#
Permanent Exhibition

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# Many Faces of Solitons

### Kanehisa Takasaki (Kyoto University)

Entrance --
KdV equation --
Modified KdV equation --
Sine-Gordon equation
Next Room

Entrance of Gallery

## What is "soliton"?

The term "soliton" was introduced in the 1960's, but the
scientific research of solitons had started in the 19th
century when
John Scott-Russell observed a large solitary wave
in a canal near Edinburgh. In the days of Scott Russell,
there was much debate concerning the very existence of
this kind of solitary waves. Nowadays, many model equations
of nonlinear phenomena are known to possess soliton solutions.
Solitons are very stable solitary waves in a solution of
those equations. As the term "soliton" suggests, these
solitary waves behave like "particles". When they are located
mutually far apart, each of them is approximately a traveling wave
with constant shape and velocity. As two such solitary waves
get closer, they gradually deform and finally merge into a
single wave packet; this wave packet, however, soon splits
into two solitary waves with the same shape and velocity before
"collision".

The stability of solitons stems from the delicate balance of
"nonlinearity" and "dispersion" in the model equations.
Nonlinearity drives a solitary wave to concentrate further;
dispersion is the effect to spread such a localized wave.
If one of these two competing effects is lost,
solitons become unstable and, eventually, cease to exist.
In this respect, solitons are completely different from
"linear waves" like sinusoidal waves. In fact, sinusoidal
waves are rather unstable in some model equations of soliton
phenomena. Computer simulations show that they soon break
into a train of solitons.

In this exhibition, we present several model equations of
soliton phenomena along with soliton solutions. These
equations are the most fundamental examples of the so called
"integrable systems". Besides soliton solutions, these model
equations exhibit many remarkable properties, which invoked
a wide range of researches since the 1960's.

## Related Links

**
Helliot-Watt University (Edinburgh) Solitons Home Page**
**
Solitons and Soliton Collisions**,
Andrey E. Miroshnichenko (Tver State University)
**
D.J. Korteweg and G. De Vries **
(
Korteweg-de Vries Institute)
**
Soliton-Lab List of Related Links**