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Symmetries and Recursion
Operators for Classical and
Supersymmetric
Di¬erential 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 di¬erential operators 5
1.3. In¬nite 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 di¬usion equation 34
4.1. Case 1: p = 0, k = 0 35
4.2. Case 2: p = 0, k = 0, q = 1 35
4.3. Case 3: p = 0, k = 0, q = 1 36
4.4. Case 4: p = ’4/5, k = 0 36
4.5. Case 5: p = ’4/5, p = 0, k = 0 36
4.6. Case 6: p = ’4/5, k = 0, q = 1 36
4.7. Case 7: p = 0, p = ’4/5, k = 0, q = 1 37
4.8. Case 8: p = 0, p = ’4/5, q = p + 1 37
4.9. Case 9: p = 0, p = ’4/5, q = 1, q = p + 1 37
5. The nonlinear Dirac equations 37
5.1. Case 1: = 0, »’1 = 0 39
5.2. Case 2: = 0, »’1 = 0 43
5.3. Case 3: = 0, »’1 = 0 43
5.4. Case 4: = 0, »’1 = 0 43
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 ¬elds 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. C-di¬erential operators 63
2. Higher symmetries and conservation laws 67
2.1. Symmetries 67
2.2. Conservation laws 72
3. The Burgers equation 80
3.1. De¬ning 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 in¬nite 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 in¬nity of the hierarchies 144
7.6. Nonlocal symmetries 146
8. B¨cklund transformations and recursion operators
a 149

Chapter 4. Brackets 155
1. Di¬erential calculus over commutative algebras 155
1.1. Linear di¬erential operators 155
CONTENTS vii

1.2. Jets 159
1.3. Derivations 160
1.4. Forms 164
1.5. Smooth algebras 168
2. Fr¨licher“Nijenhuis bracket
o 171
2.1. Calculus in form-valued derivations 171
2.2. Algebras with ¬‚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. C-cohomologies of partial di¬erential equations 187
2. Spectral sequences and graded evolutionary derivations 196
3. C-cohomologies of evolution equations 208
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 Schr¨dinger equation
o 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 Fr¨licher“Nijenhuis bracket
o 249
2. Graded extensions 251
2.1. General construction 251
2.2. Connections 252
2.3. Graded extensions of di¬erential equations 253
2.4. The structural element and C-cohomologies 253
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 di¬erential geometry 349
1. Super (graded) calculus 350
2. Classical di¬erential 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



Interest to the so-called completely integrable systems with in¬nite num-
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 in¬nite
series of symmetries and/or conservation laws, inverse scattering problem
formulation, L ’ A pair representation, existence of prolongation structures,
etc. And though no satisfactory de¬nition of complete integrability was yet
invented, a need of testing a particular system for these properties appeared.
Probably, one of the most e¬cient tests of this kind was ¬rst proposed
by Lenard [19] who constructed a recursion operator for symmetries of the
KdV equation. It was a strange operator, in a sense: being formally integro-
di¬erential, its action on the ¬rst classical symmetry (x-translation) is well-
de¬ned and produces the entire series of higher KdV equations. But applied
to the scaling symmetry, it gave expressions containing terms of the type
u dx which had no adequate interpretation in the framework of the existing
theories. And it is not surprising that P. Olver wrote “The deduction of the
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 de¬nitions, if any.
In some sense, our book deals with the problem of how to construct
a well-de¬ned concept of a recursion operator and use this de¬nition for
particular computations. As it happened, a ¬nal 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 π : E ’ M whose sections play
the role of dependent variables (unknown functions). A partial di¬erential
equation in the bundle π is a smooth submanifold E in the space J k (π) of k-
jets of π. Any such a submanifold is canonically endowed with a distribution,
the Cartan distribution. Being in general nonintegrable, this distribution
possesses di¬erent types of maximal integral manifolds a particular case of
which are (generalized) solutions of E. Thus we can de¬ne geometry of the
xi
xii PREFACE

equation E as geometry related to the corresponding Cartan distribution.
Automorphisms of this geometry are classical symmetries of E.
Dealing with geometry of di¬erential equations in the above sense, one
soon ¬nds that a number of natural constructions arising in this context is
in fact a ¬nite part of more general objects existing on di¬erential conse-
quences of the initial equation. This leads to introduction of prolongations
E l of E and, in the limit, of the in¬nite prolongation E ∞ as a submanifold
of the manifold J ∞ (π) of in¬nite jets. Using algebraic language mainly,
all ¬nite-dimensional constructions are carried over both to J ∞ (π) and E ∞
and, surprisingly at ¬rst glance, become there even more simple and elegant.
In particular, the Cartan distribution on E ∞ becomes completely integrable
(i.e., satis¬es the conditions of the Frobenius theorem). Nontrivial symme-
tries of this distribution are called higher symmetries of E.
Moreover, the Cartan distribution on E ∞ is in fact the horizontal dis-
tribution of a certain ¬‚at connection C in the bundle E ∞ ’ M (the Cartan
connection) and the connection form of C contains all vital geometrical in-
formation about the equation E. We call this form the structural element of
E and it is a form-valued derivation of the smooth function algebra on E ∞ .
A natural thing to ask is what are deformations of the structural element
(or, of the equation structure on E). At least two interesting things are
found when one answers this question.
The ¬rst one is that the deformation theory of equation structures is
closely related to a cohomological theory based on the Fr¨licher“Nijenhuis
o
bracket construction in the module of form-valued derivations. Namely, if
we denote by D1 Λi (E) the module of derivations with values in i-forms, the
Fr¨licher“Nijenhuis bracket acts in the following way:
o
[[·, ·]]fn : D1 Λi (E) — D1 Λj (E) ’ D1 Λi+j (E).
In particular, for any element „¦ ∈ D1 Λ1 (E) we obtain an operator
‚„¦ : D1 Λi (E) ’ D1 Λi+1 (E)
de¬ned by the formula ‚„¦ (˜) = [[„¦, ˜]]fn for any ˜ ∈ D1 Λi (E). Since

D1 Λ— (E) = i
i=1 D1 Λ (E) is a graded Lie algebra with respect to the
Fr¨licher“Nijenhuis bracket and due to the graded Jacobi identity, one can
o
see that the equality ‚„¦ —¦ ‚„¦ = 0 is equivalent to [[„¦, „¦]]fn = 0. The last
equality holds, if „¦ is a connection form of a ¬‚at connection. Thus, any
¬‚at connection generates a cohomology theory. In particular, natural co-
homology groups are related to the Cartan connection and we call them
i
C-cohomology and denote by HC (E).
We restrict ourselves to the vertical subtheory of this cohomological the-
0
ory. Within this restriction, it can be proved that the group HC (E) coincides
1
with the Lie algebra of higher symmetries of the equation E while HC (E)
consists of the equivalence classes of in¬nitesimal deformations of the equa-
tion structure on E. It is also a common fact in cohomological deformation
2
theory [20] that the group HC (E) contains obstructions to continuation of
PREFACE xiii

in¬nitesimal deformations up to formal ones. For partial di¬erential 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 de¬ned in D1 Λ— (E) is inherited by the
i 1
groups HC (E). In particular, the group HC (E) is an associative algebra
0
with respect to this operation while contraction with elements of HC (E)
is a representation of this algebra. In e¬ect, having a nontrivial element
1 0
R ∈ HC (E) and a symmetry s0 ∈ HC (E) we are able to obtain a whole
in¬nite series sn = Rn s0 of new higher symmetries. This is just what is
expected of recursion operators!
Unfortunately (or, perhaps, luckily) a straightforward computation of
the ¬rst C-cohomology groups for known completely integrable equations
(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 in¬nite jets and in¬nite 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 E ∞ is a ¬ber bundle „ : W ’ E ∞ such that the
˜
total space W is endowed with and integrable distribution C and the dif-
˜
ferential „— isomorphically projects any plane of the distribution C to the
corresponding plane of the Cartan distribution C on E ∞ . Coordinates along
the ¬bers of „ depend on coordinates in E ∞ in an integro-di¬erential way
and are called nonlocal.
Geometry of coverings is described in the same terms as geometry of
in¬nite prolongations, and we can introduce the notions of symmetries of
W (called nonlocal symmetries of E), the structural element, C-cohomology,
etc. For a given equation E, we can choose an appropriate covering and may
1
be lucky to extend the group HC (E). For example, for the KdV equation it
su¬ces to add the nonlocal variable u’1 = u dx, where u is the unknown
function, and to obtain the classical Lenard recursion operator as an ele-
ment of the extended C-cohomology group. The same e¬ect one sees for the
Burgers equation. For other integrable systems such coverings may be (and
usually are) more complicated.
To ¬nish this short review, let us make some comments on how recursion
operators can be e¬ciently computed. To this end, note that the module
D(E) of vector ¬elds on E ∞ splits into the direct sum D(E) = D v (E)•CD(E),
where D v (E) are π-vertical ¬elds and CD(E) consists of vector ¬elds lying in
the Cartan distribution. This splitting induces the dual one: Λ(E) = Λ1 (E)•
h
1 (E). Elements of Λ1 (E) are called horizontal forms while elements of
CΛ h
CΛ1 (E) are called Cartan forms (they vanish on the Cartan distribution).
By consequence, we have the splitting Λi (E) = p q
p+q=i C Λ(E) — Λ (E),
xiv PREFACE

where
Λq (E) = Λ1 (E) § · · · § Λ1 (E) .
C p Λ(E) = CΛ1 (E) § · · · § CΛ1 (E), h h
h
p times q times

This splitting generates the corresponding splitting in the groups of C-
p,q
i
cohomologies: HC (E) = p+q=i HC (E) and nontrivial recursion operators
1,0
are elements of the group HC (E).
The graded algebra C — Λ(E) = p
p≥0 C Λ(E) may be considered as the
algebra of functions on a super di¬erential equation related to the initial
equation E in a functorial way. This equation is called the Cartan (odd )
covering of E. An amazing fact is that the symmetry algebra of this covering
—,0 —,0
is isomorphic to the direct sum HC (E) • HC (E). Thus, due to the general
p,0
theory, to ¬nd an element of HC (E) we have just to take a system of forms
„¦ = (ω 1 , . . . , ω m ), where ω j ∈ C p Λ(E) and m = dim π, and to solve the
equation E ω = 0, where E is the linearization of E restricted to E ∞ . In
particular, for p = 1 we shall obtain recursion operators, and the action
of the corresponding solutions on symmetries of E is just contraction of a
symmetry with the Cartan vector-form „¦.


This scheme is exposed in details below. Though some topics can be
found in other books (see, e.g., [60, 12, 80, 5, 81, 101]; the collections [39]
and [103] also may be recommended), we included them in the text to make
the book self-contained. We also decided to include a lot of applications in
the text to make it interesting not only to those ones who deal with pure
theory.

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