Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations

397 Pages·2000·1.757 MB·English

by  I.S. Krasil'shchik, P.H. Kersten

Symmetries and Recursion
Operators for Classical and
Supersymmetric
Difierential Equations
by
I.S. Krasil’shchik
Independent University of Moscow
and Moscow Institute of Municipal Economy,
Moscow, Russia
and
P.H.M. Kersten
Faculty of Mathematical Sciences,
University of Twente,
Enschede, The Netherlands
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Contents
Preface xi
Chapter 1. Classical symmetries 1
1. Jet spaces 1
1.1. Finite jets 1
1.2. Nonlinear difierential operators 5
1.3. Inflnite jets 7
2. Nonlinear PDE 12
2.1. Equations and solutions 12
2.2. The Cartan distributions 16
2.3. Symmetries 21
2.4. Prolongations 28
3. Symmetries of the Burgers equation 30
4. Symmetries of the nonlinear difiusion equation 34
4.1. Case 1: p = 0, k = 0 35
4.2. Case 2: p = 0, k = 0, q = 1 35
6
4.3. Case 3: p = 0, k = 0, q = 1 36
6 6
4.4. Case 4: p = 4=5, k = 0 36
¡
4.5. Case 5: p = 4=5, p = 0, k = 0 36
6 ¡ 6
4.6. Case 6: p = 4=5, k = 0, q = 1 36
¡ 6
4.7. Case 7: p = 0, p = 4=5, k = 0, q = 1 37
6 6 ¡ 6
4.8. Case 8: p = 0, p = 4=5, q = p+1 37
6 6 ¡
4.9. Case 9: p = 0, p = 4=5, q = 1, q = p+1 37
6 6 ¡ 6 6
5. The nonlinear Dirac equations 37
5.1. Case 1: † = 0, ‚¡1 = 0 39
5.2. Case 2: † = 0, ‚¡1 = 0 43
6
5.3. Case 3: † = 0, ‚¡1 = 0 43
6
5.4. Case 4: † = 0, ‚¡1 = 0 43
6 6
6. Symmetries of the self-dual SU(2) Yang{Mills equations 43
6.1. Self-dual SU(2) Yang{Mills equations 43
6.2. Classical symmetries of self-dual Yang{Mills equations 46
6.3. Instanton solutions 49
6.4. Classical symmetries for static gauge flelds 51
6.5. Monopole solution 52
Chapter 2. Higher symmetries and conservation laws 57
1. Basic structures 57
v
vi CONTENTS
1.1. Calculus 57
1.2. Cartan distribution 59
1.3. Cartan connection 61
1.4. -difierential operators 63
C
2. Higher symmetries and conservation laws 67
2.1. Symmetries 67
2.2. Conservation laws 72
3. The Burgers equation 80
3.1. Deflning equations 80
3.2. Higher order terms 81
3.3. Estimating Jacobi brackets 82
3.4. Low order symmetries 83
3.5. Action of low order symmetries 83
3.6. Final description 83
4. The Hilbert{Cartan equation 84
4.1. Classical symmetries 85
4.2. Higher symmetries 87
4.3. Special cases 91
5. The classical Boussinesq equation 93
Chapter 3. Nonlocal theory 99
1. Coverings 99
2. Nonlocal symmetries and shadows 103
3. Reconstruction theorems 105
4. Nonlocal symmetries of the Burgers equation 109
5. Nonlocal symmetries of the KDV equation 111
6. Symmetries of the massive Thirring model 115
6.1. Higher symmetries 116
6.2. Nonlocal symmetries 120
6.2.1. Construction of nonlocal symmetries 121
6.2.2. Action of nonlocal symmetries 124
7. Symmetries of the Federbush model 129
7.1. Classical symmetries 129
7.2. First and second order higher symmetries 130
7.3. Recursion symmetries 135
7.4. Discrete symmetries 138
7.5. Towards inflnite number of hierarchies of symmetries 138
7.5.1. Construction of Y+(2;0) and Y+(2;0) 139
7.5.2. Hamiltonian structures 140
7.5.3. The inflnity of the hierarchies 144
7.6. Nonlocal symmetries 146
8. Ba˜cklund transformations and recursion operators 149
Chapter 4. Brackets 155
1. Difierential calculus over commutative algebras 155
1.1. Linear difierential operators 155
CONTENTS vii
1.2. Jets 159
1.3. Derivations 160
1.4. Forms 164
1.5. Smooth algebras 168
2. Fro˜licher{Nijenhuis bracket 171
2.1. Calculus in form-valued derivations 171
2.2. Algebras with (cid:176)at connections and cohomology 176
3. Structure of symmetry algebras 181
3.1. Recursion operators and structure of symmetry algebras 182
3.2. Concluding remarks 184
Chapter 5. Deformations and recursion operators 187
1. -cohomologies of partial difierential equations 187
C
2. Spectral sequences and graded evolutionary derivations 196
3. -cohomologies of evolution equations 208
C
4. From deformations to recursion operators 217
5. Deformations of the Burgers equation 221
6. Deformations of the KdV equation 227
7. Deformations of the nonlinear Schro˜dinger equation 231
8. Deformations of the classical Boussinesq equation 233
9. Symmetries and recursion for the Sym equation 235
9.1. Symmetries 235
9.2. Conservation laws and nonlocal symmetries 239
9.3. Recursion operator for symmetries 241
Chapter 6. Super and graded theories 243
1. Graded calculus 243
1.1. Graded polyderivations and forms 243
1.2. Wedge products 245
1.3. Contractions and graded Richardson{Nijenhuis bracket 246
1.4. De Rham complex and Lie derivatives 248
1.5. Graded Fro˜licher{Nijenhuis bracket 249
2. Graded extensions 251
2.1. General construction 251
2.2. Connections 252
2.3. Graded extensions of difierential equations 253
2.4. The structural element and -cohomologies 253
C
2.5. Vertical subtheory 255
2.6. Symmetries and deformations 256
2.7. Recursion operators 257
2.8. Commutativity theorem 260
3. Nonlocal theory and the case of evolution equations 261
3.1. The GDE(M) category 262
3.2. Local representation 262
3.3. Evolution equations 264
3.4. Nonlocal setting and shadows 265
viii CONTENTS
3.5. The functors K and T 267
3.6. Reconstructing shadows 268
4. The Kupershmidt super KdV equation 270
4.1. Higher symmetries 271
4.2. A nonlocal symmetry 273
5. The Kupershmidt super mKdV equation 275
5.1. Higher symmetries 276
5.2. A nonlocal symmetry 278
6. Supersymmetric KdV equation 280
6.1. Higher symmetries 281
6.2. Nonlocal symmetries and conserved quantities 282
7. Supersymmetric mKdV equation 290
8. Supersymmetric extensions of the NLS 293
8.1. Construction of supersymmetric extensions 293
8.2. Symmetries and conserved quantities 297
8.2.1. Case A 297
8.2.2. Case B 303
9. Concluding remarks 307
Chapter 7. Deformations of supersymmetric equations 309
1. Supersymmetric KdV equation 309
1.1. Nonlocal variables 309
1.2. Symmetries 310
1.3. Deformations 312
1.4. Passing from deformations to \classical" recursion operators 313
2. Supersymmetric extensions of the NLS equation 315
2.1. Case A 316
2.2. Case B 318
3. Supersymmetric Boussinesq equation 320
3.1. Construction of supersymmetric extensions 320
3.2. Construction of conserved quantities and nonlocal variables 321
3.3. Symmetries 322
3.4. Deformation and recursion operator 323
4. Supersymmetric extensions of the KdV equation, N = 2 324
4.1. Case a = 2 325
¡
4.1.1. Conservation laws 326
4.1.2. Higher and nonlocal symmetries 328
4.1.3. Recursion operator 330
4.2. Case a = 4 331
4.2.1. Conservation laws 331
4.2.2. Higher and nonlocal symmetries 334
4.2.3. Recursion operator 335
4.3. Case a = 1 337
4.3.1. Conservation laws 337
4.3.2. Higher and nonlocal symmetries 341
CONTENTS ix
4.3.3. Recursion operator 347
Chapter 8. Symbolic computations in difierential geometry 349
1. Super (graded) calculus 350
2. Classical difierential geometry 355
3. Overdetermined systems of PDE 356
3.1. General case 357
3.2. The Burgers equation 360
3.3. Polynomial and graded cases 371
Bibliography 373
Index 379
x CONTENTS
Preface
To our wives, Masha and Marian
Interesttotheso-calledcompletelyintegrablesystemswithinflnitenum-
ber of degrees of freedom aroused immediately after publication of the fa-
mous series of papers by Gardner, Greene, Kruskal, Miura, and Zabusky
[75, 77, 96, 18, 66, 19] (see also [76]) on striking properties of the
Korteweg{de Vries (KdV) equation. It soon became clear that systems of
such a kind possess a number of characteristic properties, such as inflnite
series of symmetries and/or conservation laws, inverse scattering problem
formulation, L A pair representation, existence of prolongation structures,
¡
etc. And though no satisfactory deflnition of complete integrability was yet
invented,aneedoftestingaparticularsystemforthesepropertiesappeared.
Probably, one of the most e–cient tests of this kind was flrst proposed
by Lenard [19] who constructed a recursion operator for symmetries of the
KdVequation. Itwasastrangeoperator, inasense: beingformallyintegro-
difierential, its action on the flrst classical symmetry (x-translation) is well-
deflned and produces the entire series of higher KdV equations. But applied
to the scaling symmetry, it gave expressions containing terms of the type
udxwhichhadnoadequateinterpretationintheframeworkoftheexisting
theories. And it is not surprising that P. Olver wrote \The deduction of the
R
form of the recursion operator (if it exists) requires a certain amount of in-
spired guesswork..." [80, p. 315]: one can hardly expect e–cient algorithms
in the world of rather fuzzy deflnitions, if any.
In some sense, our book deals with the problem of how to construct
a well-deflned concept of a recursion operator and use this deflnition for
particular computations. As it happened, a flnal solution can be explicated
in the framework of the following conceptual scheme.
We start with a smooth manifold M (a space of independent variables)
and a smooth locally trivial vector bundle …: E M whose sections play
!
the role of dependent variables (unknown functions). A partial difierential
equation in the bundle … is a smooth submanifold in the space Jk(…) of k-
E
jetsof…. Anysuchasubmanifoldiscanonicallyendowedwithadistribution,
the Cartan distribution. Being in general nonintegrable, this distribution
possesses difierent types of maximal integral manifolds a particular case of
which are (generalized) solutions of . Thus we can deflne geometry of the
E
xi
xii PREFACE
equation as geometry related to the corresponding Cartan distribution.
E
Automorphisms of this geometry are classical symmetries of .
E
Dealing with geometry of difierential equations in the above sense, one
soon flnds that a number of natural constructions arising in this context is
in fact a flnite part of more general objects existing on difierential conse-
quences of the initial equation. This leads to introduction of prolongations
l of and, in the limit, of the inflnite prolongation 1 as a submanifold
E E E
of the manifold J1(…) of inflnite jets. Using algebraic language mainly,
all flnite-dimensional constructions are carried over both to J1(…) and 1
E
and, surprisinglyatflrstglance, becomethereevenmoresimpleandelegant.
In particular, the Cartan distribution on 1 becomes completely integrable
E
(i.e., satisfles the conditions of the Frobenius theorem). Nontrivial symme-
tries of this distribution are called higher symmetries of .
E
Moreover, the Cartan distribution on 1 is in fact the horizontal dis-
E
tribution of a certain (cid:176)at connection in the bundle 1 M (the Cartan
C E !
connection) and the connection form of contains all vital geometrical in-
C
formation about the equation . We call this form the structural element of
E
and it is a form-valued derivation of the smooth function algebra on 1.
E E
A natural thing to ask is what are deformations of the structural element
(or, of the equation structure on ). At least two interesting things are
E
found when one answers this question.
The flrst one is that the deformation theory of equation structures is
closely related to a cohomological theory based on the Fro˜licher{Nijenhuis
bracket construction in the module of form-valued derivations. Namely, if
we denote by D ⁄i( ) the module of derivations with values in i-forms, the
1
E
Fro˜licher{Nijenhuis bracket acts in the following way:
[[ ; ]]fn: D ⁄i( ) D ⁄j( ) D ⁄i+j( ):
1 1 1
¢ ¢ E £ E ! E
In particular, for any element › D ⁄1( ) we obtain an operator
1
2 E
@ : D ⁄i( ) D ⁄i+1( )
› 1 1
E ! E
deflned by the formula @ (£) = [[›;£]]fn for any £ D ⁄i( ). Since
› 1
D ⁄⁄( ) = 1 D ⁄i( ) is a graded Lie algebra w2ith respeEct to the
1 E i=1 1 E
Fro˜licher{Nijenhuis bracket and due to the graded Jacobi identity, one can
see that the eLquality @ @ = 0 is equivalent to [[›;›]]fn = 0. The last
› ›
–
equality holds, if › is a connection form of a (cid:176)at connection. Thus, any
(cid:176)at connection generates a cohomology theory. In particular, natural co-
homology groups are related to the Cartan connection and we call them
-cohomology and denote by Hi( ).
C C E
Werestrictourselvestothe verticalsubtheoryof thiscohomologicalthe-
ory. Withinthisrestriction, itcanbeprovedthatthegroupH0( )coincides
C E
with the Lie algebra of higher symmetries of the equation while H1( )
E C E
consists of the equivalence classes of inflnitesimal deformations of the equa-
tion structure on . It is also a common fact in cohomological deformation
E
theory [20] that the group H2( ) contains obstructions to continuation of
C E
PREFACE xiii
inflnitesimal deformations up to formal ones. For partial difierential equa-
tions, triviality of this group is, roughly speaking, the reason for existence
of commuting series of higher symmetries.
The second interesting and even more important thing in our context
is that the contraction operation deflned in D ⁄⁄( ) is inherited by the
1
E
groups Hi( ). In particular, the group H1( ) is an associative algebra
C E C E
with respect to this operation while contraction with elements of H0( )
C E
is a representation of this algebra. In efiect, having a nontrivial element
H1( ) and a symmetry s H0( ) we are able to obtain a whole
R 2 C E 0 2 C E
inflnite series s = ns of new higher symmetries. This is just what is
n 0
R
expected of recursion operators!
Unfortunately (or, perhaps, luckily) a straightforward computation of
the flrst -cohomology groups for known completely integrable equations
C
(the KdV equation, for example) leads to trivial results only, which is not
surprising at all. In fact, normally recursion operators for nonlinear inte-
grable systems contain integral (nonlocal) terms which cannot appear when
one works using the language of inflnite jets and inflnite prolongations only.
The setting can be extended by introduction of new entities | nonlocal
variables. Geometrically, this is being done by means of the concept of a
covering. A covering over 1 is a flber bundle ¿: W 1 such that the
total space W is endowedEwith and integrable distrib!utioEn ~ and the dif-
ferential ¿ isomorphically projects any plane of the distribuCtion ~ to the
⁄
C
corresponding plane of the Cartan distribution on 1. Coordinates along
C E
the flbers of ¿ depend on coordinates in 1 in an integro-difierential way
E
and are called nonlocal.
Geometry of coverings is described in the same terms as geometry of
inflnite prolongations, and we can introduce the notions of symmetries of
W (called nonlocal symmetries of ), the structural element, -cohomology,
E C
etc. For a given equation , we can choose an appropriate covering and may
E
be lucky to extend the group H1( ). For example, for the KdV equation it
C E
su–ces to add the nonlocal variable u = udx, where u is the unknown
¡1
function, and to obtain the classical Lenard recursion operator as an ele-
R
ment of the extended -cohomology group. The same efiect one sees for the
C
Burgers equation. For other integrable systems such coverings may be (and
usually are) more complicated.
Toflnishthisshortreview, letusmakesomecommentsonhowrecursion
operators can be e–ciently computed. To this end, note that the module
D( )ofvectorfleldson 1 splitsintothedirectsumD( ) = Dv( ) D( ),
E E E E 'C E
where Dv( ) are …-vertical flelds and D( ) consists of vector flelds lying in
E C E
theCartandistribution. Thissplittinginducesthedualone: ⁄( ) = ⁄1( )
E h E '
⁄1( ). Elements of ⁄1( ) are called horizontal forms while elements of
C E h E
⁄1( ) are called Cartan forms (they vanish on the Cartan distribution).
C E
By consequence, we have the splitting ⁄i( ) = p⁄( ) ⁄q( ),
E p+q=iC E › E
L