Backlund Transformation
February 6, 2015
Chapter 1
Basic Concepts
1.1
Soliton
A soliton is a self-reinforcing solitary wave that maintains its shape while it propagates at a
constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in
a medium. Solitons arise as the solutions of a widespread class of weakly nonlinear dispersive
partial differential equations describing physical systems.
1.2
Pseudosphere
The pseudosphere is the constant negative-Gaussian curvature surface of revolution generated by a tractrix about its asymptote. The cartesian parametric equations are:
x = sech u cos v
y = sech u sin v
z = u - tanh u
for u (, ) and v [0, 2)
1.3
Lax Pair
In the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or
operators that satisfy a corresponding differential equation, called the Lax equation. A Lax
pair is a pair of matrices or operators L(t) and P (t) dependent on time and acting on a
fixed Hilbert Space, and satisfying Laxs equation:
dL
= [P, L]
dt
where[P, L] = P L - LP is the commutator.
1.3.1
Example
The Korteweg-de Vries equation is
ut = 6uux - uxxx
It can be reformulated as the Lax equation
1
dL
dt
= [P, L]
where
2
L = - x
2 + u
3
P = -4 x3 + 3(u x
+
1.4
u
)
x
Hirotas D-Operator
In order to apply Hirotas method it is necessary that the equation is quadratic and that
the derivatives only appear in combinations that can be expressed using Hirotas D-operator
defined by:
Dxn f g = (x1 x2 )n f (x1 )g(x2 )|x2 =x1 =x
For example,
Dx f g = fx g f gx
Dx Dt f g = fxt g fx gt ft gx + f gxt
Thus D operates on a product of two functions like the Leibniz rule, except that it is antisymmetric,
Dxn f g = (1)n Dxn g f
Some useful properties of the Hirotas D-Operator:
Property 1:
2
logf
x2
1
(Dx2 f
2f 2
f)
Property 2:
Dt [(Dx a b) cd + ab (Dx c d)] = (Dt Dx a d)cb ad(Dt Dx c b)
+(Dt a d)(Dx c b) (Dx a d)(Dt c b)
Property 3:
(Dx4 a a)cc aa(Dx4 c c) = 2Dx (Dx3 a c) ca
+6Dx (Dx2 a c) (Dx c a)
Property 4:
(Dx2 a b)cd + ab(Dx2 c d) = (Dx2 a d)cb + ad(Dx2 c b)
2(Dx a c)(Dx b d)
(1.1)
1.5
History
Two ideas seeded the invention of Backlund transformations
Lies work on contact transformations.
Geometric study of pseudospherical surfaces.
Chapter 2
B
acklund Transformation
2.1
Definition
Backlund transformation is an invariant transformation that transforms one solution to another of a differential equation. It usually involves partial derivatives, and is easily solvable
once the initial solution is given. The most common use of Backlund transformations is to
obtain multi-soliton solutions to integrable systems.
2.2
Uses of B
acklund transformation
If a Backlund transformation can be found, then the solution of a nonlinear partial differential
equation can be used to obtain either a different solution to the same partial differential
equation, or to obtain a different nonlinear partial differential equation.
2.3
Idea
From a solution of a nonlinear partial differential equation, we can sometimes find a relationship that will generate the solution of:
A different partial differential equation (i.e., a Backlund transformation)
The same partial differential equation (i.e., an auto-Backlund transformation)
2.4
Procedure to determine B
acklund transformation
The first step is to determine a Backlund transformation between two partial differential
equations. There are various methods available that can be used and some will be illustrated
in the examples to follow but usually finding the Backlund transformation is an extremely
difficult task. After the Backlund transformation is determined, it will utilize a solution of
one of the partial differential equations to determine a solution to other partial differential
equation.
2.5
Examples
2.5.1
Cauchy-Reimann Equations
Consider the Cauchy-Riemann System:
ux = vy , uy = -vx
If u and v are solutions of the Cauchy-Riemann equations, then u is also a solution of the
Laplace Equation uxx + uyy = 0 and so is v. This follows straightforwardly by differentiating
the equations with respect to x and y and using the fact that
uxy = uyx and vxy = vyx
If u is a solution of the Laplace Equation, then there exists a function v which solves
the Cauchy-Riemann equations together with u.
To further illustrate this example let us consider
u(x, y) = xy. The Cauchy-Riemann equations become:
vy = y and vx = x
The solution is
v(x, y) = 12 (y 2 x2 ) + c
which solves the Laplace equation along with u.
2.5.2
Sine-Gordon Equation
Consider the Sine-Gordon equation:
uxy = sin u
(2.1)
where u is a real-valued function of two variables x and y. Let us assume that a Backlund
transformation transforms u into v where v also satisfies the same Sine-Gordon Equation
vxy = sin v
(2.2)
and our goal is to find the Backlund transformation. Let w+ = 21 (u + v) and w = 21 (u v),
then w+ and w satisfy
+
wxy
= sin w+ cos w and wxy
= cos w+ sin w
(2.3)
We assume that the Backlund transformation has the trial form
wx = f (w+ )
(2.4)
and by the second equation in (2.3) we have
f 0 wy+ = cos w+ sin w
(2.5)
By substituting (2.5) into the first equation of (2.3), we have
+
w f
wx sin
[ f 0 cos w+ + sin w+ ] + cos w [ ff0 cos w+ - sin w+ ] = 0
f0
which can be satisfied if
f
cos w+ + sin w+ = 0 and ff0 cos w+ - sin w+ = 0
f0
A solution of this over determined system can be found
f = a sin w+
where a is the arbitrary constant called a Backlund parameter. (2.4) and (2.5) now take the
form
wx = a sin w+ and wy+ =
1
a
sin w
(2.6)
(2.6) is the Backlund transformation of the Sine-Gordon equation (2.1). After experimenting
with different forms of assumptions like (2.4), one may find a Backlund transformation for
a given equation.
Since an auto-Backlund transformation is known, Eq. (2.6), and is also given in terms of
u and v by equations (2.7) and (2.8), we can try to find the solution of the Sine-Gordon
Equation.
)
vx = ux + 2a sin( v+u
2
(2.7)
vy = uy +
2
a
sin( vu
)
2
(2.8)
Differentiating equation (2.7) w.r.t. y and equation (2.8) w.r.t. y we have,
vxy = uxy + 2 sin( vu
) cos( v+u
)
2
2
(2.9)
vxy = uxy +
2 sin( v+u
) cos( vu
)
2
2
(2.10)
Adding equations (2.9) and (2.10) we get,
vxy = sin v
Starting with the solution u(x, y) = 0 of equation (2.1), we can use the auto-Backlund
transformation to determine another solution; equations (2.7) and (2.8) become
vx = 2a sin v2 and vy = a2 sin v2
This system of equations is easily solved to yield a new solution of the Sine-Gordon equation
tan v4 = Cexp(ay + xa )
This solution may be used to determine another solution ,and so on.
6
Chapter 3
B
acklund transformation determining
methods
3.1
Chens Method
Chens method is an efficient method for determining the Backlund transformation. This
procedure will be illustrated by considering Korteweg-de Vries equation.
ut + 6uux + uxxx = 0
its Lax pair is given by
L = , t = A
where
L = x2 + u, A = -4x3 - 3(ux + x u)
Let v = x /, one gets
vx + v 2 + u =
(3.1)
2
-vt = 4vxxx + 12vvxx + 12v vx +
12vx2
+ 6ux v + 6uvx + 3uxx
(3.2)
Eliminating u, v satisfies the modified Korteweg-de Vries which is given by
vt - 6v 2 vx + 6vx + vxxx = 0
Relation (3.1) is the Miura transformation between KdV and modified KDV. If v solves the
modified KdV, so does -v, then one can find u solving KdV such that
-vx + v 2 + u =
(3.3)
2
vt = -4vxxx + 12vvxx - 12v vx +
12vx2
- 6
ux v - 6
uvx + 3
uxx
(3.4)
u u). Let wx = 12 u and wx = 12 u, then from
Subtracting (3.3) from (3.1), we get vx = 12 (
(3.1) and (3.2), we get the Backlund transformation for KdV
(w + w)x = - (w w)2
(w w)
t = 4(w
w)xxx + 12(w
w)(w w)xx
2
+12(w w) (w w)x + 12(w
w)2x + 12(w w)wxx
+12wx (w w)x + 6wxxx
where is the Backlund parameter.
3.2
Hirotas Bilinear Operator Method
Hirota introduced certain bilinear operators to convert the nonlinear wave equations into
bilinear equations for which Backlund transformations can be constructed. Hirotas Doperator was discussed in section 1.4.
Consider the KdV equation
ut + 6uux + uxxx = 0
let u = 2(lnf )xx , we get
ut + 6uux + uxxx = x [ f12 Dx (Dt + Dx3 )f f ]
Thus, if f solves the associated bilinear equation
Dx (Dt + Dx3 )f f = 0
(3.5)
then u solves the KdV equation. The Backlund transformation can be determined by the
following procedure.
We have a Backlund transformation for solutions of eq. (3.5). Let f be any solution of eq.
(3.5). A different solution f 0 is then defined by a Backlund transformation:
(Dt + 3k 2 Dx + Dx3 )f 0 f = 0
(3.6)
and
Dx2 f 0 f = k 2 f 0 f
(3.7)
where k is an arbitrary parameter. We will try to prove that f defined by eqs. (3.6) and
(3.7) satisfies eq (3.5).
By means of property 2 and property 3 as mentioned in section 1.4, we have
0
[Dx (Dt + Dx3 )f f ]f 0 f 0 f f [Dx (Dt + Dx3 )f 0 f 0 ] = 2Dx [(Dt + Dx3 )f f 0 ] f 0 f
+6Dx (Dx2 f f 0 ) (Dx f 0 f )
By the use of eq (3.7), the last term is converted to
6k 2 Dx (Dx f f 0 ) f 0 f
Hence, we have
[Dx (Dt + Dx3 )f f ]f 0 f 0 f f [Dx (Dt + Dx3 )f 0 f 0 ] =
2Dx [(Dt + 3k 2 Dx + Dx3 )f f 0 ] f f 0
8
which vanishes by the virtue of eq. (3.43). Thus we have verified that f 0 satisfies eq. (3.5)
provided that f satisfies the same equation.
Using property 4 from section 1.4, we have
(Dx2 f 0 f 0 )f f + f 0 f 0 (Dx2 f f ) = (Dx2 f 0 f )f f 0 + f 0 f (Dx2 f f 0 )
2(Dx f 0 f )(Dx f 0 f )
Dividing this equation by a factor of 2(f 0 f )2 and using eq 3.7 and property 1 from section
1.4, we have
wx0 + wx = -k 2 + (w0 w)2
(3.8)
0
where w and w are defined by,
w0 = - x
logf 0
(3.9)
and
logf
w = - x
(3.10)
and satisfy the differential equations
0
=0
wt0 - 6(wx0 )2 + wxxx
(3.11)
and
wt - 6(wx )2 + wxxx = 0
(3.12)
Differentiating equation (3.8) twice wrt x and using eqs (3.11) and (3.12), we have
wt0 + wt = 4[k 2 wx0 + wx2 + wx (w0 w)2 + wxx (w0 w)]
(3.13)
Equations (3.8) and (3.13) give the required Backlund transformation.
Bibliography
[1] Wikipedia: Backlund Transformation
[2] Wikipedia: Solitons
[3] Wikipedia: Lax Pair
[4] Hietarinta J. (2005), Hirotas bilinear method and soliton solutions
[5] Hirota R. (1974), A New Form of Backlund Transformations and Its Relation to the
Inverse Scattering Problem
[6] Wolfram Mathworld: Pseudosphere
[7] Thompson R. (2012), Backlund Transformation Part one, Mathematical Physics Seminar, University of Minnesota
[8] Li C.Y., Yurov A., Lie-Backlund-Darboux Transformation
[9] Zwillinger D. (1997), Handbook of Differential Equations 3rd edition, Academic Press
10
