120
L. Parameswar
2 The Soliton and Its History
In nature not all waves disperse or spread and hence diminish over distances. But
there are many cases of fairly permanent and powerful waves that have the capacity
to travel extraordinary distances without diminishing in size or shape, and which
are described by nonlinear equations of dispersive type. This kind of a wave was
first observed by the Scottish naval architect “Scott Russel” in 1834. In 1830s Scott
Russel carried out investigations on the shape of hulls of ships and observed the
speed and forces needed to propel them. In 1834, riding on a horseback, he observed
the “Great wave of translation” in the union canal and reported his observations to
the British Association [2].
I believe I shall best introduce the phenomenon by describing the circumstances of my own
acquaintance with it. 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-defined 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 figure 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.
This rolling pile of water is a solitary wave which maintained its shape and speed
much larger than the convectional wave. Scott Russel also performed laboratory
experiments [2] generating solitary waves by dropping a weight at one end of the
water channel and obtained the relation, c
2
= g(h + a) where h is the undisturbed
depth of water, a is the amplitude of the wave, g the acceleration due to gravity and
c the speed of the wave.
A simple nonlinear dispersive wave equation whose modern version is given by
U t + 6 UU xx + U xxx = 0
( 1 )
where (UU xx ) is then nonlinear term and (U xxx ) is the dispersion term.
The solution is written as
U (x, t) =
c
2
sech
2
(
√ (c)(x − ct)
2
)
(2)
A delicate balance between the nonlinear term (UU xx ) and dispersion term in equation [3] results in a solitary wave pulse which moves with uniform velocity proportional to the amplitude. Another important observation is that after interaction, such
solitary waves emerge unaffected in their amplitude and velocity, except for phase
shifts. A solitary wave can arise when the nonlinearity balances linear dispersion.
In appropriate nonlinear systems, these solitary waves can interact elastically like
particles without changing their shape and velocities.
L. Parameswar
2 The Soliton and Its History
In nature not all waves disperse or spread and hence diminish over distances. But
there are many cases of fairly permanent and powerful waves that have the capacity
to travel extraordinary distances without diminishing in size or shape, and which
are described by nonlinear equations of dispersive type. This kind of a wave was
first observed by the Scottish naval architect “Scott Russel” in 1834. In 1830s Scott
Russel carried out investigations on the shape of hulls of ships and observed the
speed and forces needed to propel them. In 1834, riding on a horseback, he observed
the “Great wave of translation” in the union canal and reported his observations to
the British Association [2].
I believe I shall best introduce the phenomenon by describing the circumstances of my own
acquaintance with it. 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-defined 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 figure 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.
This rolling pile of water is a solitary wave which maintained its shape and speed
much larger than the convectional wave. Scott Russel also performed laboratory
experiments [2] generating solitary waves by dropping a weight at one end of the
water channel and obtained the relation, c
2
= g(h + a) where h is the undisturbed
depth of water, a is the amplitude of the wave, g the acceleration due to gravity and
c the speed of the wave.
A simple nonlinear dispersive wave equation whose modern version is given by
U t + 6 UU xx + U xxx = 0
( 1 )
where (UU xx ) is then nonlinear term and (U xxx ) is the dispersion term.
The solution is written as
U (x, t) =
c
2
sech
2
(
√ (c)(x − ct)
2
)
(2)
A delicate balance between the nonlinear term (UU xx ) and dispersion term in equation [3] results in a solitary wave pulse which moves with uniform velocity proportional to the amplitude. Another important observation is that after interaction, such
solitary waves emerge unaffected in their amplitude and velocity, except for phase
shifts. A solitary wave can arise when the nonlinearity balances linear dispersion.
In appropriate nonlinear systems, these solitary waves can interact elastically like
particles without changing their shape and velocities.