Proc. Nati. Acad. Sci. USA
Vol. 77, No. 9, pp. 5025-5027, September 1980
Applied Mathematical Sciences
Backlund transformations and nonlinear differential difference
equations
(nonlinear evolution equations/Schrbdinger spectral problem/soliton)
D. LEVI* AND R. BENGURIAt
Physics Department, The Rockefeller University, New York, New York 10021
Communicated by Kenneth M. Case, June 2,1980
ABSTRACT It is shown that any Bicklund transformation
of a nonlinear differential equation integrable by the multichannel Schrbdinger eigenvalue problem can be written in the
form V1 = U'V - VU. This allows us to interpret the Bicklund
transformation formally as a nonlinear differential difference
equation for which we can immediately construct the soliton
solutions.
Auto-&icklund transfonnations (BT) are an important property
of any integrable nonlinear evolution equation and have been
derived for a wide class of equations. BT are equations that
relate different solutions of the same nonlinear evolution
equation; the application of the BT allows one to construct the
soliton solutions of the nonlinear evolution equations. In this
note we want to show that, for the multichannel (i.e., matrix)
Schr6dinger spectral problem (MSP), all the BT defined
through the Wronskian technique (1) can be written in a very
simple and unique way (see Eq. 12). We consider the MSP only
because the proof of our statement becomes simpler and also
because MSP includes a wide class of other problems, in particular the Zakharov-Shabat problem, which can be obtained
from MSP by means of the reduction techniques.t Eq. 12 seems
appropriate for studying the group properties of the BT, and
work on this is in progress. On the other hand, it can be formally
intepreted as a functional differential equation (DDE) (or
elsewhere also called differential difference equations), which
is integrable and for which we shall provide the one-soliton
solution.
Backlund transformations for the MSP
We shall summarize some of the results contained in ref. 1. The
MSP is defined by the differential equation
Vxx = (Q - k2)i,
[1]
in which Q is an X X q hermitian matrix that depends on x e
R and 61 is a vector that depends on x and keC. Both 41 and Q
may depend parametrically on some other variable. We assume
Q vanishes exponentially as x I co. For Eq. 1, we can define
the continuum part of the spectrum by the following asymptotic
boundary condition for the matrix solution A, for ke R, built
up of X7 linearly independent vector solutions of Eq. 1,
e-ikx+ R(k)ekx,
4I(x,k)
k)
'I'x
[2a]
> T(k)e-ikx.
x
--I
-
co
(-4k2) 3
dxTI'T(x,k)F(x)YI(x,k)
= 3' dx I'T(x,k)AF(x)T(x,k), [3]
in which he T is the transpose of the matrix
erator A is defined by
AF(x) = F..(x) - 2[Q'(x)F(x) + F(x)Q(x)]
V'. Here the op-
+r
dx'F(x') [4]
with the operator r defined as
rF(x) = Q'(x)F(x) + F(x)Qx(x)
+ 3
cho dx'[Q'(x)Q'(x')F(x') - Q'(x)F(x')Q(x')
-
[5]
Q'(x')F(x')Q(x) + F(x')Q(x')Q(x)].
From the Wronskian identity and the asymptotic behavior of
' and T' given in Eq. 2, we get (see equation 3.2.1 of ref. 1)
(2ik)[MR(k) - R'(k)M] = 3
dxI'T(x,k)
X [MQ(x) -
Q'(x)M]*(x,k) [6a]
and (see equation 3.2.3 of ref. 1)
(-4k2)[NR(k) + R'(k)N] = f
dx IT(x,k)(rN)i&(x,k),
[6b]
in which M and N are independent of x but otherwise are arbitrary n X n matrices. Let f and g be arbitrary entire scalar
functions. § Then from Eqs. 3 and 6 we get
(2ik)f(-4k2)[MR(k) - R'(k)M]
= 3' dxt'IT(x,k)[f(A){MQ(x) -Q'(x)Mj]4'(x,k) [7a]
_co
Abbreviations: BT, Bicklund transformation(s); MSP, multichannel
Schrodinger spectral problem; DDE, differential difference equation(s).
* Permanent address: Instituto di Fisica dell'Universita di Roma, 00185
Roma, Italy.
t On leave from the Department of Physics, Universidad de Chile,
Santiago, Chile.
*Calogero, F. & Degasperis, A. (1979) Reduction Technique for
Matrix Nonlinear Evolution Equations Solvable by the Spectral
Transform, preprint, Istituto di Fisica, Universita di Rotna.
§ One can also considerf and g as being matrix-valued entire functions.
For simplicity we consider the case when they are scalar func-
-D
+
4f(x,k)
(equation 3.1.3 in ref. 1) one obtains, for an arbitrary matrix
F(x), which vanishes asymptotically together with all its derivatives, the following relationship,
0
Let A' and T be two matrix solutions of Eq. 1 (with potential
Q' and Q, respectively, and reflection coefficients R' and R,
respectively). Then through the generalized Wronskian identity
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tions.
5025
5026
Applied Mathematical Sciences: Levi and Benguria
and
(-4k2)g(-4k2)[NR(k) + R'(k)N]
Proc. Natl. Acad. Sci. USA 77 (1980)
Eq. 17b is identically satisfied when we substitute the expressions for A and B given by Eqs. 16. In fact, we get
clo
dx"
*T(x,k)[g(A)rN]T(x,k). [7b]
= 5
By using Eqs. 7, Calogero and Degasperis (1) have shown that
the BT for the MSP can be written as
[8a]
f(A)[MQ(x) - Q'(x)M] + g(A)rN = 0,
and the corresponding reflection coefficients R and R' are related by
R'(k) = [f(-4k2)M - 2ik g(-4k2)N]R(k)
X [f(-4k2)M + 2ik g(-4k2)N]-l. [8b]
One can write the MSP given by Eq. 1 in the form
ox(xk)=U(Qk)=(xk),
in which U is a 2,q X 2n matrix defined by
U(Qk)
=
(-kI Q)
-2D(x) - 2Q'(x)
dyD(y)
dyD(y)Q(x) - 2g(-4k2)Q'(x)N
+ 2
+ 2g(-4k2)NQ(x) -2g(-4k2)[NQ(x) -Q'(x)N] +
-2 3
2D,(x)
dy[D(y)Q(x) - Q'(x)D(y)] 0.
As for Eq. 17a, when we substitute the expression for A and B
given by Eqs. 16, it becomes
Dx(x) - 2[D(x)Q(x) + Q'(x)D(x)]
[9]
+ 3
+ 3J
dy[D(y)Qx(x) + Q'(x)D(y)]
dy
dz[D(z)Q(y)Q(x) - Q'(y)D(z)Q(x)
[10]
Q'(x)D(z)Q(y) + Q'(x)Q'(y)D(z)]
4k2D(x) - g(-4k2)[NQx(x) - Q'(x)N]
-
in which I is the identity i X t7 matrix and k is the 27 vector
constructed from i in the following way
=
(1
-
ik¢,)
g, can be written as
[12]
VX = U(Q',k)V - VU(Q,k).
Proof: Let us write V in terms of two q X i matrices A and
B,Ax + ikBx + BQ
[13]
V (A + Bx-ikB
B
~~A + ikB
in which A and B are given by
00
00c
dy 3 dz $D(z)Q(y) - Q'(y)D(z)l
f(-4k2)M + g(-4k2) 3 dy [NQ(y) - Q'(y)N] [14a]
and
B
=
23
dy D(y) + 2g(-4k2)N,
[14b]
with
+
j(x)(rN), [15]
= f(A) - f(-4k2)
[16a]
D(x) = Q(x)[MQ(x) - Q'(x)M]
in which
A + 4k2
+
[11]
THEOREM. Let U(Q,k) [and U(Q',k)] be the 2q X 2i1 matrix
defined by Eq. 10, in which Q (and Q', respectively) is the
matrix potential and k the eigenvalue for the MSP given by
Eq. 1; then we can explicitly construct a 2q X 2q matrix
V(Q,Q',f,gk) such that the BT (Eq. 8a), parametrizd by f and
A -D(x) + J
+
and
-
-
f(-4k2)[Q'(x)M - MQ(x)]
g(-4k2) 3
dy[Q'(x)NQ(y) - Q'(x)Q'(y)N
NQ(y)Q(x) + Q'(y)NQ(x)] + 2g(-4k2)NQx(x) = 0.
Recalling Eqs. 4 and 5, we can write the previous expression
as
[A + 4k2]D(x) + f(-4k2)[Q'(x)M - MQ(x)]
+ g(-4k2)rN = 0. [18]
Recalling Eqs. 16, Eq. 18 finally reads
f(A)[MQ(x) - Q'(x)M] + g(A)rN = 0,
which is exactly the BT given by Eq. 8.
Differential difference equations
It is known (2) that one can apply the generalized inverse
method of Zakharov and Shabat to solve some DDE. In order
to do that, one writes the DDE as the compatibility condition
for a linear problem
[19a]
4I',(n,x) = U(n,x)%t(n,x)
[19b]
4I'(n + 1,x) = V(nx)q(nx),
in which n is an integer, x is a real variable, and 4V, U, and V are
matrix-valued functions depending on a spectral parameter k.
The compatibility condition
VX(n,x) = U(n + 1,x)V(n,x) - V(n,x)U(n,x) [20]
is equivalent to the DDE. We note that Eq. 20 has the same
structure as Eq. 12. In particular, if we formally substitute
Q = qn
[21]
in Eqs. 8 (i.e., in the BT), we get a DDE that can be solved by
the inverse method; the U and V corresponding to this DDE
are exactly the ones defined by Eqs. 1 and 13, respectively.
The solutions that one obtains for the MSP through the inverse spectral technique are also solutions to the DDE when the
evolution in time is determined by the matrix V or, that is, the
Q= qn+I
(ajx)
=
g(A) - g(-4k2)
[16b]
A + 4k2[1b
By Eqs. 10 and 13, Eq. 12 becomes
AXX-Q'A + AQ + 2Bx(Q-k2)+ BQx = 0 [17a]
BXX-Q'B + BQ + 2AX =0.
[17b]
Proc. Natl. Acad. Sci. USA 77 (1980)
Applied Mathematical Sciences: Levi and Benguria
reflection coefficient evolves according to the following
equation:
R(n + 1) = [f(-4k2)M - 2ikg(-4k2)N]R(n)
X [f(-4k2)M + 2ikg(-4k2)N]-l. [22]
Then, following ref. 1, the one-soliton solution to the DDE
corresponding to a purely imaginary eigenvalue k = ip of the
MSP, given by Eq. 1, reads
q(n,x) = -2 p2 Icosh[pfx - t(n)1]-2 P(n),
[23]
in which P(n) is an idempotent matrix given by
(CoIE(n- n)ICo) fM + 2pgN]n-nop(no)
X JM - 2pgN]-(n-no), [23a]
with
(Col
=-ICo)
P~o(ColICo)
P(no)
and
[23b]
E(n) = LfM + 2pgN]-[fM - 2pgN]n,
[23c]
4=o+-1Ilog{(coIEln no)J.
[23d]
-
The vector ICo) is given by the asymptotic behavior of i1, as
x - + co, corresponding to the initial problem. It is important
to notice that the value k = ip is a pole of the reflection coefficient of the BT of the MSP but it is not a pole of Eq. 22, which
gives us the evolution of the DDE.
An example
To conclude this note, we exhibit the matrix V corresponding
to the simplest BT contained in Eq. 8; then we write down the
corresponding DDE together with its one-soliton solution. The
simplest BT is obtained by setting f = constant = p, g = constant = 1/2, and M = N = l:
W'2(x) + Wx(x) + 2pfW'(x) - W(x)j
+ 1/2[W'(x) - W(x)]2 = 0, [24]
in which we have introduced W defined by
W(x)
dx'Q(x').
From Eq. 14 we derive
A(x) = -pI + 1/2fW(x) - W'(x)j
[25]
[26a]
5027
and
B(x) = I,
since D = 0. Therefore, V reads
[26b]
[27]
-1/2(Wx + W')
(-p + ik)I + 1/2(W - W')J
One can easily check at this point that V. = U'V - VU, with
-(P + ik)I + 1/2(W
-
+ W9)
I
U given by Eq. 10, is equivalent to Eq. 24. The DDE that one
obtains from Eq. 24 is
[w(n + 1,x) + w(n,x)]x + 2h1w(n + 1,x) - w(n,x)}
+ 1/2fw(n + 1,x) - w(n,x)12 = 0. [28]
As we have said in the general framework, the DDE given by
Eq. 28 can be put into the Zakharov-Shabat form, with U and
V being
- kx(nx)
U(n,x) = (-ikI
[29]
ikW
V(nx)
=
-(h + ik)I + 1/2{w(n,x) - w(n
+
1,x)l
-1/2fw.(n,x) + w.(n + 1,x)
-(+ h - ik)I + 1/2tw(n,x) - w(n + 1,x)j< [30]
The one-soliton solution to Eq. 28 is given by
w(n,x) = -2 1i - tghfp(x -(n))JjP(no),
in which P(no) is given by Eq. 23b and
t(n) = to + 2p (n -nO) log {
pJ
This work was supported in part by the National Science Foundation
under Grant MCS 78-20455 (to R.B.) and in part by a North Atlantic
Treaty Organization Fellowship (to D.L.).
1. Calogero, F. & Degasperis, A. (1977) II Nuovo Cimento, 39B,
1-54.
2. Levi, D., Ragnisco, 0. & Bruschi, M. (1980) "Extension of the
Zakharov-Shabat generalized inverse method to solve differential
difference and difference-difference equations," presented at the
Jadwisin Workshop on Solitons (Warsaw, 1979), Il Nuovo Cimento, in press.