Solitons are nonlinear waves. As a preliminary definition, a soliton
is considered as solitary, traveling
wave pulse solution of nonlinear partial differential equation (PDE). The
nonlinearity will play a significant role. For most dispersive evolution
equations these solitary waves would scatter inelastically and lose 'energy' due
to the radiation. Not so for the solitons: after a fully nonlinear interaction,
the solitary waves remerge, retaining their identities with same speed and
shape. It should have remarkable stability properties. Stability plays a
important role in soliton physics.
The beginning of soliton physics in often dated back to the month of August
1834 when John Scott Russell observed the "great wave of translation''. He
describes what he saw in [1]:
- "I believe I shall best introduce this phenomenon by describing the
circumstances of my own first acquaintance with it. I was observing the motion
of a boat which was rapidly along a narrow channel by a pair of hoses, 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 dimension 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 foot and 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 first chance interview with that singular and
beautiful phenomenon which I have called the Wave of Translation a name which
it now very generally bears."
Due to the work of Stokes, Boussinesq, Rayleigh, Korteweg,
de Vries, and many others we know that the "great wave of translation'' is a
special form of a surface water wave.
The equation describing the
(unidirectional) propagation of waves on the surface of a shallow channel was
derived by Korteweg and de Vries in 1895. After performing a Galilean and
variety of scaling transformations, the KdV equation can be written in
simplified form:
One soliton solution of this nonlinear PDE:
where is
the speed of the soliton and is
the phase. This clearly represents the solitary wave observed by John Scott
Russell and shows that the peak amplitude is exactely half the speed. Thus
larger solitary waves have greater speeds. This suggest a numerical experiment:
we start with two solitary wave solutions, with centers well separeted and
different amplitude.
Figure:
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The KdV equation can admits also a Multi-soliton solution.
[1] J. Scott Russell. Report on
waves, Fourteenth meeting of the British Association for the Advancement of
Science, 1844.