t

X(t)

D

’ D+

n

x

A weak solution.

7.4. SOLITONS 209

We therefore have

n n

dx dt un ‚t • + un+1 ‚x • + dx dt un ‚t • + un+1 ‚x • .

0=

n+1 n+1

D’ D+

(7.102)

Let «

™

’X

1

n= , (7.103)

™ ™

|X|2 |X|2

1+ 1+

be the unit outward normal to D’ , then, using the divergence theorem, we

have

n n

dx dt un ‚t • + un+1 ‚x • dx dt ’• ‚t un + ‚x un+1

=

n+1 n+1

D’ D’

n

™

dt • ’X(t)un + un+1

+ ’

n+1 ’

‚D’

(7.104)

Here we have written the integration measure over the boundary as

™

1 + |X|2 dt.

ds = (7.105)

Performing the same manoeuvre for D+ , and observing that • can be any

smooth function, we deduce that

n

i) ‚t un + n+1 ‚x un+1 = 0 within D± .

™+ n

ii) X(un ’ un ) = n+1 (un+1 ’ un+1 ) on X(t).

’ + ’

The reasoning here is identical to that in chapter one, where we considered

variations at endpoints to obtain natural boundary conditions. We therefore

end up with the same equations for the motion of the shock as before.

The notion of weak solutions is widely used in applied mathematics, and it

is the principal ingredient of the ¬nite element method of numerical analysis

in continuum dynamics.

7.4 Solitons

A localized disturbance in a dispersive medium soon falls apart, since its

various frequency components travel at di¬ering speeds. At the same time,

non-linear e¬ects will distort the wave pro¬le. In some systems, however,

these e¬ects of dispersion and non-linearity can compensate each other and

210 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

give rise to solitons, stable solitary waves which propagate for long distances

without changing their form. Not all equations possessing wave-like solutions

also possess solitary wave solutions. The best known example of equations

that do, are:

1) The Korteweg-de-Viries (KdV) equation, which in the form

‚3u

‚u ‚u

= ’ 3,

+u (7.106)

‚t ‚x ‚x

has a solitary wave solution

u = 2±2 sech2 (±x ’ ±3 t) (7.107)

which travels at speed ±2 . The larger the amplitude, therefore, the

faster the solitary wave travels. This equation applies to steep waves

in shallow water.

2) The non-linear Shr¨dinger (NLS) equation with attractive interactions

o

1 ‚2ψ

‚ψ

’ »|ψ|2ψ,

=’

i (7.108)

2

‚t 2m ‚x

where » > 0. It has solitary-wave solution

√

±

ψ = eikx’iωt sech ±(x ’ U t), (7.109)

m»

where

1 ±

ω = mU 2 ’

k = mU, . (7.110)

2 2m

In this case, the speed is independent of the amplitude, and the moving

solution can be obtained from a stationary one by means of a Galilean

boost. (You should remember how this works from homework set zero!)

The nonlinear equation for the stationary wavepacket may be solved

by observing that

(’‚x ’ 2sech2 x)ψ0 = ’ψ0

2

(7.111)

where ψ0 (x) = sech x. This is the bound-state of the P¨schl-Teller

o

equation that we have met several times in the homework. The non-

linear Schrodinger equation describes many systems, including the dy-

namics of tornados, where the solitons manifest as the knot-like kinks

sometimes seen winding their way up thin funnel clouds6 .

6

H.Hasimoto, J. Fluid Mech. 51 (1972) 477.

7.4. SOLITONS 211

3) The sine-Gordon (SG) equation is

‚ 2 • ‚ 2 • m2

’ 2+ sin β• = 0. (7.112)

‚t2 ‚x β

This has solitary-wave solutions

4

tan’1 e±mγ(x’U t) ,

•= (7.113)

β

1

where γ = (1 ’ U 2 )’ 2 and |U | < 1. Again, the velocity is not related

to the amplitude, and the moving soliton can be obtained by boost-

ing a stationary soliton. The boost is now a Lorentz transformation,

and so we only get subluminal solitons, whose width is Lorentz con-

tracted by the usual relativistic factor of γ. The sine-Gordon equation

describes, for example, the evolution of light pulses whose frequency is

in resonance with an atomic transition in the propagation medium7 .

In the case of the sine-Gordon soliton, the origin of the solitary wave is

particularly easy to understand, as it can be realized as a “twist” in a chain

of coupled pendulums. The handedness of the twist determines whether we

take the + or ’ sign in the solution given above.