The Diffiety Institute Preprint Series

Preprint DIPS–2/98 January 26, 1998

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Symmetries and recursion operators for soliton equations

by

I.S. Krasil’shchik

Symmetries and recursion operators for soliton equations

I.S. Krasil’shchik^{∗} Moscow Institute for Municipal Economy and the Diffiety Institute ^{†}

June 4, 1998

Abstract

The theory of symmetries for nonlinear differential equations is exposed using naïve coordinate language. As an example, the Burgers equation is considered. Methods for computing recursion operators are given and nonlocal problems arising in this context are discussed.

Key words and phrases: nonlinear differential equations, symmetries, recursion operators, coverings

1991 AMS Subject Classification: 58F37.

It is known that all soliton equations possess several structures which make of them very interesting objects for mathematical investigations. These structures are:

• infinite series of commuting symmetries and conservation laws,

• Hamiltonian (and usually, bi-Hamiltonian) maps,

• recursion operators.

The first equation which was found to possess all these properties was the celebrated Korteweg-de Vries equation

u_{t} = u_{xxx} + uu_{x}.  (1)

∗This work was partially supported by the Russian Foundation for Base Research Grant N 97-01-00462 and by the INTAS Grant N 96-0793.

†For correspondence: 1st Tverskoy-Yamskoy per. 14, Apt.45, 125047 Moscow, Russia. E.mail:

josephk@glasnet.ru

It was shown, in particular, that soliton solutions of (1) were invariant (with respect to higher symmetries) solutions. Later similar results were obtained for other soliton equations.

In fact, geometrical theory of differential equations^{1} originates from classical works by S. Lie and A. Bäcklund and much later was revised and put into the modern differential geometry framework in [3, 5, 6].

In this paper we do not intend to expose the general theory of symmetries for differential equations, but try to clarify practical aspects of computations. As an example, we choose the Burgers equation

u_{t} = u_{xx} + uu_{x}  (2)

for which the whole conceptual scheme looks quite transparent. Besides symmetries, we explain how to compute recursion operators using the methods developed in [1, 2]. Recursion operators are a very powerful instrument for symmetry computation, but as a rule need the use of nonlocal variables. Hence, we discuss the latter ones as well.

1 Symmetries of differential equations.

Let us analyse the notion of a symmetry and deduce efficient formulae for computations.

What is a symmetry? Consider a differential equation^{2} E given by

F (x, u, …, \frac{∂^{|σ|}u}{∂x^{σ}}, …) = 0,  (3)

where x = (x_{1}, …, x_{n}) are independent variables, u = (u^{1}, …, u^{m}) are unknown functions (dependent variables) and \frac{∂^{|σ|}u}{∂x^{σ}} = \frac{∂^{|σ|}u}{∂x_{1}^{i_{1}}...∂x_{n}^{in}} for σ = (i_{1}, …, i_{n}), |σ| = i_{1} + … + i_{n} ≤ k, k being the order of E.

Let

\frac{∂u}{∂τ} = Φ(x, u, …, \frac{∂^{|σ|}u}{∂x^{σ}}, …)  (4)

be an evolution equation and suppose (3) and (4) to be compatible. Let u = u(x) be a solution of (3) and u_{τ} = u(x, τ) be a solution of (4) and (3) with initial data u_{0} = u. Then {u_{τ}} is a curve in the space of solutions of equation (3). Taking an arbitrary solution^{3} of (3), we get a correspondence Φ_{τ} : u ↦ u_{τ}. Hence, evolution equation (4) determines a one-parameter family of transformations on the solution space of (3). Thus we can give a preliminary Definition 1 Symmetry is an evolution equation which preserves solutions.

1Of which the theory of symmetries is a natural component.

2When we say a differential equation we in fact mean a system of equations and thus F in (3) is a vector function F = (F^{1}, …, F^{r}).

3We assume the initial value problem for (4) to be uniquely solvable.

Of course, this is not an operational definition: the concept of a solution space, as well as of an action on this space, remains rather vague. To make them exact, we should answer the next question:

What is a differential equation? Groups of transformations arise in differential geometry as maps of main objects (curves, surfaces, etc.) preserving basic structures (e.g., Riemannian metrics). Thus, if one is going to use differential geometry techniques in application to differential equations, one should put these equations into appropriate conceptual framework. This is being done as follows.

Consider the space J^{k}(n, m) with coordinates

(x_{1}, …, x_{n}, u^{1}, …, u^{m}, …, u_{σ}^{1}, …, u_{σ}^{m}, …),

where, as above, σ = (i_{1}, …, i_{n}) and |σ| ≤ k. This space is called the space of jets of the order k. Then an equation E of the form (3) can be understood as a subset in J^{k}(n, m) defined by relations

F^{1}(x, u, …, u_{σ}, …) = 0, …, F^{r}(x, u, …, u_{σ}, …) = 0.

If all F^{i} are smooth functions and the system F^{1}, …, F^{r} is of the maximal rank, then this subset is a smooth^{4} submanifold of the codimension r. Thus, differential equations are smooth submanifolds in jet spaces.

Let f = (f^{1}(x), …, f^{m}(x)) be smooth vector function. Then f determines an n-dimensional submanifold in J^{k}(n, m) defined by

u_{σ}^{j} = \frac{∂^{|σ|}f^{j}}{∂x^{σ}} , j = 1, …, m, |σ| ≤ k,

(here and below we formally assume u_{(0,...,0)} = u). This submanifold is called the k-jet of f and is denoted by j_{k}(f). It easily seen that f satisfies equation (3) if and only if j_{k}(f) lies in corresponding submanifold in J^{k}(n, m). In other words, solutions of E are identified with jets lying in the equation manifold. Hence, the following definition is justified:

Definition 2 Let E ⊂ J^{k}(n, m) be a differential equation. A symmetry of E is a diffeomorphism φ : J^{k}(n, m) → J^{k}(n, m), such that (i) φ(E) = E and (ii) φ(j_{k}(f)) is of the form j_{k}(f_{φ}) for any smooth function f.

This definition, being perfectly correct, can not be considered as a working one though: to compute symmetries one needs to know solutions, and it does not make sense!

4By smooth we always mean of class C^{∞}.

Infinite prolongations and the Cartan distribution. To overcome this problem, first note the following. Let f(x) be a solution of (3). Differentiate F^{j}(f(x)) along x_{α}:

\frac{∂F^{j}(f(x))}{∂x_{α}} =

= \frac{∂F^{j}}{∂x_{α}} (f(x)) + \frac{∂F^{j}}{∂u^{1}} (f(x))\frac{∂f^{1}}{∂x_{α}} + … + \frac{∂F^{j}}{∂u^{m}} (f(x))\frac{∂f^{m}}{∂x_{α}} + …

… + \frac{∂F^{j}}{∂u_{σ}^{1}} (f(x))\frac{∂^{|σ|+1}f^{1}}{∂x^{σ+1_{α}}} + … + \frac{∂F^{j}}{∂u_{σ}^{m}} (f(x))\frac{∂^{|σ|+1}f^{m}}{∂x^{σ+1_{α}}} + …

where σ + 1_{α} =^{def} (i_{1}, …, i_{α} + 1, …, i_{n}). Define operators

D_{α}^{[k]} = \frac{∂}{∂x_{α}} + ∑_{j=1}^{m} ∑_{|σ|≤k} u_{σ+1_{α}}^{j} \frac{∂}{∂u_{σ}^{j}}  (5)

Then one can easily see that the system

F = 0, D_{1}^{[k]}(F) = 0, …, D_{n}^{[k]}(F) = 0  (6)

possesses the same solutions as the initial equation (3). If E ⊂ J^{k}(n, m) is the submanifold corresponding to (3), then a submanifold in J^{k+1}(n, m) corresponds to (6), which we denote by E^{1} and call the first prolongation of E.

Now, we define by induction E^{l+1} = (E^{l})^{1} and see that the set E^{l} lies in J^{k+l}(n, m); we call E^{l} the l-th prolongation of E. It is reasonable to consider all prolongations of E simultaneously. What geometrical object does correspond to this construction?

Let J^{∞}(n, m) be the space with coordinates

(x_{1}, …, x_{n}, u^{1}, …, u^{m}, …, u_{σ}^{1}, …, u_{σ}^{m}, …),

where |σ| is unlimited. This space is called the space of infinite jets and the whole series of prolongations of the equation E determines a submanifold E^{∞} in J^{∞}(n, m) which is called the infinite prolongation of the equation E. Infinite prolongations are expressed by relations

D_{σ}(F) = 0, |σ| ≥ 0,

where D_{σ} = D_{1}^{i_{1}} ∘ … ∘ D_{n}^{i_{n}} and

D_{α} = \frac{∂}{∂x_{α}} + ∑_{j,σ} u_{σ+1_{α}}^{j} \frac{∂}{∂u_{σ}^{j}}  (7)

are infinite counterparts of (5) and are called total derivatives.

Expressions of the form (7) may be viewed at as vector fields on the space J^{∞}(n, m). Thus, at any point θ ∈ J^{∞}(n, m) an n-dimensional plane C_{θ} arises spanned by vectors of these fields. It means that there exists a distribution C : θ ↦ C_{θ} on J^{∞}(n, m). We call it the Cartan distribution. Its importance to the theory of differential equations is explained by the following

Theorem 1 [3] (i) If E^{∞} ⊂ J^{∞}(n, m) is the infinite prolongation, then C_{θ} is tangent to E^{∞} at any point θ ∈ E^{∞}.

A submanifold in E^{∞} is a solution of E if and only if it is an integrable manifold of the Cartan distribution.

Hence, E^{∞} carries the structure which completely determines solutions of E, and this structure is the Cartan distribution.

Definition 3 (final) A diffeomorphism φ : E^{∞} → E^{∞} is called a (higher) symmetry of E, if it preserves the Cartan distribution, i.e. if φ_{∗}(C_{θ}) = C_{φ(θ)} for any θ ∈ E^{∞}.

This definition is in a complete agreement with the general geometrical approach. The only problem consists in efficient computation of symmetries. And in fact, everything becomes efficient, if one chooses

Infinitesimal approach. Let M be a smooth manifold and D be distribution on M. Suppose that D is spanned by vector fields X_{1}, …, X_{n}. Assume that a one-parameter group of symmetries {A_{τ}} of the distribution D is given, i.e. (A_{τ})_{∗}D_{θ} = D_{A_{τ} (θ)} for any point θ ∈ M. It is equivalent to the following system of equations

(A_{τ})_{∗}(X_{α}) = λ_{α}^{1}X_{1} + … + λ_{α}^{n}X_{n}, α = 1, …, n,  (8)

where λ_{α}^{β} are depending on τ smooth functions on M with det ∥λ_{α}^{β}∥ ≠ 0. Let X be the vector field corresponding to \{A_{τ}\}, i.e. \frac{dA_{τ}}{dτ} |_{τ=0} = X. Differentiating (8) with respect to τ at τ = 0 one obtains

[X, X_{α}] = μ_{α}^{1}X_{1} + … + μ_{α}^{n}X_{n}, α = 1, …, n,  (9)

where μ_{α}^{β} are smooth functions on M.

Definition 4 A vector field X satisfying (9) is called an infinitesimal symmetry of the distribution D.

Symmetries of D form a Lie algebra with respect to commutator of vector fields: if X and Y satisfy (9), then [X, Y] satisfies it as well. Denote this algebra by s(D).

Note now that any vector field X lying in D is a symmetry of D and, by definition, is tangent to any integral manifold of this distribution. It means, if A_{τ} is a one-parameter group corresponding to X, then any transformation A_{τ} moves an integrable manifold of D along itself and thus trivially acts on the space of integral manifolds as a whole. In this sense, such symmetries are trivial. Respectively, field lying in D are trivial infinitesimal symmetries and we denote the set of these symmetries by s_{0}(D).

Obviously, s_{0}(D) is an ideal in the Lie algebra s(D): if X ∈ s(D) and Y ∈ s_{0}(D), then [X, Y] ∈ s_{0}(D). It means that one can consider the quotient Lie algebra

sym(D) =^{def} s(D)/s_{0}(D),

and its elements are naturally identified with nontrivial symmetries of D. We apply now this construction to the Cartan distribution C.

How to compute infinitesimal symmetries? Consider the space of infinite jets J^{∞}(n, m) and the infinite prolongation E^{∞} of equation E ⊂ J^{k}(n, m). Let C be the Cartan distribution on J^{∞}(n, m) and C_{E} be its restriction onto E^{∞}.

Definition 5 Elements of sym(C) are called infinitesimal symmetries of the infinite jet space and their set is denoted by sym_{n,m}. Elements of sym(C_{E}) are called infinitesimal symmetries of the equation E and their set is denoted by sym(E).

Both sym(C_{E}) and sym(E) are Lie algebras.

In the sequel we omit the adjective ”infinitesimal”, since from now on we work with infinitesimal symmetries only.

In principal, there exist two approaches to computation of symmetries of equation E. The first, exterior, one consists of finding those symmetries X ∈ sym_{n,m} which are tangent to E^{∞}. The second, interior, is the direct computation of sym(E). While the former seems to be preferable from computational viewpoint, the latter is, at first glance, more justified. But, as it happens, these approaches are equivalent.

Proposition 1 [3] Any symmetry X ∈ sym(E) is a restriction onto E of some symmetry X′ ∈ sym_{n,m}.

Thus we can start with description of sym_{n,m}.

First note the following. Any vector field on J^{∞}(m, n) is represented

X = ∑_{α} A_{α} \frac{∂}{∂x_{α}} + ∑_{β,σ} B_{σ}^{β} \frac{∂}{∂u_{σ}^{β}} ,  (10)

where A_{α}, B_{σ}^{β} are smooth functions on J^{∞}(n, m). We say that the field X is vertical, if it vanishes on functions depending on x only. Hence, vertical fields are of the form ∑_{β,σ} B_{σ}^{β} \frac{∂}{∂u_{σ}^{β}}. Consider an arbitrary field (10). Then the field X − ∑_{α} X_{α}D_{α}, where D_{α}, α = 1, …, n are total derivatives (7), is a vertical field.

Hence, the algebra D(J^{∞}(n, m)) of all vector fields on J^{∞}(n, m) splits into the direct sum

D(J^{∞}(n, m)) = D^{v}(J^{∞}(n, m)) ⊕ s_{0}(C)

D^{v}(J^{∞}(n, m)) being the algebra of vertical fields. This presentation induces a splitting of the algebra s(C) and we obtain

s(C) = sym_{n,m} ⊕s_{0}(C).  (11)

From the latter it easily follows

Proposition 2 Elements of sym_{n,m} are identified with vertical vector fields X such that

[X, D_{α}] = 0  (12)

for all α = 1, …, n.

Using this result, let us obtain an explicit description of symmetries on J^{∞}(n, m). Let X = ∑_{β,σ} B_{σ}^{β} \frac{∂}{∂u_{σ}^{β}} be a vertical field. Then

[X, D_{α}] = [∑_{β,σ} B_{σ}^{β} \frac{∂}{∂u_{σ}^{β}} , \frac{∂}{∂x_{α}} + ∑_{β,σ} u_{σ+1_{α}}^{β} \frac{∂}{∂u_{σ}^{β}} ] =

∑_{β,σ} B_{σ+1_{α}}^{β} \frac{∂}{∂u_{σ}^{β}} − ∑_{β,σ} D_{α}(B_{σ}^{β})\frac{∂}{∂u_{σ}^{β}} ,

from where one has B_{σ+1_{α}}^{β} = D_{α}(B_{σ}^{β}), or

B_{σ}^{β} = D_{σ}(B_{(0,...,0)}^{β}).

Let B^{β} =^{def} B_{(0,...,0)}^{β}.

Theorem 2 [3] Any symmetry X ∈ sym_{n,m} is of the form

� _{B} = ∑_{β,σ} D_{σ}(B^{j})\frac{∂}{∂u_{σ}^{β}} ,  (13)

where B = (B^{1}, …, B^{m}), B^{β} being arbitrary functions on J^{∞}(n, m).

Definition 6 Vector fields � _{B} are called evolution fields, B being called the generating function.

Remark 1 Suppose, a field � _{B} is integrable^{5}, i.e. possesses a trajectory in J^{∞}(n, m). Then evolution of u_{σ}^{β} along these trajectories would be governed by equations

\frac{∂u_{σ}}{∂τ} = D_{σ}(B)

5Contrary to the finite dimensional case, this is not usually true for infinite dimensional manifolds J^{∞}(n, m) or E^{∞}.

and, in particular,

\frac{∂u}{∂τ} = B(x, …, u_{σ}, …),

which explains the name ”evolution” for fields � _{B} and is in complete agreement with the naïve Definition 1!

Recall that sym_{n,m} is a Lie algebra, i.e. is closed under commutator of vector fields. Hence, for any � _{B_{1}} and � _{B_{2}} their commutator [� _{B_{2}}, � _{B_{2}}] is again an evolution field � _{B} for some new B. This function B is called the Jacobi bracket of B_{1} and B_{2} and is denoted by {B_{1}, B_{2}}. From (13) it follows that

B^{j} = ∑_{β,σ} (D_{σ}(B_{1}^{β})\frac{∂B_{2}^{j}}{∂u_{σ}^{β}} − D_{σ}(B_{2}^{β})\frac{∂B_{1}^{j}}{∂u_{σ}^{β}} ) .  (14)

Clearly, the function algebra on J^{∞}(n, m) is a Lie algebra with respect to the Jacobi bracket.

Now we restrict everything above said onto E^{∞}. Suppose, as in (3), E is given by relations F^{1} = 0, …, F^{r} = 0. Then a field X ∈ D(J^{∞}(n, m)) is tangent to E^{∞} if and only if one has

X(F^{j}) = ∑_{β,σ} a_{σ}^{jβ}D_{σ}(F^{β})

for some functions a_{σ}^{jβ}. In particular, if X = � _{B}, the last equation rewrites as

� _{B}(F^{j}) = ∑_{β,σ} a_{σ}^{jβ}D_{σ}(F^{β}),

or, if we use the notation

ℓ_{F} (B) =^{def} � _{B}(F)  (15)

as

ℓ_{F^{j}}(B) = ∑_{β,σ} a_{σ}^{jβ}D_{σ}(F^{β}).  (16)

Definition 7 The operator

ℓ_{F}^{j} = ∑_{β} \frac{∂F}{∂u_{σ}^{j}} D_{σ}, j = 1, …, m,  (17)

is called the operator of universal linearization.

Thus, � _{B} is a symmetry of E if and only if B satisfies equation (16).

Restrict (16) onto E^{∞}. Then the right-hand side vanishes by definition, and we obtain the following important result.

Theorem 3 [3] Let E ⊂ J^{k}(n, m)be a differential equation given by relations F^{1} = 0, …, F^{r} = 0 and � _{B} be an evolution vector field. Denote by B̄ restriction of B onto E^{∞}. Then � _{B} is a symmetry of E if and only if

ℓ_{E}(B̄) = 0,  (18)

where ℓ_{E} is the restriction of ℓ_{F} = (ℓ_{F^{1}}, …, ℓ_{F^{r}}) onto E^{∞}. In other words,

sym(E) = ker(ℓ_{E}).

Note that solutions of (18) are closed under the Jacobi bracket restricted onto E^{∞}.

Equation (18) is the basis for direct computations of symmetries. We illustrate its use in the next section.

2 Example: the Burgers equation.

Consider the Burgers equation B

u_{t} = u_{xx} + uu_{x}  (19)

and choose internal coordinates on B^{∞} by setting u_{k} = u_{(k,0)}. Below we use the method described in [3].

Defining equations. Rewrite restrictions onto B^{∞} of all basic concepts in these coordinate system.

For the total derivatives we obviously obtain

D_{x} = \frac{∂}{∂x} + ∑_{k=0}^{∞} u_{i+1} \frac{∂}{∂u_{i}} ,  (20)
D_{t} = \frac{∂}{∂t} + ∑_{k=0}^{∞} D_{x}^{i}(u_{3} + u_{0}u_{1})\frac{∂}{∂u_{i}} .  (21)

The operator of universal linearization for B is then looks as

ℓ_{B} = D_{t} − u_{1} − u_{0}D_{x} − D_{x}^{2}  (22)

and, as it follows from Theorem 3, evolution field

� _{φ} = ∑_{i=1}^{∞} D_{x}^{i}(φ)\frac{∂}{∂u_{i}} ,  (23)

is a symmetry for B if and only if φ = φ(x, t, u_{0}, …, u_{k}) satisfies the equation

D_{t}φ = u_{1}φ + u_{0}D_{x}φ + D_{x}^{2}φ,  (24)

where D_{t}, D_{x} are given by (20), (21). Computing D_{x}^{2}φ we obtain

D_{x}^{2}φ = \frac{∂^{2}φ}{∂x^{2}} + 2 ∑_{i=1}^{k} u_{i+1} \frac{∂^{2}φ}{∂x∂u_{i}} + ∑_{i,j=0}^{k} u_{i+1}u_{j+1} \frac{∂^{2}φ}{∂u_{i}∂u_{j}} + ∑_{i=0}^{k} u_{i+2} \frac{∂φ}{∂u_{i}} ,

while

D_{x}^{i}(u_{0}u_{1} + u_{3}) = ∑_{α=0}^{i} \binom{i}{α}u_{α}u_{i−α+1} + u_{i+3}.

Hence, (24) transforms to

\frac{∂φ}{∂t} + ∑_{i=1}^{k} ∑_{α=1}^{i} \binom{i}{α}u_{α}u_{i−α+1} \frac{∂φ}{∂u_{i}} =

u_{1}φ + u_{0} \frac{∂φ}{∂x} + \frac{∂^{2}φ}{∂x^{2}} + 2 ∑_{i=1}^{k} u_{i+1} \frac{∂^{2}φ}{∂x∂u_{i}} + ∑_{i,j=0}^{k} u_{i+1}u_{j+1} \frac{∂^{2}φ}{∂u_{i}∂u_{j}} .  (25)

Higher order terms. Note now that the left-hand side of (25) is independent

of u_{k+1} while the right-hand one is quadratic in this variable and is of the form

u_{k+1}^{2} \frac{∂^{2}φ}{∂u_{k}^{2}} + 2u_{k+1} (\frac{∂^{2}φ}{∂x∂u_{k}} + ∑_{i=0}^{k−1} u_{i+1} \frac{∂^{2}φ}{∂u_{i}∂u_{k}} ) .

It means that

φ = Au_{k} + ψ,  (26)

where A = A(t) and ψ = ψ(t, x, u_{0}, …, u_{k−1}). Substituting (26) into equation (25) one obtains

Ȧu_{k} + \frac{∂ψ}{∂t} + ∑_{i=1}^{k−1} ∑_{α=1}^{i} \binom{i}{α}u_{α}u_{i−α+1} \frac{∂ψ}{∂u_{i}} + ∑_{i=1}^{k} \binom{k}{i}u_{i}u_{k−i+1}A =

u_{1}(Au_{k} + ψ) + u_{0} \frac{∂ψ}{∂x} + \frac{∂^{2}ψ}{∂x^{2}} + 2 ∑_{i=1}^{k−1} u_{i+1} \frac{∂^{2}ψ}{∂x∂u_{i}} +

∑_{i,j=0}^{k−1} u_{i+1}u_{j+1} \frac{∂^{2}ψ}{∂u_{i}∂u_{j}} ,

where Ȧ =^{def} \frac{dA}{dt}. Here again everything is at most quadratic in u_{k}, and equating coefficients at u_{k}^{2} and u_{k} we get

\frac{∂^{2}ψ}{∂u_{k−1}^{2}} = 0,

2 (∑_{i=0}^{k−2} u_{i+1} \frac{∂^{2}ψ}{∂u_{i}∂u_{k−1}} + \frac{∂^{2}ψ}{∂x∂u_{k−1}} ) = ku_{1}A + Ȧ.

Hence,

ψ = \frac{1}{2}(ku_{0}A + Ȧx + ȧ)u_{k−1} + O[k − 2],

where a = a(t) and O[l] denotes a function independent of u_{i}, i > l. Thus

φ = Au_{k} + \frac{1}{2}(ku_{0}A + Ȧx + ȧ)u_{k−1} + O[k − 2]  (27)

which gives the ”upper estimation” for solutions of (24).

Estimating Jacobi brackets. Let

φ = φ(t, x, u_{0}, …, u_{k}), ψ(t, x, u_{0}, …, u_{l})

be two symmetries of B. Then their Jacobi bracket restricted onto B^{∞} looks as

\{φ, ψ\} = ∑_{i=0}^{l} D_{x}^{i}(φ)\frac{∂ψ}{∂u_{i}} − ∑_{i=0}^{k} D_{x}^{j}(ψ)\frac{∂φ}{∂u_{j}} .  (28)

Suppose φ is of the form (27) and similarly

ψ = Bu_{l} + \frac{1}{2}(lu_{0}B + Ḃx + ḃ)u_{l−1} + O[l − 2]

and compute (28) for these functions temporary denoting ku_{0}A + Ȧ + a and lu_{0}B + Ḃ + b by Ā and B̄ respectively. Then we have:

\{φ, ψ\} = D_{x}^{l}(Au_{k} + \frac{1}{2}Āu_{k−1})B + \frac{1}{2}D_{x}^{l−1}(Au_{k} + \frac{1}{2}Āu_{k−1})B̄ −

D_{x}^{k}(Bu_{l} + \frac{1}{2}B̄u_{l−1})A − \frac{1}{2}D_{x}^{k−1}(Bu_{k} + \frac{1}{2}B̄u_{l−1})Ā + O[k + l − 1] =

\frac{1}{2}(lD_{x}(Ā)u_{k+l−2} + Āu_{k+l−1})B̄ + \frac{1}{2}(Au_{k+l−1} + \frac{1}{2}Āu_{k+l−2})B −

\frac{1}{2}(kD_{x}(B̄)u_{k+l−2} + B̄u_{k+l−1})Ā − \frac{1}{2}(Bu_{k+l−1} + \frac{1}{2}B̄u_{k+l−2})A +

O[k + l − 3],

or in short,

\{φ, ψ\} = \frac{1}{2}(lȦB − KḂA)u_{k+l−2} + O[k + l − 3].  (29)

Low-order symmetries. Take k = 2 and solve equation (24) directly. Then one obtains five independent solutions which are

φ_{1}^{0} = u_{1}, φ_{1}^{1} = tu_{1} + 1, φ_{2}^{0} = u_{2} + u_{0}u_{1},

φ_{2}^{1} = tu_{2} + (tu_{0} + \frac{1}{2}x)u_{1} + \frac{1}{2}u_{0},

φ_{2}^{2} = t^{2}u_{2} + (t^{2}u_{0} + tx)u_{1} + tu_{0} + x.  (30)

Action of low-order symmetries. Let us compute the action

T_{i}^{j} =^{def} {φ_{i}^{j}, •} = � _{φ_{i}^{j}} − ℓ_{φ_{i}^{j}}

of symmetries φ_{i}^{j} on other symmetries of the equation B.

For φ_{1}^{0} one has

T_{1}^{0} = � _{u_{1}} − ℓ_{u_{1}} = ∑_{i≥0} u_{i+1} \frac{∂}{∂u_{i}} − D_{x} = −\frac{∂}{∂x}.

Hence, if φ = Au_{k} + O[k − 1] is a function of the form (27), then one has

T_{1}^{0}φ = −\frac{1}{2}Ȧu_{k−1} + O[k − 2].

Consequently, if φ is a symmetry, then, since sym(B) is closed under the Jacobi bracket,

(T_{1}^{0})^{k−1}φ = (−\frac{1}{2})^{k−1} \frac{d^{k−1}A}{dt^{k−1}}u_{1} + O[0]

is a symmetry as well. But from (30) one sees that first-order symmetries are linear in t. Thus, we obtain

Proposition 3 If φ = Au_{k} + O[k − 1] is a symmetry of the Burgers equation, then A is a kth degree polynomial in t.

Final description. Note that direct computations show that the equation B possesses a third-order symmetry of the form

φ_{3}^{0} = u_{3} + \frac{3}{2}u_{0}u_{2} + \frac{3}{2}u_{0}^{2} + \frac{3}{4}u_{0}^{2}u_{1}.

Using the actions T_{2}^{2} and T_{3}^{0}, one can see that

((T_{2}^{2})^{i} ∘ (T_{3}^{0} ∘ T_{2}^{2})^{k−1})u_{1} = (−\frac{3}{2})^{k−1} \frac{k!(k − 1)!}{(k − i)!}u_{k} + O[k − 1]  (31)

is a symmetry, since u_{1} is the one.

Theorem 4 The symmetry algebra sym(B) for the Burgers equation B = {u_{t} = uu_{x} + u_{xx}}, as a vector space, is generated by elements of the form

φ_{k}^{i} = t^{i}u_{k} + O[k − 1], k ≥ 1, i = 0, …, k,

which are polynomial in all variables. For the Jacobi bracket one has

\{φ_{k}^{i}, φ_{l}^{j}\} = \frac{1}{2}(li − kj)φ_{k+l−2}^{i+j−1} + O[k + l − 3].  (32)

The algebra sym(B) is simple and has φ_{1}^{0}, φ_{2}^{2} and φ_{3}^{0} as its generators.

Proof. It only remains to prove that all φ_{k}^{i} are polynomials and that sym(B) is simple. The first fact follows from (31) and from the obvious observation that coefficients of both T_{2}^{2} and T_{3}^{0} are polynomials.

Let us prove that sym(B) is a simple Lie algebra. To do this, introduce an order in the set {φ_{k}^{i}} defining

Φ_{\frac{k(k+1)}{2} +i} =^{def} φ_{k}^{i}.

Then any symmetry may be represented as ∑_{α=1}^{s} λ_{α}Φ_{α}, λ ∈ R. Let I ⊂ sym(B) be an ideal and Φ = Φ_{s} + ∑_{α=1}^{s−1} λ_{α}Φ_{α} be its element. Assume Φ_{s} = φ_{k}^{i} for some k ≥ 1 and i ≤ k.

Note now that

T_{1}^{1} = ∑_{α≥0} D_{x}^{α}(tu_{1} + 1)\frac{∂}{∂u_{α}} − tD_{x} = \frac{∂}{∂u_{0}} − t\frac{∂}{∂x}

and

T_{2}^{0} = ∑_{α≥0} D_{x}^{α}(u_{2} + u_{0}u_{1})\frac{∂}{∂u_{α}} − D_{x}^{2} − u_{0}D_{x} − u_{1} = −\frac{∂}{∂t}.

Therefore,

((T_{1}^{1})^{k−1} ∘ (T_{2}^{0})^{i})Φ = cφ_{1}^{0},

where the coefficient c does not vanish. Hence, I contains φ_{1}^{0}. But due to (31) the latter, together with φ_{2}^{2} and φ_{3}^{0}, generates the whole algebra.

Further details on the structure of sym(B) one can find in [3].

3 Recursion operators and nonlocalities.

In fact, we were successful in getting a complete description of symmetry algebra for the Burgers equation because sym(B) is a very rich algebra. In other cases this method may not work, e.g. for the KdV equation (??)^{6}. Below we expose another method based on computation of recursion operators.

Recursion operators for the Burgers and KdV equations. Informally speaking, a recursion operator for an evolution equation E

\frac{∂u}{∂t} = F (t, x, u, …, \frac{∂^{k}u}{∂x^{k}})  (33)

is an expression of the form

R = ∑_{i=−∞}^{s} a_{i} ∘ D_{x}^{i} ∘ b_{i},

6Though it will be helpful anyway to estimate the symmetry algebra.

a_{i}, b_{i} being functions of x, t, …, u_{k}, …, which preserves symmetries of E, i.e. R : sym(E) → sym(E). Usually, integrable equations possess such operators. For example (see [5]), the operator

D_{x} + \frac{1}{2}u_{0} + \frac{1}{2}u_{1}D_{x}^{−1}

is a recursion for the Burgers equation, while

D_{x}^{2} + \frac{2}{3}u_{0} + \frac{1}{3}u_{1}D_{x}^{−1}

is he one for the KdV equation (??).

Having a recursion operator and a starting symmetry φ_{0}, one easily obtains explicit expressions for the whole series φ_{k} = R^{k}φ_{0}. Thus, having a method for computing recursion operators, we get a powerful means to generate symmetries. Such a method was found in [1, 2] and we discuss it in action below.

Cartan forms. A Cartan form on E^{∞} is a form which vanishes on the distribution C_{E} or, equivalently, which anihilates all total derivatives D_{1}, …, D_{n} on E^{∞}. Cartan forms form a vector subspace C^{1}Λ(E^{∞}) in the space Λ^{1}(E^{∞}) of all differential one-forms on E^{∞}. For the case of equation (33), one can choose the forms

ω_{s} =^{def} du_{s} − u_{s+1}dx − D_{x}^{s}(F)dt,  (34)

s = 0, 1, …, as a basis in C^{1}Λ(E^{∞}). It can easily be checked that i_{D_{x}}ω_{s} = i_{D_{t}}ω_{s} = 0.

Below we shall need formulae for Lie actions of total derivatives on the forms ω_{s}. Let us make necessary computations in the case (33). For D_{x} one has:

L_{D_{x}}(ω_{s}) = D_{x}(du_{s} − u_{s+1}dx − D_{x}(F)dt) =

d(D_{x}u_{s}) − (D_{x}u_{s+1})dx − u_{s+1}d(D_{x}x) − D_{x}^{s+1}(F)dt − D_{x}^{s}(F)d(D_{x}t) = du_{s+1} − u_{s+2}dx − u_{s+1}d(1) − D_{x}^{s+1}(F)dt − D_{x}^{s}(F)d(0),

or

L_{D_{x}}(ω_{s}) = ω_{s+1}.  (35)

To apply D_{t} to ω_{s}, note first that

[D_{x}, D_{t}] = D_{x} ∘ D_{t} − D_{t} ∘ D_{x} = 0.  (36)

Hence, using (35) and (36), we obtain

L_{D_{t}}(ω_{s}) = D_{t}D_{x}^{s}(ω_{0}) = D_{x}^{s}(D_{t}ω_{0}).  (37)

But

D_{t}(ω_{0}) = D_{t}(du_{0} − u_{1}dx − F dt) = d(D_{t}u_{0}) − (D_{t}u_{1})dx − (D_{t}F)dt =

dF − (D_{x}F)dx − (D_{t}F)dt = \frac{∂F}{∂x} dx + \frac{∂F}{∂t} dt + ∑_{i=0}^{k} \frac{∂F}{∂u_{i}} du_{i} −

(\frac{∂F}{∂x} + ∑_{i=0}^{k} u_{i+1} \frac{∂F}{∂u_{i}} ) dx − (\frac{∂F}{∂t} + ∑_{i=0}^{k} D_{x}^{i})F)\frac{∂F}{∂u_{i}} ) ,

or

D_{t}ω_{0} = ∑_{i=0}^{k} \frac{∂F}{∂u_{i}} ω_{i}.

Hence

D_{t}ω_{s} = D_{x}^{s} (∑_{i=0}^{k} \frac{∂F}{∂x_{i}} ω_{i}) = ∑_{i=0}^{k} ∑_{α=0}^{s} \binom{s}{α}D_{x}^{s−α} (\frac{∂F}{∂u_{i}} ) ω_{i+α}.  (38)

Action of Cartan forms on symmetries. Let ω ∈ C^{1}Λ(E^{∞}) be a Cartan form and consequently

ω = a_{0}ω_{0} + a_{1}ω_{1} + … + a_{s}ω_{s},  (39)

a_{i} being functions on E^{∞}. Consider another function φ and the vector field

� _{φ} = ∑_{i=0}^{∞} D_{x}^{i}(φ)\frac{∂}{∂u_{i}} .

Hence, contracting � _{φ} with ω, we obtain

i_{� _{φ}}(ω) = a_{0}i_{� _{φ}}ω_{0} + a_{1}i_{� _{φ}}ω_{1} … + a_{s}i_{� _{φ}}ω_{s} =
a_{0}φ + a_{1}D_{x}(φ) + … + a_{s}D_{x}^{s}(φ).

Therefore, any ω of the form (39) determines a differential operator

R_{ω} = a_{0} + a_{1}D_{x} + … + a_{s}D_{x}^{s}.  (40)

Theorem 5 [2] Restrict operator (40) onto symmetries of the equation E, i.e. for any � _{φ} ∈ sym(E) set R_{ω}(� _{φ}) =^{def} � _{R_{ω}φ}. Then R_{ω} preserves the algebra sym(E), i.e. R_{ω} sym(E) ⊂ sym(E), if and only if the form ω satisfies the equation

ℓ_{E}^{[1]}ω = 0,  (41)

where ℓ_{E}^{[1]} is the extension of the universal linearizaton operator (17) to Cartan forms.

Thus equation (41) may be used to compute recursion operators. We show how it works using the example of the Burgers equation.

Basic relations for B. In this case we obtain

ℓ_{B}^{[1]}(ω) = (D_{t} − u_{1} − u_{0}D_{x} − D_{x}^{2}) (∑_{i=0}^{s} a_{i}ω_{i}) =

∑_{i=0}^{s}(ℓ_{B}(a_{i})ω_{i} + a_{i}D_{t}(ω_{i}) − u_{0}a_{i}ω_{i+1} − 2D_{x}(a_{i})ω_{i+1} − a_{i}ω_{i+2}).

Using (38) one can now reduce (41) to a system of equations on functions a_{0}, …, a_{s}. We shall not do it here in the general form, but write down the equations for s = 2. In this case one obtains

D_{x}(a_{1}) = 0,
D_{x}(a_{0}) + D_{x}^{2}(a_{1}) + u_{0}D_{x}(a_{1}) = u_{1}a_{1} + D_{t}(a_{1}),
D_{x}^{2}(a_{0}) + u_{0}D_{x}(a_{0}) + u_{1}a_{0} = D_{t}(a_{0}).  (42)

A ”disappointing” result. Now, if one tries to solve (42) or the equation ℓ_{B}^{[1]}ω = 0 in its general form (see [1] for the proof), one will get the only solution

ω = a_{0}ω_{0}, a_{0} ∈ R.

Corresponding operator R_{ω} just multiplies symmetries by the constant a_{0} and produces no nontrivial recursions.

And this is not unnatural, since the known recursion operator for the Burgers equation contains the term D_{x}^{−1} (see above), which could not appear in our setting.

Nonlocal setting. Introduce a new variable u_{−1} =^{def} D_{x}^{−1}(u_{0}) = ∫ u_{0} dx and set

D_{x}u_{−1} = u_{0}, D_{t}u_{−1} = ∫ D_{t}(u_{0}) dx = u_{1} + \frac{1}{2}u_{0}^{2}.

Then we get ”new” total derivatives

D̃_{x} = u_{0} \frac{∂}{∂u_{−1}} + D_{x}, D̃_{t} = (u_{1} + \frac{1}{2}u_{0}^{2})\frac{∂}{∂u_{−1}} + D_{t}

and the Cartan form

ω_{−1} = du_{−1} − u_{0}dx − (u_{1} + \frac{1}{2}u_{0}^{2})dt.

Let us solve the equation

ℓ̃_{B}ω = 0,  (43)

where ℓ̃_{B} is the operator ℓ_{B} in which ”old” total derivatives D_{x}, D_{t} are substituted by D̃_{x} and D̃_{t} respectively.

Proposition 4 [2] Equation (43) possesses two independent solutions

ω_{0}, ω = ω_{1} + \frac{1}{2}u_{0}ω_{0} + \frac{1}{2}u_{1}ω_{−1}.

The operator R_{ω_{0}} is the identity, while R_{ω} coincides with the classical recursion operator for the Burgers equation.

Remark 2 Try to compute symmetries in this setting: a very intersting result awaits you!

General scheme. Below we expose the way of action in the case of a general evolution equation.

1. Solve the equation

ℓ_{E}φ = 0

for low-order φ.

2. Solve the equation

ℓ_{E}^{[1]}ω = 0

for low-order Cartan forms. Usually you will get trivial solutions only. 3. Extend the setting with nonlocal variables. To do this, solve the equation

ℓ_{E}^{∗}ψ = 0,  (44)

where ℓ_{E}^{∗} is the formally adjoint^{7} to ℓ_{E}. The solutions of (44) are generating functions of conservation laws for the equation E (see [7] for details).

4. Enlarge the setting by adding nonlocal variables D_{x}^{−1}ψ = ∫ ψ dx for any solution ψ; equation (44) guarantees the new setting to be well-defined.

5. Solve the equation

ℓ̃_{E}ω = 0

in the new setting and get the recursion operator.

For integrable equations this scheme works perfectly.

Acknowledgements. I am grateful to the organizers of the 1st Non-Orthodox School on Nonlinearity & Geometry for the opportunity to give this lecture.

References

[1] I.S. Krasil’shchik, Some new cohomological invariants for nonlinear differential equations, Diff. Geom. Appl. 2 (1992) 307–350.

[2] I.S. Krasil’shchik and P.H.M. Kersten, Deformations and recursion operators for evolution equations, In: Pràstaro A., Rassias Th.M. (Eds.), Geometry in Partial Differential Equations (1993) World Scientific, Singapore.

7If Δ = ∑_{i=0}^{s} a_{i}D_{x}^{i}, then Δ^{∗} = ∑_{i=0}^{s}(−1)^{i}D_{x} ∘ a_{i}.

[3] I.S. Krasil’shchik, Lychagin V.V. and A.M. Vinogradov, Geometry of Jet Spaces and Nonlinear Differential Equations, Advanced Studies in Contemporary Mathematics 1 (1986) Gordon and Breach, New York, London.

[4] I.S. Krasil’shchik and A.M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math. 15 (1989) 161–209.

[5] P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics 107 (1986) Springer-Verlag, New York.

[6] L.V. Ovsyannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.

[7] A.M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math. 2 (1984) 21–78.