Table Of ContentA Simple Construction of Recursion Operators
for Multidimensional Dispersionless Integrable Systems
A. Sergyeyev
Mathematical Institute, Silesian University in Opava,
Na Rybn´ıˇcku 1, 746 01 Opava, Czech Republic
E-mail: [email protected]
6
1
0
2 November 19, 2016
v
o
N
We present a simple novel construction of recursion operators for integrable multidimensional
2 dispersionless systems that admit Lax-type representations whose operators are linear in the spec-
2
tral parameter and do not involve the derivatives with respect to the latter. New examples of
] recursion operators obtained using our technique include inter alia those for the general heavenly
P
equation, which describes a class of anti-self-dual solutions of the vacuum Einstein equations, and
A
a six-dimensional equation resulting from a system of Ferapontov and Khusnutdinova.
.
h
t
a
m Introduction
[
Thefieldofintegrablesystemsisadynamicsubject closelyrelatedtomanypartsofmodernmathematics
3
v and featuring manifold applications, cf. e.g. [1, 3, 5, 7, 9, 12, 20, 36, 37, 40, 47] and references therein.
5
A characteristic feature of integrable systems is that they never show up alone – instead, they always
5
9 belong to integrable hierarchies, and the latter are typically constructed using the recursion operators
1
(ROs), cf.e.g.[1,3,12,21,36,39,40]andreferencestherein. Inparticular, theROsformultidimensional
0
. integrabledispersionlesssystemsarenowasubjectofintenseresearch, cf.e.g.[1,12,24,28,30,33,34,35,
1
0 40]andreferencestherein. Infact, theoverwhelming majorityofintegrablesystems infourormoreinde-
5 pendent variable known to date, including those relevant for applications, like the (anti-)self-dual Yang–
1
Mills equations or the (anti-)self-dual vacuum Einstein equations, is dispersionless, i.e., can be rewritten
:
v
as first-order homegeneous quasilinear systems, cf. e.g. [1, 12, 45] and Example 5 below for details.
i
X
The study of a hierarchy associated to a given integrable system, and hence of its recursion operator,
r
a is important for many reasons, inter alia because known exact solutions of integrable systems, like, say,
multisoliton and finite-gap solutions, always happen to be invariant under a certain combination of the
flows in the hierarchy, and this is instrumental in the actual construction of the solutions in questions,
cf. e.g. [12, 36].
Another feature making recursion operators relevant for applications is that existence of a (formal)
recursion operator can in many situations be employed as an efficient integrability test or as a means
of proving nonintegrability, see e.g. [32] and references therein for general theory and [46] for a recent
application.
There are several methods for construction of the ROs for the integrable dispersionless systems: the
partner symmetry approach [24], the method based on adjoint action of the Lax operators [30], and the
approach using the Cartan equivalence method [33, 34, 35].
Below we present another method for construction of ROs for integrable multidimensional disper-
sionless systems. Roughly speaking, it consists in constructing a Lax-type representation for the system
under study with the following additional properties: i) the Lax operators are linear in the spectral
parameter and ii) the solution of the associated linear system is a (nonlocal) symmetry of the nonlinear
1
system under study. Taking the coefficients at the powers of spectral parameter in the operators con-
stituting the newly constructed Lax-type representation yields, under certain technical conditions, the
RO. Asimilar approach, with symmetries replaced by cosymmetries andProposition 1 by Proposition 2,
allows one to look for the adjoint ROs.
Tothebestofauthor’sknowledge, thekeyideaofourapproach, i.e., thattheLax-typerepresentation
employed for theconstruction oftherecursion operatorshould beconstructed fromtheoriginal Lax-type
representation but is by no means obliged to be identical with the latter, has not yet appeared in the
literature. This kind of flexibility noticeably enhances the applicability of our method. On the other
hand, the extraction of the RO from a Lax-type representation which is linear in the spectral parameter
was already explored to some extent e.g. in [41].
The paper is organized as follows. In Section 1 we recall some basic facts concerning the geometric
theory of PDEs. In Section 2 we state some general results on the construction of recursion operators.
Section 3, which is the core of the paper, explores the construction of special Lax-type representations
that give rise to ROs through the results of Section 2. Section 4 gives a selection of examples illustrating
our approach, and Section 5 provides a brief discussion.
1 Preliminaries
Let = 0 be a system of m PDEs F = 0, I = 1,...,m, in d independent variables xi, i = 1,...,d, so
I
F
we put ~x = (x1,...,xd), for an unknown N-component vector function u = (u1,...,uN)T, where the
superscript ‘T’ indicates the transposed matrix. Let
uα = ∂i1+···+inuα/∂(x1)i1 ∂(xn)in
i1...in ···
and uα uα. In what follows all functions are tacitly assumed to be meromorphic in all their
00···0 ≡
arguments.
As e.g. in [40, 17], xi and uα are considered here and below as independent quantities and can
i1...in
be viewed as coordinates on an abstract infinite-dimensional space (a jet space). By a local function we
shall mean a function of ~x, u and of a finite number of the derivatives of u. To indicate locality of a
function f we shall write it as f(~x,[u]).
We denote by
N ∞
∂ ∂
Dxj = ∂xj +X X uαi1...ij−1(ij+1)ij+1...in∂uα
α=1i1,...,in=0 i1...in
the total derivatives with respect to xj, cf. e.g. [40, 17]. For the sake of simplicity the extensions of D
xi
to nonlocal variables are again denoted by D , as e.g. in [30].
xi
The system = 0 together with all its differential consequences
F
Di1 Din = 0, i ,i ,...,i = 1,...,d, n = 1,2,...,
x1··· xnF 1 2 n
defines a diffiety Sol , a submanifold of the infinite jet space, see e.g. [17] and references therein for
F
details. This submanifold can be informally thought of as a set of all formal solutions of = 0, which
F
motivates the choice of notation. Below all equations will be required to hold only on Sol , or a certain
F
differential covering thereof, see e.g. [17] for details on the latter, rather than e.g. on the whole jet space
unless otherwise explicitly stated.
The directional derivative (cf. e.g. [3, 30] and references therein) along U = (U1,...,UN)T is the
vector field on the jet space
N ∞
∂
∂U = X X (Dxi11···DxinnUα)∂uα .
α=1i1,...,in=0 i1...in
2
The total derivatives as well as the directional derivative can be applied to (possibly vector or matrix)
local functions P.
Recall,cf.e.g.[40,17],thatalocalN-componentvectorfunctionU isa(characteristicofa)symmetry
for the system = 0 if U satisfies the linearized version of this system, namely, ℓ (U) = 0. We stress
F
F
that by our blanket assumption this is required to hold on Sol only rather than on the whole jet space.
F
Informally this definition means that the flow u = U leaves Sol invariant.
τ F
Here
N ∞
∂f
ℓf = X X ∂uα Dxi11 ···Dxinn
α=1i1,...,in=0 i1...in
is the operator of linearization and ℓF = (ℓF1,...,ℓFm)T. Notice an important identity ℓf(U) = ∂U(f)
which holds for any local f and U, see e.g. [17].
Finally, recall that if we have an operator in total derivatives
q d d
Q = Q0 +XX···XQj1...jkDxj1 ···Dxjk,
k=1j1=1 jk=1
where Q0 and Qj1...jk are s s′-matrix-valued local functions, its formal adjoint reads (cf. e.g. [40])
×
q d d
Q† = (Q0)T +XX···X(−1)kDxj1 ···Dxjk ◦(Qj1...jk)T.
k=1j1=1 jk=1
For integrable systems in more than two independent variables their symmetries usually depend, in
addition to local variables, described above, also on nonlocal variables. In other words, we are forced
to consider solutions U of ℓ (U) = 0 involving nonlocal variables. A number of authors refers to such
F
objects as to the shadows of nonlocal symmetries, see e.g. [17] and references therein for more details
on this terminology; cf. also [5] and references therein. However, as in the present paper we shall not
deal with full nonlocal symmetries in the sense of [17], for the sake of simplicity in what follows we shall
refer to solutions U of ℓ (U) = 0 involving nonlocal variables just as to nonlocal symmetries.
F
A cosymmetry, also known as adjoint symmetry, see e.g. [5], γ is a quantity which is dual to a
symmetry: it has m rather than N components and satisfies (cf. e.g. [3, 17] for details) the system
ℓ† (γ) = 0.
F
Note that cosymmetries, just like symmetries, may depend on nonlocal variables in addition to the local
ones.
A recursion operator (RO) for our system = 0 is then, roughly speaking, an operator that sends
F
any symmetry of = 0 into another symmetry thereof [39, 40]. However, this definition works well
F
only when the RO contains no nonlocal terms, as often is the case for linear and linearizable equations,
cf. e.g. [4, 40] and references therein.
In order to properly handle nonlocal terms in ROs it is more appropriate to view the RO as a
B¨acklund auto-transformation for ℓ (G) = 0, see [30, 41, 42] and references therein for details, and
F
below we adhere to this point of view. Likewise, an adjoint recursion operator is from this perspective
a B¨acklund auto-transformation for ℓ† (γ) = 0, cf. e.g. [17].
F
As a closing remark, recall that a partial differential system is called dispersionless (cf. e.g. [28] and
references therein) if it can be written in the form of a quasilinear homogeneous first-order system
d n ∂uα
XXAj (u) = 0, (1)
Iα ∂xj
j=1 α=1
where I = 1,...,m, m N, and u = (u1,...,uN)T.
≥
3
This class of systems is quite rich: for instance, quasilinear scalar second-order PDEs which do not
explicitly involve the dependent variable u,
d d
XXfij(~x,∂u/∂x1,...,∂u/∂xd)∂2u/∂xi∂xj = 0,
i=1 j=i
can be brought into the form (1) by putting u = (∂u/∂x1,...,∂u/∂xd)T.
2 Some remarks on recursion operators
Consider the following differential operators in total derivatives
d d
A = A0 + AjD , B = B0 + BjD , i = 1,2,
i i P i xj i i P i xj
j=1 j=1 (2)
d d
L = L0 + LkD , M = M0 + MkD ,
P xk P xk
k=1 k=1
where Aj = Aj(~x,[u]) and Bj = Bj(~x,[u]) for i = 1,2 and j = 1,...,d are scalar functions, A0 =
i i i i i
A0(~x,[u]) and B0 = B0(~x,[u]) are N N matrices, L0 = L0(~x,[u]) and Lk = Lk(~x,[u]) for k = 1,...,d
i i i ×
are N m matrices, while M0 = M0(~x,[u]) and Mk = Mk(~x,[u]) for k = 1,...,d are m N matrices.
× ×
Proposition 1. Let the operators A ,B ,L,M of the form (2) be such that
i i
i) [A ,A ] = 0, (3)
1 2
ii) (A B A B ) = L ℓ , (4)
1 2 2 1 F
− ◦
iii) ℓ = M (B A B A ), (5)
F 1 2 2 1
◦ −
iv) there exist two distinct indices p,q 1,...,d such that we can express D U˜ and D U˜ from the
xp xq
∈ { }
relations
A (U˜) = B (U), i = 1,2. (6)
i i
Then relations (6) define a recursion operator for = 0, i.e., whenever U = (U1,...,UN)T is a
(possibly nonlocal) symmetry for = 0, so is U˜ = (U˜1F,...,U˜N)T defined by (6).
F
Proof. First of all, if U = (U1,...,UN)T is a symmetry for = 0, then the system (6) for U˜ is
F
compatible by virtue of i)–iii).
We now have ℓ (U˜) = 0 by virtue of (5), and hence U˜ is a shadow of symmetry for our system. (cid:3)
F
As an aside note that the condition (5) in a slightly different form has appeared in [11, 15, 16], and
was given there a certain geometric interpretation.
In complete analogy with the above we can also readily prove the counterpart of Proposition 1 for
adjoint recursion operators.
Proposition 2. Suppose that A ,B ,L,M are as before, but with A0 and B0 being m m matrices.
i i i i ×
Further assume that these operators are such that
i) [A ,A ] = 0, (7)
1 2
ii) ℓ† = L (B A B A ), (8)
F ◦ 1 2 − 2 1
iii) (A B A B ) = M ℓ† , (9)
1 2 − 2 1 ◦ F
iv) there exist two distinct indices p,q 1,...,d such that we can express D γ˜ and D γ˜ from the
xp xq
∈ { }
system
A (γ˜) = B (γ), i = 1,2. (10)
i i
Then (10) defines an adjoint recursion operator for = 0, i.e., whenever γ = (γ1,...,γm)T is a
F
cosymmetry for = 0, then so is γ˜ defined by (10).
F
4
3 Recursion operators from Lax-type representations
We start with the following observation which is readily checked by straightforward computation.
Proposition 3. Suppose that the operators A and B define a RO (resp. an adjoint RO) as in Propo-
i i
sition 1 (resp. as in Proposition 2). Then the operators = λA B , i = 1,2, where λ is a spectral
i i i
L −
parameter, satisfy [ , ] = 0, i.e., they constitute a Lax-type representation for = 0.
1 2
L L F
Thus, if our system admits an (adjoint) RO, it also admits a Lax-type representation which is linear
in the spectral parameter.
Hence, a natural source of A ,B and L,M satisfying the conditions of Proposition 1 is provided by
i i
the Lax-type representations for = 0 of the form
F
ψ = 0, i = 1,2, (11)
i
L
with linear in λ such that ψ is a nonlocal symmetry of = 0, i.e., we have ℓ (ψ) = 0, cf. e.g.
i F
L F
[41]. Then putting = λB A or = λA B gives us natural candidates for A and B which
i i i i i i i i
L − L −
then should be checked against the conditions of Proposition 1, and, if the latter hold, yield a recursion
operator for = 0.
F
Even if it is impossible to construct the recursion operator from (11), one still can construct infinite
series of nonlocal symmetries for = 0 expanding ψ into formal Taylor or Laurent series in λ, cf. e.g.
F
[41, 44].
ItisimportanttostressthattheLaxrepresentationwiththeoperators employed intheabovecon-
i
L
structiondoesnothavetobetheoriginalLaxrepresentationofoursystem. Ingeneral, weshouldcustom
tailor the operators constituting the Lax pair for the construction in question, so that the solutions
1,2
L
of the associated linear problem (11) are symmetries, i.e., satisfy the linearized version of our system.
The natural building blocks for these are the original Lax operators X , their formal adjoints
i i
Xi†, and the operators adXi = [Xi,·], butLin general one has to twist them, cf. e.g. Example 4 below.
More explicitly, suppose that the system under study, i.e., = 0, admits a Lax representation of
F
the form
X ψ = 0, i = 1,2, (12)
i
where X are linear in the spectral parameter λ but such ψ does not satisfy ℓ (ψ) = 0, i.e., ψ is not a
i F
(nonlocal) symmetry.
Then we can seek for a nonlocal symmetry of = 0 of the form
F
Φ = Φ(~x,[u,ψ,χ,~ζ]), (13)
i.e., for a vector function of ~x, and of u, ψ, χ, and ζ~ and a finite number of the derivatives of u, ψ, χ,
~
and ζ.
Here χ satisfies the system
X †χ = 0, i = 1,2,
i
d
Z = ζj∂/∂xj satisfies
P
j=1
[X ,Z] = 0, i = 1,2,
i
and ζ~ = (ζ1,...,ζd)T.
Moreover, we should also require that there exist the operators L which are linear in λ and such
i
that
L Φ = 0, i = 1,2.
i
Then one should extract A and B for Proposition 1 from these L rather than from the original X .
i i i i
5
~
The study of specific examples strongly suggests that the nonlocal variables ψ, χ and ζ usually do
not mix and enter Φ linearly, so it typically suffices to look for nonlocal symmetries Φ of = 0 in one
F
of the following forms,
r r d
Φα = Xaαsψs +XXaksDxk(ψs), (14)
s=1 s=1 k=1
r r d
Φα = Xbα,sχs +XXbα,s,kDxk(χs), (15)
s=1 s=1 k=1
d d
Φα = Xcαjζj + X cαl,kDxk(ζl), (16)
j=1 k,l=1
where α = 1,...,N, p N, aα, aα,k, bα,s, bα,s,k, cα and cα,k are local functions, and r is the number
∈ s s j l
of components of ψ, instead of general form (13). Moreover, in many cases it is possible to restrict
oneself to considering only zero-order terms in the nonlocal variables, i.e., use the simpler Ansa¨tze
r r d
Φα = aαψs, Φα = bα,sχ , and Φα = cαζj instead of (14), (15) and (16).
P s P s P j
s=1 s=1 j=1
A similar approach, with symmetries replaced by cosymmetries and Proposition 1 by Proposition 2
can, of course, be applied to the construction of adjoint recursion operators.
Finally, let us point out that the assumption of linear dependence of the Lax operators on λ is
not as restrictive as it seems: a great many of known today multidimensional dispersionless hierarchies
include systems with this property and, moreover, by Proposition 3 if a system admits an RO of the
form described in Proposition 1 then it necessarily possesses a Lax-type representation whose operators
are linear in λ.
Moreover, in some cases the nonlinearity in λ can be removed upon a proper rewriting of the Lax
pair.
Consider for example the Pavlov equation [43]
u +u +u u u u = 0, (17)
yy xt x xy y xx
−
which possesses a Lax pair of the form [43] (cf. also [26])
ψ = ( u λ)ψ , ψ = ( λ2 λu +u )ψ (18)
y x x t x y x
− − − −
which is quadratic in λ.
However, rewriting the second equation of (18) as
ψ = λ( λ u )ψ +u ψ
t x x y x
− −
and substituting ψ for ( λ u )ψ , we obtain
y x x
− −
ψ = λψ +u ψ ,
t y y x
and thus (18) can be rewritten in the form linear in λ:
ψ = ( u λ)ψ , ψ = λψ +u ψ . (19)
y x x t y y x
− − −
4 Examples
Example 1
It isreadilychecked thatif ψ satisfies (19)thenΦ = 1/ψ isanonlocal symmetry for(17), i.e., U = 1/ψ
x x
satisfies the linearized version of (17),
U +U +u U u U +u U u U = 0. (20)
yy xt x xy y xx xy x xx y
− −
6
Now by virtue of (19) Φ = 1/ψ also satisfies a pair of linear equations of the form L Φ = 0, i = 1,2,
x i
where
L = D +(λ u )D +u , L = D +λD u D +u .
1 y x x xx 2 t y y x xy
− − −
Now, as Φ = 1/ψ satisfies (20) we can identify L with from (11) and put L = λA B , where
x i i i i i
L −
A = D , A = D , B = D +u D u , B = D +u D u .
1 x 2 y 1 y x x xx 2 t y x xy
− − −
Then it is easily seen that these operators satisfy the conditions of Proposition 1 for suitably chosen L
and M, so we arrive at the recursion operator for the Pavlov equation given by the formulas
U˜ = U +u U u U, U˜ = U +u U u U,
x y x x xx y t y x xy
− − −
which maps a (possibly nonlocal) symmetry U to a new nonlocal symmetry U˜. This is nothing but the
recursion operator found in [26] rewritten as a B¨acklund auto-transformation for (20).
Example 2
Consider the general heavenly equation
au u +bu u +cu u = 0, a+b+c = 0, (21)
xy zt xz yt xt yz
where a,b,c are constants. Here d = 3, N = 1, x1 = x, x2 = y, x3 = z, x4 = t, u1 = u. Note that this
equation describes inter alia a class of anti-self-dual solutions of the Einstein field equations as shown
in [22].
By definition, a (nonlocal) symmetry U of (21) satisfies the linearized version of (21), that is,
au U +au U +bu U +bu U +cu U +cu U = 0. (22)
xy zt zt xy xz yt yt xz xt yz yz xt
Eq.(21) admits [10, 22] a Lax-type representation with the operators
u u
xz xt
L = (1+cλ)D D cλ D ,
1 x t z
− u − u
zt zt
(23)
u u
yz yt
L = (1 bλ)D D +bλ D .
2 y t z
− − u u
zt zt
It is readily seen that ψ that satisfies L ψ = 0, i = 1,2, also satisfies (22), i.e., it is a nonlocal symmetry
i
for (21), so in spirit of Proposition 3 let
u u
xt yt
A = cD c D , A = bD +b D ,
1 x z 2 y z
− u − u
zt zt
u u
xz yz
B = D + D , B = D + D .
1 x t 2 y t
− u − u
zt zt
Then all conditions of Proposition 1 are readily seen to be satisfied, e.g. we have B A B A =
1 2 2 1
−
(1/u )ℓ , where F = au u +bu u +cu u .
zt F xy zt xz yt xt yz
Hence the relations
u U +cu U˜ u U u U bu U˜ u U
U˜ = xz t xt z − zt x, U˜ = yz t − yt z − zt y, (24)
x y
cu − bu
zt zt
˜
where U is a (possibly nonlocal) symmetry of (21), define a new symmetry U and thus a novel recursion
operator (24) for (21), i.e., a B¨acklund auto-transformation for (22). This operator has first appeared in
the very first version of the present paper (arXiv:1501.01955v1) and was later rediscovered, in a slightly
different form, in [23].
7
Example 3
Consider the Mart´ınez Alonso–Shabat [29] equation
u = u u u u (25)
yt z xy y xz
−
where d = 4, N = 1, x1 = x, x2 = y, x3 = z, x4 = t, u1 = u. Eq.(21) admits [35] a Lax-type
representation with the operators
L = D λu D , L = D λu D +λD . (26)
1 y y x 2 z z x t
− −
Consider the action of operators
= ad = [L , ]
Li Li i ·
on operators of the form wD and spell it out as
x
ad (wD ) ((λB A )w)D ,
Li x ≡ i − i x
so we have
A = D , A = D , B = u D u , B = u D D u .
1 y 2 z 1 y x xy 2 z x t xz
− − −
All conditions of Proposition 1 are again satisfied, so the relations
U˜ = u U u U,
y y x xy
−
(27)
U˜ = u U u U U ,
z z x xz t
− −
where U is any (possibly nonlocal) symmetry of (25), define a new symmetry U˜, i.e., Eq.(27) provides
a recursion operator for (25). This operator was already found in [35].
Applying (27) to the simplest symmetry U = u we obtain (modulo an arbitrary function of x and
x
t resulting from the integration) a nonlocal symmetry of the form U˜ = w u2/2, where the nonlocal
− x
variable w is defined by the relations
w = u u , w = u u u .
y y xx z z xx xt
−
This example suggests that Eq.(27) can be further simplified: a new symmetry U˜ can be constructed
by putting
U˜ = W u U,
x
−
where W is defined by the relations
W = u U +u U ,
y y x x y
W = u U +u U U .
z z x x z t
−
Thus, the recursion operators produced within our approach are not necessarily in the simplest possible
form.
Example 4
Consider a system [13, 14]
m = n +nm mn , n = m +mn nm , (28)
t x r r z y s s
− −
where x,y,z,r,s,t are independent and m,n are dependent variables.
Eq.(28) can be written as a condition of commutativity for the following pair of vector fields,
[D mD λD +λmD ,D nD λD +λnD ] = 0. (29)
z s x r y s t r
− − − −
8
where λ is a spectral parameter, and is therefore integrable.
By virtue of (28) there exists a potential u such that
m = u /u , n = u /u . (30)
z s y s
Substituting this into (28) gives a single second-order equation for u,
u u u u u u +u u u u +u u = 0, (31)
s zt z st s xy y sx y rz z ry
− − −
which can be written as a commutativity condition [L ,L ] = 0 for
1 2
u u u u
z z y y
L = D D λD +λ D , L = D D λD +λ D .
1 z s x r 2 y s t r
− u − u − u − u
s s s s
Let χ satisfy L†χ = 0, i = 1,2. Then ζ = u /χ is a nonlocal symmetry for (31) and we can readily
i s
obtain a linear system for ζ of the form ζ = 0, ζ = 0 from that for χ.
1 2
L L
Upon spelling out as = A λB and checking that the conditions of Theorem 1 are satisfied
i i i i
L L −
for suitable L and M we arrive at the following novel RO for (31):
u u (u u )
U˜ = yU˜ yU +U st − ry U,
y s r t
u − u − u
s s s
(32)
u u (u u )
˜ z ˜ z rz sx
U = U U +U + − U.
z s r x
u − u u
s s s
Here again U is a (in general nonlocal) symmetry of (31), and U˜ is a new nonlocal symmetry for (31).
Example 5
Consider (see e.g. [1, 2, 31] and references therein) the following Lax operators:
L = D +∂ K λD , L = D +∂ K λD . (33)
1 y x˜ x˜ 2 x y˜ y˜
− −
The commutativity condition [L ,L ] = 0 yields the system
1 2
∂ ∂ K ∂ ∂ K [∂ K,∂ K] = 0. (34)
x x˜ y y˜ x˜ y˜
− −
Here K takes values in a (matrix) Lie algebra g and is known as the Yang K-matrix.
Note that (34) can be rewritten in the dispersionless form of the type (1) upon introducing (cf. e.g.
[31, 42]) an additional dependent variable J taking values in G = exp(g). Namely, the dispersionless
form of (34) reads
∂ J = J∂ K, ∂ J = J∂ K. (35)
x y˜ y x˜
The compatibility condition ∂ ∂ J = ∂ ∂ J yields an equation
x y y x
∂ (J−1∂ J) = ∂ (J−1∂ J), (36)
x˜ x y˜ y
while (34) is identically satisfied by virtue of (35). In other words, (35) defines a B¨acklund transforma-
tion relating (34) and (36).
If we define the gauge potentials by setting A = ∂ K, A = ∂ K, A = 0, A = 0, then they
x y˜ y x˜ x˜ y˜
satisfy the anti-self-dual Yang–Mills equations on R4 with Euclidean metric with coordinates Xi such
that √2x = X1 +iX2, √2x˜ = X1 iX2, √2y = X3 iX4, √2y˜ = X3 +iX4, where i = √ 1, cf. e.g.
− − −
[31]. Conversely, any sufficiently smooth solution of the anti-self-dual Yang–Mills equations can, up to
9
a suitable gauge transformation, be obtained in this way [31]. In other words, the anti-self-dual Yang–
Mills equations, which play an important role in modern physics, cf. e.g. [31] and references therein,
can, in a suitable gauge, be written in dispersionless form, exactly as claimed in Introduction.
It is natural to assume that in the Lax pair equations L ψ = 0 we have ψ G = exp(g). However,
i
∈
such ψ cannot be a symmetry for (34), since a symmetry must live in g just like K. However, we can
get a Lax representation for Φ g using the adjoint action: [L ,Φ] = [L ,Φ] = 0, i.e., we use the Lax
1 2
∈
operators = ad L .
i i
L
Then all conditions of Proposition 1 are satisfied for A and B obtained by spelling out as
i i i
L
= λA B , which yields the following RO for (34):
i i i
L −
V = U +[∂ K,U], V = U +[∂ K,U], (37)
x˜ y x˜ y˜ x y˜
where U and V are symmetries for (34). This RO is known from the literature, see [6, 42, 2].
If in the above construction we take g = diff(RN), i.e., we put
N
K = Xui∂/∂zi,
i=1
where u = u (x,y,x˜,y˜,z1,...,zN) are scalar functions, then (34) becomes [28] the Manakov–Santini
i i
system [25], an integrable system in (N + 4) independent variables, and (37) provides a recursion
operator for this system which was found earlier in [30] (see also [28]) by a different method.
Further examples of recursion operators obtained using our approach can be found in [19].
5 Concluding remarks
We presented above a method ofconstruction for recursion operatorsand adjoint recursion operatorsfor
a broad class of multidimensional integrable systems that can be written as commutativity conditions
for a pair of vector fields, or, even more broadly, of linear combinations of vector fields with zero-
order matrix operators, linear in the spectral parameter and free of derivatives in the latter. This
method is based on constructing a special Lax-type representation for the system under study from the
original Lax-type representation, see Section 3 for details. Note that if a system under study admits
several essentially distinct Lax representations then our approach applied to these Lax representations
in principle can give rise to several essentially distinct recursion operators.
To the best of author’s knowledge, the method in question works for all known examples of mul-
tidimensional dispersionless integrable systems admitting recursion operators in the form of B¨acklund
auto-transformation for linearized systems, including e.g. the ABC equation [48]
au u +bu u +cu u = 0, a+b+c = 0,
x yt y xt t xy
the simplest (2+1)-dimensional equation of the so-called universal hierarchy of Mart´ınez Alonso and
Shabat [29],
u u u +u u = 0,
yy y tx x ty
−
the complex Monge–Amp`ere equation (see [38] for its recursion operator), first, second and modified
heavenly equations, etc. In particular, this method enabled us to find hitherto unknown recursion
operators for the general heavenly equation of Doubrov and Ferapontov (21) and for the equation
(31). Note that our approach, when applicable, is in general computationally less demanding than the
methods of [30] and [34].
While all examples to which we have applied our approach so far are dispersionless, Propositions 1
and 2 and the method described in Section 3 do not explicitly require this to be the case, so it would
be interesting to find out whether it is possible to extend our approach to the construction of recursion
operators to other classes of integrable systems.
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