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达布变换基础及应用.pdf
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DARBOUX TRANSFORMATIONS FROM REDUCTIONS
OF THE KP HIERARCHY
J J C NIMMO
Department of Mathematics, University of Glasgow,
Glasgow G12 8QW, Scotland
ABSTRACT
The use of effective Darboux transformations for general classes Lax pairs is dis-
cussed. The general construction of “binary” Darboux transformations preserving
certain properties of the operator, such as self-adjointness, is given. The classes of
Darboux transformations found include the multicomponent BKP and CKP reduc-
tions of the KP hierarchy.
1.
Introduction
Darboux transformations define a mapping between the solutions of a linear dif-
ferential equations and a similar equation containing different coefficients. Since in-
tegrable nonlinear evolution equations frequently arise as the compatibility condition
for a pair for such equations, Darboux transformations may be used to construct fam-
ilies of exact solutions of the nonlinear equations. Typically these are multi-soliton
solutions. A good introduction to this topic, including the following example, is given
in the monograph by Matveev and Salle1.
As an example which motivates the work to be presented, consider the Lax pair∗
L = i∂y + ∂2 + u, M = ∂t + 4∂3 + 3∂u + 3u∂ + 3i∂−1(uy).
for the variant of the Kadomtsev-Petviashvili equation known as KPI
(uu + 6uux + uxxx)x) − 3uyy = 0,
in which u is a real variable. This means that [L, M] = 0 if and only if u satisfies
KPI.
For all non-zero θ such that L(θ) = M(θ) = 0, a Darboux transformation is
defined by the operator G = θ∂θ−1 in the sense that
∗Here and below ∂ = ∂/∂x and ∂y = ∂/∂y and so on.
L(ψ) = M(ψ) = 0 =⇒ eL(G(ψ)) = fM (G(ψ)) = 0,
where eL and fM are the operators obtained from L and M by replacing u by eu =
u + 2(log θ)xx. This result is readily proved by observing that
i.e.
eLG = GL and fM G = GM,
eL = GLG−1 and fM = GM G−1.
In this way the Darboux transformation manifests itself as a (differential) gauge
transformation. It also follows that
[L, M] = 0 =⇒ [eL, fM ] = 0,
auto-B¨acklund transformation.
There is a problem with this transformation however. For almost all u, since θ is
i.e. eu satisfies KPI whenever u does. Hence the Darboux transformation induces an
the solution of a complex equation, eu is not real and so we do not obtain solutions of
KPI. At the root of the problem is the fact that, while L and M are self-adjoint, eL and
fM are not. In order to overcome this problem one may use a “binary” transformation
(to be defined in the next section) which does preserve the self-adjointness of L and
M.
This paper is concerned with the use of binary transformations to preserve the
structure of two classes of operators with matrix coefficients and arbitrary order.
2. The structure of the binary transformation
For an (matrix) operator L, let S = {θ, non-singular : L(θ) = 0} (and define eS,
S† for operators eL, L† etc.). A (formally invertible) gauge transformation Gθ, for
θ ∈ S, defines a mapping
Consider also the (formal) adjoint operator G†
θ. (Taking the formal adjoint is, as
Gθ: S → eS, where eL = GθLG−1
θ .
usual, the linear operation defined by
(a∂i)† = (−1)i∂ia†,
for a matrix a, where a† denotes the Hermitian conjugate of a.) Since eL† = G†−1
we have
θ L†G†
θ,
ically, we may identify this subset in terms of θ and denote a member as i(θ). For
example, in the classical case when Gθ = θ∂θ−1 and G†
∂θ†, we find that
θ = −θ†−1
G†
θ: eS† → S†.
θ we obtain some nontrivial solution in eS†. Typ-
By determining the kernel of G†
G†
θ(ρ) = 0 ⇐⇒ ρ = θ†−1
c,
where c is independent of x.
We represent this situation in the diagram below.
Gθ
G†
θ
S
θ
S†
.
eS
eS†
i(θ)
G−1
ˆθ
Gθ: S → bS.
To describe the general form of the binary transformation we consider operators
L, eL and bL and the corresponding sets of non-singular solutions matrices S, eS and
bS. Let θ ∈ S and bθ ∈ bS be such that Gθ: S → eS and Gˆθ: bS → eS. Then we get the
mapping
The difficulty with this definition of a transformation is that to define it we need
one of the solutions we are trying to determine, namely bθ! To overcome this, we
use the fact that there corresponds to bθ ∈ bS a solution i(bθ) ∈ eS† and then use the
(φ)) for any φ ∈ S†. This is shown in
θ
mapping G†−1
the diagram below.
θ
Gθ
: S† → eS† to obtain bθ = i−1(G†−1
/ eS
eS†
/ i(bθ)
S†
φ
G†
S
θ
G ˆθ
bS
In this way we obtain the definition of a general binary transformation.
Definition Consider an operator L and gauge operator Gθ, where θ ∈ S, such that
G†
θ(i(θ)) = 0. For each φ ∈ S†, define
Gθ,φ = G−1
ˆθ
Gθ,
(φ)). Then
θ
where bθ = i−1(G†−1
where bL = Gθ,φLG−1
θ,φ, is called a binary transformation.
Gθ,φ: S → bS,
In the next section we will consider two concrete examples of such a binary transfor-
mation.
Now suppose that the operator L has a constraint of the form
L†R† = RL,
/
/
_
_
o
o
/
o
o
A
A
o
o
/
where R is in some formally invertible (matrix differential) operator†. We wish to find
binary transformations that preserve this constraint. That is—using the notation of
the above definition—we want bL to satisfy the constraint whenever L does.
Examples for the choice of R include
• R = I. L is self-adjoint. An example of the application of this is quoted in the
introduction.
• R = iI. L is skew-adjoint. This corresponds to the CKP reduction of the
KP hierarchy2 and the reduction of the Kuperschmidt “k = 0” non-standard
hierarchy3, 4.
• R = ∂. This corresponds to the BKP5 or the Kuperschmidt “k = 1” reduction.
The binary transformation we will discuss in the next section will preserve general-
izations of these three reductions.
Let the gauge transformation G, such that bL = GLG−1, preserve the constraint
L† = RLR†−1. Then
which means that
bL† − RbLR†−1
= 0
G†−1
L†G† − RGLG−1R†−1
= G†−1
RLR†−1
G† − RGLG−1R†−1
= 0.
This leads to the single condition RG = G†−1
R.
Note that the relation between L and its adjoint imposes a relationship between
the solution sets S and S†. In particular, for each θ ∈ S, R†(θ) ∈ S†. Hence, in the
case of a binary transformation G = Gθ,φ, we may make the choice φ = R†(θ).
3. Darboux transformations for general operators
In this section we describe two classes of Darboux transformation for general
classes of matrix differential operators of arbitrary order. The first is originally due
to Matveev6 and has also been considered recently by Oevel7. We will present a very
simple proof of this result. The second was found by Oevel & Rogers8 in the case of
scalar operators in the context of Sato theory. We will derive a more general version
here.
In both cases, the results are remarkably general. There is however a serious
drawback. There is, in this general case, absolutely no guarantee that the transformed
operator we have the same “form” as the original and so only in special cases does
one get a transformation that induces an auto-B¨acklund transformation.
†It is tempting to look for a constraint of the form L†S = RL but this in fact corresponds to
two constraints since on taking adjoints L†R† = S†L.
First, consider
L = ∂t +
nXi=0
ui∂i and eL = ∂t +
nXi=0eui∂i,
where ui and eui are N × N (not necessarily constant) matrices. Let the operator G
be such that
Hence G must satisfy
nXi=0
[G, L]G−1 =
eL = GLG−1 = L + [G, L]G−1.
(eui − ui)∂i.
nXi=0
[G, L]G−1 = [G, L]θ∂−1θ−1 =
ai∂i−1θ−1,
Taking G = θ∂θ−1, where θ is a non-singular N × N matrix, and hence G−1 =
θ∂−1θ−1, we get
for some matrices ai. For i = 1, . . . , n, ai = eui−1 − ui−1 and in order that G define a
Darboux transformation we must have
a0 = 0.
This condition gives [G, L](θ) = 0 i.e. G(L(θ)) = 0 since G(θ) = 0. Hence we only
need require that L(θ) = θC, for some x-independent matrix C. Note that if L(θ) = 0
then for θ′ = θ exp(∂−1
t (C)), L(θ′) = θ′C. Also, Gθ′ = Gθ and so we may suppose,
without loss of generality, that C = 0‡. Thus we find that θ ∈ S.
The second case we consider is
L = ∂t +
nXi=1
ui∂i and eL = ∂t +
nXi=1eui∂i,
where ui and eui are again N × N matrices. Note that the multiplicative term in L
and eL is omitted.
As in the first case, a gauge operator G must satisfy
[G, L]G−1 =
[G, L]G−1 =
nXi=1eui − ui.
nXi=0
ai∂i,
There are now two simple choices. First, let G = G(1)
matrix. Then
θ = θ−1, an (invertible) N × N
‡Note that if L is an ordinary differential operator, then taking C 6= 0 is a genuine generalization.
For example, this is exploited in the classical “discrete eigenvalue adding” Darboux transformation
for the time-independent Schr¨odinger operator.
and so a0 = [G, L](θ) = G(L(θ)) = 0, i.e. θ ∈ S.
Second, let G = G(2)
ρ = ρ−1
x ∂, where ρx is an invertible N × N matrix. Now
[G, L]G−1 = [G, L]∂−1ρx =
ai∂i−1ρx,
n+1Xi=1
[G, L](∂−1(ρx)) = G(L(ρ)) = 0. Thus L(ρ) = C, an
and we must have a1 = 0, i.e.
x-independent matrix. Again, we may take C = 0 without loss of generality and so
ρ ∈ S.
As in the scalar case8, it is the composition of the two gauge transformations
which is of most interest, and we take Gθ = G(2)
G(1)
θ = (θ−1)−1
x ∂θ−1.
G
(1)
θ (1)
4. Binary transformations and reductions
To determine the binary transformations Gθ,φ corresponding to the two Darboux
transformations found above we must determine two additional things: the mapping
i: ˆS → eS† and then the element bθ ∈ bS in terms of θ and φ.
First consider L = ∂t +Pn
∂(θ†i(θ)) = 0 is satisfied by the choice i(θ) = θ†−1. Further,
i=0 ui∂i, Gθ = θ∂θ−1. Here the condition G†
−θ†−1
θ(i(θ)) =
θ
bθ = G†−1
= −θ†−1
(φ)†−1
∂−1(θ†φ)†−1
= −θΩ−1,
where Ω = ∂−1(φ†θ). It may be shown that for all operators L = Pn
exact in the sense that dΩ = φ†θdx + A(u1, . . . , un, θ, φ)dt7.
i=0 ui∂i, Ω is
In this case the binary transformation is
Gθ,φ = G−1
ˆθ
Gθ = θΩ−1∂−1Ω∂θ−1
= θΩ−1∂−1(∂Ω − Ωx)θ−1
= 1 − θΩ−1∂−1φ†.
For discussion of the reduction we will also need
= 1 − φΩ†−1
G†−1
∂−1θ†.
Now suppose that L satisfies the constraint L†R† = RL where R = A, a (not
necessarily constant) matrix. Then we may choose φ = R†(θ) = A†θ. The condition
RGθ,φ = G†−1
θ,φ R now gives
A − AθΩ−1∂−1φ† = A − φΩ†−1
∂−1θ†A ⇐⇒ AθΩ−1∂−1θ†A = A†θΩ†−1
∂−1θ†A
⇐⇒ A† = ±A.
This establishes the following theorem.
Theorem 1 Let the matrix operator L =Pn
L†A = AL,
i=0 ui∂i satisfy the constraint
where A is an Hermitian or skew-Hermitian matrix. Then the binary transformation
where Ω = ∂−1(θ†Aθ), preserves the above constraint, i.e.
G = 1 − θΩ−1∂−1θ†A
bL†A = AbL.
For the second case, L =Pn
x ∂θ−1 and hence i(θ) = (θ†−1)x.
To determine the binary transformation it is notationally convenient to write an
element of S† as φx rather than φ as we did above. Also, it is necessary to introduce
two integrals
i=1 ui∂i, Gθ = (θ−1)−1
bL = GLG−1 satisfies
where
Now
and so
Ω = ∂−1(φ†θx) and Ω′ = ∂−1(φ†
xθ),
Ω + Ω′ = φ†θ.
i(bθ) = (bθ†−1
)x = G†−1
θ
(φx)
= −(θ†−1
)x∂−1(θ†φx)
(bθ−1)x = −∂−1(φ†
bθ = −Ω′θ−1 + ∂−1(Ω′
= φ† − Ω′θ−1−1
= θΩ−1.
xθ−1)−1
Integrating by parts and taking inverses, we get
xθ)(θ−1)x = −Ω′(θ−1)x.
We may now obtain
Gθ,φx = bθ∂−1(bθ−1)x(θ−1)−1
= −θΩ−1∂−1Ω′∂θ−1
= 1 − θΩ−1∂−1φ†∂,
x ∂θ−1
and in a similar way
G†−1
θ,φx = 1 − ∂φΩ′†−1
∂−1θ†.
Suppose that L satisfies the constraint L†R† = RL where R = A∂, A a matrix,
θ,φxR
and choose φx = R†(θ) = −(A†θ)x, i.e. φ = −A†θ. The condition RGθ,φx = G†−1
is
A∂ − A∂θΩ−1∂−1θ†A∂ = A∂ − ∂A†θΩ′†−1
∂−1θ†A∂ ⇐⇒ A† = ±A and Ax = 0.
With these conditions on A, Ω = ±Ω′†.
This establishes a second theorem.
Theorem 2 Let the matrix operator L =Pn
L†A∂ + A∂L = 0,
i=1 ui∂i satisfy the constraint
where A is an x-independent Hermitian or skew-Hermitian matrix. Then the binary
transformation
G = 1 − θΩ−1∂−1θ†A∂
where§ Ω = ∂−1(θ†Aθx), preserves the above constraint, i.e. bL = GLG−1 satisfies
bL†A∂ + A∂bL = 0.
5. Examples
5.1. Davey-Stewartson I
This system has Lax pair
L = ∂y + α∂ + Q, M = i∂t + α∂2 +
1
2
(Q∂ + ∂Q + αQy) + D,
where Q = 0 u
Let A = 1
ǫ¯u 0 ! and D = U 0
0 −ǫ !. Then
0
0 V ! is real.
L†(iA)† = (iA)L, M †A† = AL.
Hence we may use Theorem 1 (with A = I) to obtain a binary transformation. This
transformation has been used to obtain a wide class of solutions including dromions9.
5.2. Sawada-Kotera equation
The equation is a reduction of the BKP equation and so has Lax pair admitting
the BKP reduction:
L = (∂2 + 3u)∂, M = ∂t + (9∂5 − 15∂u∂ + 30(∂2u + u∂2) + 15u2)∂,
§For a better notation we have replaced Ω with −Ω in the statement of the theorem
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