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The material of the book is arranged as follows.
In Chapter 1 we deal with spaces of ¬nite jets and partial di¬erential
equations as their submanifold. The Cartan distribution on J k (π) is intro-
duced and it maximal integral manifolds are described. We describe auto-
morphisms of this distribution (Lie“B¨cklund transformations) and derive
a
de¬ning relations for classical symmetries. As applications, we consider clas-
sical symmetries of the Burgers equation, of the nonlinear di¬usion equation
(and obtain the so-called group classi¬cation in this case), of the nonlinear
Dirac equation, and of the self-dual Yang“Mills equations. For the latter,
we get monopole and instanton solutions as invariant solutions with respect
to the symmetries obtained.
Chapter 2 is dedicated to higher symmetries and conservation laws. Ba-
sic structures on in¬nite prolongations are described, including the Cartan
connection and the structural element of a nonlinear equation. In the con-
text of conservation laws, we brie¬‚y expose the results of A. Vinogradov
on the C-spectral sequence [102]. We give here a complete description for
higher symmetries of the Burgers equation, the Hilbert“Cartan equation,
and the classical Boussinesq equation.
PREFACE xv

In Chapter 3 we describe the nonlocal theory. The notion of a covering
is introduced, the relation between coverings and conservation laws is dis-
cussed. We reproduce here quite important results by N. Khor kova [43] on
the reconstruction of nonlocal symmetries by their shadows. Several appli-
cations are considered in this chapter: nonlocal symmetries of the Burgers
and KdV equation, symmetries of the massive Thirring model and symme-
tries of the Federbush model. In the last case, we also discuss Hamiltonian
structures for this model and demonstrate the existence of in¬nite number
of hierarchies of symmetries. We ¬nish this chapter with an interpretation
of B¨cklund transformations in terms of coverings and discuss a de¬nition
a
of recursion operators as B¨cklund transformations belonging to M. Marvan
a
[73].
Chapter 4 starts the central topic of the book: algebraic calculus of form-
valued derivations. After introduction of some general concepts (linear dif-
ferential operators over commutative algebras, algebraic jets and di¬erential
forms), we de¬ne basic constructions of Fr¨licher“Nijenhuis and Richardson“
o
Nijenhuis brackets [17, 78] and analyze their properties. We show that to
any integrable derivation X with values in one-forms, i.e., satisfying the
condition [[X, X]]fn = 0, a complex can be associated and investigate main
properties of the corresponding cohomology group. A source of examples
for integrable elements is provided by algebras with ¬‚at connections. These
algebras can be considered as a model for in¬nitely prolonged di¬erential
equation. Within this model, we introduce algebraic counterparts for the
notions of a symmetry and a recursion operator and prove some results
describing the symmetry algebra structure in the case when the second co-
homology group vanishes. In particular, we show that in this case in¬nite
series of commuting symmetries arise provided the model possesses a non-
trivial recursion operator.
Chapter 5 can be considered as a speci¬cation of the results obtained
in Chapter 4 to the case of partial di¬erential equations, i.e., the algebra
in question is the smooth function algebra on E ∞ while the ¬‚at connection
is the Cartan connection. The cohomology groups arising in this case are
C-cohomology of E. Using spectral sequence techniques, we give a com-
plete description of the C-cohomology for the “empty” equation, that is for
the spaces J ∞ (π) and show that elements of the corresponding cohomol-
ogy groups can be understood as graded evolutionary derivations (or vector
¬elds) on J ∞ (π). We also establish relations between C-cohomology and
deformations of the equation structure and show that in¬nitesimal defor-
1,0
mations of a certain kind (elements of HC (E), see above) are identi¬ed
with recursion operators for symmetries. After deriving de¬ning equations
for these operators, we demonstrate that in the case of several classical
systems (the Burgers equation, KdV, the nonlinear Schr¨dinger and Boussi-
o
nesq equations) the results obtained coincide with the well-known recursion
xvi PREFACE

operators. We also investigate the equation of isometric immersions of two-
dimensional Riemannian surfaces into R3 (a particular case of the Gauss“
Mainardi“Codazzi equations, which we call the Sym equation) and prove its
complete integrability, i.e., construct a recursion operator and in¬nite series
of symmetries.
Chapter 6 is a generalization of the preceding material to the graded
case (or, in physical terms, to the supersymmetric case). We rede¬ne all
necessary algebraic construction for graded commutative algebras and in-
troduce the notion of a graded extension of a partial di¬erential equation. It
is shown that all geometrical constructions valid for classical equations can
be applied, with natural modi¬cations, to graded extensions as well. We
describe an approach to the construction of graded extensions and consider
several illustrative examples (graded extensions of the KdV and modi¬ed
KdV equations and supersymmetric extensions of the nonlinear Schr¨dinger
o
equation).
Chapter 7 continues the topics started in the preceding chapter. We
consider here two supersymmetric extensions of the KdV equations (one-
and two-dimensional), new extensions of the nonlinear Schr¨dinger equation,
o
and the supersymmetric Boussinesq equation. In all applications, recursion
operators are constructed and new in¬nite series of symmetries, both local
and nonlocal, are described.
Finally, in Chapter 8 we brie¬‚y describe the software used for
computations described in the book and without which no serious ap-
plication could be obtained.


Our collaboration started in 1991. It could not be successful without
support of several organizations among which:
• the University of Twente,
• NWO (Nederlandse Organisatie voor Wetenschappelijk Onderzoek),
• FOM (Fundamenteel Onderzoek der Materie / Samenwerkingsver-
band Mathematische Fysica),
• INTAS (International Association for the promotion of co-operation
with scientists from the New Independent States of the former Soviet
Union).
We are also grateful to Kluwer Academic Publishers and especially to Pro-
fessor Michiel Hazewinkel for the opportunity to publish this book.

Joseph Krasil shchik and Paul Kersten,
Moscow“Enschede
CHAPTER 1


Classical symmetries

This chapter is concerned with the basic notions needed for our exposi-
tion ” those of jet spaces and of nonlinear di¬erential equations. Our main
purpose is to put the concept of a nonlinear partial di¬erential equation
(PDE) into the framework of smooth manifolds and then to apply powerful
techniques of di¬erential geometry and commutative algebra. We completely
abandon analytical language, maybe good enough for theorems of existence,
but not too useful in search for main underlying structures.
We describe the geometry of jet spaces and di¬erential equations (its
geometry is determined by the Cartan distribution) and introduce classical
symmetries of PDE. Our exposition is based on the books [60, 12]. We also
discuss several examples of symmetry computations for some equations of
mathematical physics.

1. Jet spaces
We expose here main facts concerning the geometrical approach to jets
(¬nite and in¬nite) and to nonlinear di¬erential operators.

1.1. Finite jets. Traditional approach to di¬erential equations consists
in treating them as expressions of the form
‚u ‚u
F x1 , . . . , xn , ,..., , . . . = 0, (1.1)
‚x1 ‚xn
where x1 , . . . , xn are independent variables, while u = u(x1 , . . . , xn ) is an
unknown function (dependent variable). Such an equation is called scalar,
but one can consider equations of the form (1.1) with F = (F 1 , . . . , F r )
and u = (u1 , . . . , um ) being vector-functions. Then we speak of systems of
PDE. What makes expression (1.1) a di¬erential equation is the presence of
partial derivatives ‚u/‚x1 , . . . in it, and our ¬rst step is to clarify this fact
in geometrical terms.
To do it, we shall restrict ourselves to the situation when all func-
tions are smooth (i.e., of the C ∞ -class) and note that a vector-function
u = (u1 , . . . , um ) can be considered as a section of the trivial bundle
1m : Rm — Rn = Rn+m ’ Rn . Denote Rm — Rn by J 0 (n, m) and con-
n
sider the graph of this section, i.e., the set “u ‚ J 0 (n, m) consisting of the
points
(x1 , . . . , xn , u1 (x1 , . . . , xn ), . . . , um (x1 , . . . , xn ) ,
1
2 1. CLASSICAL SYMMETRIES

which is an n-dimensional submanifold in Rn+m .
Let x = (x1 , . . . , xn ) be a point of Rn and θ = (x, u(x)) be the corre-
sponding point lying on “u . Then the tangent plane to “u passing through
the point θ is completely determined by x and by partial derivatives of u at
the point x. It is easy to see that the set of such planes forms an mn-di-
mensional space Rmn with coordinates, say, uj , i = 1, . . . , n, j = 1, . . . , m,
i
j
where ui “corresponds” to the partial derivative of the function uj with
respect to xi at x.
Maintaining this construction at every point θ ∈ J 0 (n, m), we obtain
def
the bundle J 1 (n, m) = Rmn — J 0 (n, m) ’ J 0 (n, m). Consider a point
θ1 ∈ J 1 (n, m). By doing this, we, in fact, ¬x the following data: values
of independent variables, x, values of dependent ones, uj , and values of
all their partial derivatives at x. Assume now that a smooth submanifold
E ‚ J 1 (n, m) is given. This submanifold determines “relations between
points” of J 1 (n, m). Taking into account the above given interpretation of
these points, we see that E may be understood as a system of relations on
unknowns uj and their partial derivatives. Thus, E is a ¬rst-order di¬erential
equation! (Or a system of such equations.)
With this example at hand, we pass now to a general construction.
Let M be an n-dimensional smooth manifold and π : E ’ M be a
smooth m-dimensional vector bundle1 over M . Denote by “(π) the C ∞ (M )-
module of sections of the bundle π. For any point x ∈ M we shall also
consider the module “loc (π; x) of all local sections at x.

Remark 1.1. We say that • is a local section of π at x, if it is de¬ned on
a neighborhood U of x (the domain of •). To be exact, • is a section of the
pull-back — π = π |U , where : U ’ M is the natural embedding. If •, • ∈
“loc (π; x) are two local sections with the domains U and U respectively,
then their sum • + • is de¬ned over U © U . For any function f ∈ C ∞ (M )
we can also de¬ne the local section f • over U.

For a section • ∈ “loc (π; x), •(x) = θ ∈ E, consider its graph “• ‚ E
and all sections • ∈ “loc (π; x) such that
(a) •(x) = • (x);
(b) the graph “• is tangent to “• with order k at θ.
It is easy to see that conditions (a) and (b) determine an equivalence relation
∼k on “loc (π; x) and we denote the equivalence class of • by [•]k . The
x x
k becomes an R-vector space, if we put
quotient set “loc (π; x)/ ∼x

[•]k + [ψ]k = [• + ψ]k , a[•]k = [a•]k , •, ψ ∈ “loc (π; x), a ∈ R, (1.2)
x x x x x


1
In fact, all constructions below can be carried out ” with natural modi¬cations
” for an arbitrary locally trivial bundle π (and even in more general settings). But we
restrict ourselves to the vector case for clearness of exposition.
1. JET SPACES 3

while the natural projection “loc (π; x) ’ “loc (π; x)/ ∼k becomes a linear
x
k (π). Obviously, J 0 (π) coincides with
map. We denote this space by Jx x
Ex = π ’1 (x), the ¬ber of the bundle π over the point x ∈ M .
Remark 1.2. The tangency class [•]k is completely determined by the
x
point x and partial derivatives up to order k at x of the section •. From
k
here it follows that Jx (π) is ¬nite-dimensional. It is easy to compute the
dimension of this space: the number of di¬erent partial derivatives of order
i equals n+i’1 and thus
n’1
k
n+i’1 n+k
k
dim Jx (π) =m =m . (1.3)
n’1 k
i=0

Definition 1.1. The element [•]k ∈ Jx (π) is called the k-jet of the
k
x
section • ∈ “loc (π; x) at the point x.
The k-jet of • can be identi¬ed with the k-th order Taylor expansion of
the section •. From the de¬nition it follows that it is independent of coor-
dinate choice (in contrast to the notion of partial derivative, which depends
on local coordinates).
Let us consider now the set
J k (π) = k
Jx (π) (1.4)
x∈M

and introduce a smooth manifold structure on J k (π) in the following way.
Let {U± }± be an atlas in M such that the bundle π becomes trivial over each
U± , i.e., π ’1 (U± ) U± — V , where V is the “typical ¬ber”. Choose a ba-
± , . . . , e± of local sections of π over U . Then any section of π |
sis e1 U±
±
m
1 e± + · · · + um e± and the functions
is representable in the form • = u 1 m
1 , . . . , um , where x , . . . , x are local coordinates in U , con-
x1 , . . . , xn , u 1 n ±
’1 (U ). Let us de¬ne the functions
stitute a local coordinate system in π ±
m: k (π) ’ R, where σ = (σ , . . . σ ), |σ| = σ + · · · + σ ¤ k, by
uσ x∈U± Jx 1 n 1 n

‚ |σ| uj
def
uj [•]k = , (1.5)
σ x
‚xσ
x
def
‚xσ = (‚x1 )σ1 . . . (‚xn )σn . Then these functions, together with local coor-
dinates x1 , . . . , xn , de¬ne the mapping f± : x∈U± Jx (π) ’ U± — RN , where
k

N is the number de¬ned by (1.3). Due to computation rules for partial
derivatives under coordinate transformations, the mapping
’1
: (U± © Uβ ) — RN ’ (U± © Uβ ) — RN
(f± —¦ fβ ) U± ©Uβ

is a di¬eomorphism preserving the natural projection (U± © Uβ ) — Rn ’
(U± © Uβ ). Thus we have proved the following result:
Proposition 1.1. The set J k (π) de¬ned by (1.4) is a smooth manifold
while the projection πk : J k (π) ’ M , πk : [•]k ’ x, is a smooth vector
x
bundle.
4 1. CLASSICAL SYMMETRIES

Note that linear structure in the ¬bers of πk is given by (1.2).
Definition 1.2. Let π : E ’ M be a smooth vector bundle, dim M =
n, dim E = n + m.
(i) The manifold J k (π) is called the manifold of k-jets for π;
(ii) The bundle πk : J k (π) ’ M is called the bundle of k-jets for π;
(iii) The above constructed coordinates {xi , uj }, where i = 1, . . . , n, j =
σ
1, . . . , m, |σ| ¤ k, are called the special (or adapted ) coordinate system
on J k (π) associated to the trivialization {U± }± of the bundle π.
Obviously, the bundle π0 coincides with π.
Note that tangency of two manifolds with order k implies tangency with
less order, i.e., there exists a mapping πk,l : J k (π) ’ J l (π), [•]k ’ [•]l , k ≥
x x
l. From this remark and from the de¬nitions we obtain the commutative
diagram
πk,l
J k (π) ’ J l (π)
πk



l,s
π
,s







J s (π)
πk




πl




πs







M
where k ≥ l ≥ s and all arrows are smooth ¬ber bundles. In other words,
we have
πl,s —¦ πk,l = πk,s , πl —¦ πk,l = πk , k ≥ l ≥ s. (1.6)
On the other hand, for any section • ∈ “(π) (or ∈ “loc (π; x)) we can de¬ne
the mapping jk (•) : M ’ J k (π) by setting jk (•) : x ’ [•]k . Obviously,
x
jk (•) ∈ “(πk ) (respectively, jk (•) ∈ “loc (πk ; x)).
Definition 1.3. The section jk (•) is called the k-jet of the section •.
The correspondence jk : “(π) ’ “(πk ) is called the operator of k-jet.
From the de¬nition it follows that
πk,l —¦ jk (•) = jl (•), k ≥ l,
j0 (•) = •, (1.7)
for any • ∈ “(π).
Let •, ψ ∈ “(π) be two sections, x ∈ M and •(x) = ψ(x) = θ ∈ E. It is
a tautology to say that the manifolds “• and “ψ are tangent to each other
with order k + l at θ or that the manifolds “jk (•) , “jk (ψ) ‚ J k (π) are tangent
with order l at the point θk = jk (•)(x) = jk (ψ)(x).
Definition 1.4. Let θk ∈ J k (π). An R-plane at θk is an n-dimensional
plane tangent to some manifold of the form “jk (•) such that [•]k = θk .
x
1. JET SPACES 5

Immediately from de¬nitions we obtain the following result.
Proposition 1.2. Let θk ∈ J k (π) be a point in a jet space. Then the
¬ber of the bundle πk+1,k : J k+1 (π) ’ J k (π) over θk coincides with the set
of all R-planes at θk .
For a point θk+1 ∈ J k+1 (π), we shall denote the corresponding R-plane
at θk = πk+1,k (θk+1 ) by Lθk+1 ‚ Tθk (J k (π)).

1.2. Nonlinear di¬erential operators. Since J k (π) is a smooth
manifold, we can consider the algebra of smooth functions on J k (π). De-
note this algebra by Fk (π). Take another vector bundle π : E ’ M and
— —
consider the pull-back πk (π ). Then the set of sections of πk (π ) is a mod-
ule over Fk (π) and we denote this module by Fk (π, π ). In particular,
Fk (π) = Fk (π, 1M ), where 1M is the trivial one-dimensional bundle over
M.
def
The surjections πk,l and πk generate the natural embeddings νk,l =
def
— —
πk,l : Fl (π, π ) ’ Fk (π, π ) and νk = πk : “(π ) ’ Fk (π, π ). Due to (1.6),
we have the equalities
νk,l —¦ νl,s = νk,s , νk,l —¦ νl = νk , k ≥ l ≥ s. (1.8)
Identifying Fl (π, π ) with its image in Fk (π, π ) under νk,l , we can consider
Fk (π, π ) as a ¬ltered module,
“(π ) ’ F0 (π, π ) ’ . . . ’ Fk’1 (π, π ) ’ Fk (π, π ), (1.9)
over the ¬ltered algebra
C ∞ (M ) ’ F0 (π) ’ . . . ’ Fk’1 (π) ’ Fk (π). (1.10)
Let F ∈ Fk (π, π ). Then we have the correspondence
def
∆(•) = jk (•)— (F ),
∆ = ∆F : “(π) ’ “(π ), • ∈ “(π). (1.11)
Definition 1.5. A correspondence ∆ of the form (1.11) is called a (non-
linear ) di¬erential operator of order2 ¤ k acting from the bundle π to
the bundle π . In particular, when ∆(f • + gψ) = f ∆(•) + g∆(ψ) for all
•, ψ ∈ “(π) and f, g ∈ C ∞ (M ), the operator ∆ is said to be linear.
From (1.9) it follows that operators ∆ of order k are also operators of
all orders k ≥ k, while (1.8) shows that the action of ∆ does not depend on
the order assigned to this operator.
Example 1.1. Let us show that the k-jet operator jk : “(π) ’ “(πk )
(see De¬nition 1.3) is di¬erential. To do this, recall that the total space of

the pull-back πk (πk ) consists of points (θk , θk ) ∈ J k (π) — J k (π) such that

2
For the sake of briefness, we shall use the words operator of order k below as a
synonym of the expression operator of order ¤ k.
6 1. CLASSICAL SYMMETRIES

πk (θk ) = πk (θk ). Consequently, we may de¬ne the diagonal section ρk of
def

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