What are the criteria for solitary wave to exist?

[hanson]

It seems to me that solitary waves do not exit very often in nature?
What I am referring to is the solitary traveling wave with the conventional sech^2(x) profile.
When would such solitary wave arise in water or other flows?

 

[AlphaNumeric]

Soliton solutions along with a huge chunk of stuff on Lax pairs etc was covered in a course I did a few years ago. As a kind of physical motivation for the material, the lecturer outlined where the ideas came from. The first 2 or 3 pages of the printed notes http://www.hep.phys.soton.ac.uk/~g.j.weatherill/lecturenotes/II/IntegrableSystems.pdf cover it.

 

[Chris Hillman]

Solitons in nature versus theory

It seems to me that solitary waves do not exit very often in nature?
What I am referring to is the solitary traveling wave with the conventional sech^2(x) profile. When would such solitary wave arise in water or other flows?

This point is clearly addressed in the books I recommended, including Drazin and Johnson: Solitons: an Introduction. The answer is that the sech soliton arises as an exact solution of the KdV equation, which arises as an approximation in hydrodynamics which is suitable for studying shallow water waves. The KdV equation admits many other wave solutions; the sech soliton arises for certain initial conditions. Physically these correspond to modeling something the wave created in a narrow channel when you suddenly displace water at one end (e.g. by pushing down a wooden block). You can easily make such waves in your basement using say a rain gutter. With more experience, solitons can also sometimes be spotted in steps leading into bodies of water such as lakes.