A sine-Gordon solitary wave as a twist in a ribbon of coupled pendulums.

Exercise: Find the expression for the sine-Gordon soliton, by ¬rst showing

that the static sine-Gordon equation

‚ 2 • m2

’ 2+ sin β• = 0 (7.114)

‚x β

implies that

1 2 m2

• + 2 cos β• = const., (7.115)

2 β

7

See G. L. Lamb, Rev. Mod. Phys. 43(1971) 99, for a nice review.

212 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

and solving this equation (for a suitable choice of the constant) by separation

of variables. Next, show that if f (x) is solution of the static equation, then

f (γ(x ’ U t)), γ = (1 ’ U 2 )’1/2 , |U | < 1 is a solution of the time-dependent

equation.

The existence of solitary-wave solutions is interesting in its own right. It

was the fortuitous observation of such a wave by Scott Russell on the Union

Canal, near Hermiston in England, that founded the subject8 . Even more

remarkable was Scott Russell™s subsequent discovery (made in a specially

constructed trough in his garden) of what is now called the soliton property:

two colliding solitary waves interact in a complicated manner yet emerge

from the encounter with their form unchanged, having su¬ered no more than

a slight time delay. Each of the three equations given above has exact multi-

soliton solutions which show this phenomenon.

After languishing for more than a century, soliton theory has grown to

be a huge subject. It is, for example, studied by electrical engineers who

use soliton pulses in ¬bre-optic communications. No other type of signal

can propagate though thousands of kilometers of undersea cable without

degredation. Solitons, or “quantum lumps” are also important in particle

physics. The nucleon can be thought of as a knotted soliton (in this case

called a “skyrmion”) in the pion ¬eld, and gauge-¬eld monopole solitons

appear in many string and ¬eld theories. The soliton equations themselves

are aristrocrats among partial di¬erential equations, with ties into almost

every other branch of mathematics.

Exercise: Lax pair for the non-linear Schr¨dinger equation. Let L be the

o

matrix di¬erential operator

χ—

i‚x

L= , (7.116)

χ i‚x

8

“I was observing the motion of a boat which was rapidly drawn along a narrow channel

by a pair of horses, when the boat suddenly stopped - not so the mass of water in the

channel which it had put in motion; it accumulated round the prow of the vessel in a state

of violent agitation, then suddenly leaving it behind, rolled forward with great velocity,

assuming the form of a large solitary elevation, a rounded, smooth and well-de¬ned heap

of water, which continued its course along the channel apparently without change of form

or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate

of some eight or nine miles an hour, preserving its original ¬gure some thirty feet long and

a foot to a foot and a half in height. Its height gradually diminished, and after a chase of

one or two miles I lost it in the windings of the channel. Such, in the month of August

1834, was my ¬rst chance interview with that singular and beautiful phenomenon which I

have called the Wave of Translation.” ”John Scott Russell, 1844

7.4. SOLITONS 213

and let P the matrix

χ—

i|χ|2

P= . (7.117)

’i|χ|2

’χ

Show that the equation

™

L = [L, P ] (7.118)

is equivalent to the non-linear Shr¨dinger equation

o

iχ = ’χ ’ 2|χ|2 χ.

™ (7.119)

Physics Illustration: Solitons in Optical Fibres. We wish to transmit picosec-

ond pulses of light with a carrier frequency ω0 . Suppose that the dispersive

properties of the ¬bre are such that the associated wavenumber for frequen-

cies near ω0 can be expanded as

1

k = ∆k + k0 + β1 (ω ’ ω0 ) + β2 (ω ’ ω0 )2 + · · · . (7.120)

2

Here, β1 is the reciprocal of the group velocity, and β2 is a parameter called

the group velocity dispersion (GVD). The term ∆k parameterizes the change

in refractive index due to non-linear e¬ects. It is proportional to the square

of the electric ¬eld. Let us write the electric ¬eld as

E(x, t) = A(x, t)eik0 z’ω0 t , (7.121)

where A(x, t) is a slowly varying envelope function. When we transform from

Fourier variables to space and time we have

‚ ‚

(ω ’ ω0 ) ’ i (k ’ k0 ) ’ ’i

, , (7.122)

‚t ‚z

and so the equation determining A becomes

‚A β2 ‚ 2 A

‚A

’i ’

= iβ1 + ∆kA. (7.123)

2 ‚t2

‚z ‚t

If we set ∆k = γ|A2 |, where γ is normally positive, we have

β2 ‚ 2 A

‚A ‚A

’ γ|A|2 A.

i + β1 = (7.124)

2

‚z ‚t 2 ‚t

214 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

We may get rid of the ¬rst-order time derivative by transforming to a frame

moving at the group velocity. We do this by setting

„ = t ’ β1 z,

ζ=z (7.125)

and using the chain rule, as we did for the Galilean transformation in home-

work set 0. The equation for A ends up being

β2 ‚ 2 A

‚A

’ γ|A|2 A.

i = (7.126)

2 ‚„ 2

‚ζ

This looks like our non-linear Schr¨dinger equation, but with the role of

o

space and time interchanged! Also, the coe¬cient of the second derivative

has the wrong sign so, to make it coincide with the Schr¨dinger equation we

o

studied earlier, we must have β2 < 0. When this condition holds, we are

said to be in the “anomalous dispersion” regime ” although this is rather

a misnomer since it is the group refractive index , Ng = c/vgroup , that is

decreasing with frequency, not the ordinary refractive index. For pure SiO2

glass, β2 is negative for wavelengths greater than 1.27µm. We therefore have

anomalous dispersion in the technologically important region near 1.55µm,

where the glass is most transparant. In the anomalous dispersion regime we

have solitons with

√

β2 ±

A(ζ, „ ) = ei±|β2 |ζ/2 sech ±(„ ), (7.127)

γ