + 3c(3)u2 u1
+ 6c(3)uu2 + 6c(3)u2 + 4c(3)u3 )
1
+ 3tx2 c(4)u1
+ 2tx(2c(7)u1 + 3c(4)u + 2c(3)uu1 + 2c(3)u2 )
+ t(4c(9)u1 + 4c(7)u + 4c(6)uu1 + 4c(6)u2 + 6c(4)
+ 2c(3)u2 + 3c(2)u2 u1 + 6c(2)uu2 + 6c(2)u2 + 4c(2)u3 )
1
+ x2 (3c(4) + c(3)u1 )
+ 2x(2c(7) + c(6)u1 + c(3)u + c(2)uu1 + c(2)u2 )
+ 4c(9) + 4c(8)u1 + 2c(6)u + 4c(5)uu1 + 4c(5)u2 ’ 6c(3)
+ c(2)u2 + 3c(1)u2 u1 + 6c(1)uu2 + 6c(1)u2 + 4c(1)u3 )/4$ (8.86)
1


and

69 : for i := 1 : 9 do write vec(i) := df(f(1), c(i));
vec(1) := (3u2 u1 + 6uu2 + 6u2 + 4u3 )/4$
1

vec(2) := (t(3u2 u1 + 6uu2 + 6u2 + 4u3 ) + 2x(uu1 + u2 ) + u2 )/4$
1

vec(3) := (t2 (3u2 u1 + 6uu2 + 6u2 + 4u3 )
1
+ 4tx(uu1 + u2 ) + 2tu2 + x2 u1 + 2xu ’ 6)/4$

vec(4) := (t3 (3u2 u1 + 6uu2 + 6u2 + 4u3 )
1
+ 6t2 x(uu1 + u2 ) + 3t2 (u2 + 4u1 ) + 3tx2 u1 + 6txu + 6t + 3x2 )/4$

vec(5) := uu1 + u2 $

vec(6) := (2t(uu1 + u2 ) + xu1 + u)/2$

vec(7) := t2 (uu1 + u2 ) + txu1 + tu + x$

vec(8) := u1 $

vec(9) := tu1 + 1$
70 : (8.87)

The previous application demonstrates in a nice way how calculations con-
cerning symmetries and other invariants of partial di¬erential equations are
performed.
3. OVERDETERMINED SYSTEMS OF PDE 371

We ¬nish this section with the remark that it is possible to run the
program automatically on this system (8.66). Doing this, the complete con-
struction does take 0.3 seconds. Most problems need however the researcher
as operator in the construction of the general solution.
3.3. Polynomial and graded cases. A very often arising situation
is the construction of symmetries and of conservation laws for equations
admitting scaling symmetry.
Let us take for example:
Example 8.6. The KdV equation is given by:
ut = uux + uxxx , (8.88)
which as we have seen in Section 5 of Chapter 3 admits a scaling symmetry
‚ ‚ ‚
S = ’x ’ 3t + 2u + ··· (8.89)
‚x ‚t ‚u
This means that in physical terms all variables are of appropriate dimen-
sions, whereas in mathematical terms it means that all variables are graded 2 ,
i.e.,
degree(x) ≡ [x] = ’1, [t] = ’3, [u] = 2, [ux ] = 3, [ut ] = 5, . . . . (8.90)
This grading means that all objects are graded too, and for the generat-
ing functions of symmetries and conservation laws only those functions are
of interest which are of a speci¬ed degree in the variables.
Example 8.7. Suppose that in the previous example we are interested
to have the most general functions F and G of degree 5 and 7 respectively,
with respect to the graded variables u, ux , uxx , uxxx , uxxxx , uxxxxx which
are of degree 2, 3, 4, 5, 6, 7 respectively. The result will be:
G = c3 uxxxxx + c4 uuxxx + c5 ux uxx + c6 u2 ux .
F = c1 uxxx + c2 uux ,
(8.91)
If, however, we are in the situation that F is of degree 5 with respect to the
graded variables p1 , u, ux , uxx , uxxx , uxxxx , uxxxxx which are of degree 1, 2,
3, 4, 5, 6, 7 respectively, then the result will be:
F = c1 uxxx + c2 p1 uxx + (c3 u + c4 p2 )ux + c5 p3 u + c6 p5 , (8.92)
1 1 1

while for G we have the general presentation
G = c1 uxxxxx + c2 p1 uxxxx + (c3 u + c4 p2 )uxxx
1
+ (c5 ux + c6 p1 u + c7 p3 )uxx + c8 p1 u2
1 x
+ (c9 u2 + c10 p2 u + c11 p4 )ux + c12 p1 u3
1 1
+ c13 p3 u2 + c14 p5 u + c15 p7 . (8.93)
1 1 1

2
The term graded here means that some weights can be assigned to all variables in
such a way that the equation becomes homogeneous with respect to these weights.
372 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

Procedures are availabe to construct the most general presentation of
a function of a speci¬ed degree, with respect to a speci¬ed list of graded
variables.
Once one knows that all objects are graded, the conditions (1.37) do lead
to polynomial equations with respect to the jet variables, the coe¬cients
of which have to vanish. This process does lead to just algebraic linear
equations for the constants in the original expressions (8.92) and (8.93).
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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
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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