text: In 1834, John Scott Russell describes his wave of translation. The discovery is described here in Scott Russell's own words:
  Scott Russell spent some time making practical and theoretical investigations of these waves. He built wave tanks at his home and noticed some key properties:
  * The waves are stable, and can travel over very large distances (normal waves would tend to either flatten out, or steepen and topple over)
  * The speed depends on the size of the wave, and its width on the depth of water.
  * Unlike normal waves they will never merge – so a small wave is overtaken by a large one, rather than the two combining.
  * If a wave is too big for the depth of water, it splits into two, one big and one small.
  Scott Russells experimental work seemed at odds with Isaac Newtons and Daniel Bernoullis theories of hydrodynamics. George Biddell Airy and George Gabriel Stokes had difficulty accepting Scott Russells experimental observations because they could not be explained by the existing water wave theories. Additionnal observations were reported by Henry Bazin in 1862 after experiments carried out in the canal de Bourgogne in France. Their contemporaries spent some time attempting to extend the theory but it would take until the 1870s before Joseph Boussinesq and Lord Rayleigh published a theoretical treatment and solutions. In 1895 Diederik Korteweg and Gustav de Vries provided what is now known as the Korteweg–de Vries equation, including solitary wave and periodic cnoidal wave solutions.
  In 1965 Norman Zabusky of Bell Labs and Martin Kruskal of Princeton University first demonstrated soliton behavior in media subject to the Korteweg–de Vries equation (KdV equation) in a computational investigation using a finite difference approach.  They also showed how this behavior explained the puzzling earlier work of Fermi, Pasta, Ulam, and Tsingou.
  In 1967, Gardner, Greene, Kruskal and Miura discovered an inverse scattering transform enabling analytical solution of the KdV equation. The work of Peter Lax on Lax pairs and the Lax equation has since extended this to solution of many related soliton-generating systems.
  Solitons are, by definition, unaltered in shape and speed by a collision with other solitons. So solitary waves on a water surface are near-solitons, but not exactly – after the interaction of two (colliding or overtaking) solitary waves, they have changed a bit in amplitude and an oscillatory residual is left behind.
  Solitons are also studied in quantum mechanics, thanks to the fact that they could provide a new foundation of it through de Broglie's unfinished program, known as "Double solution theory" or "Nonlinear wave mechanics". This theory, developed by de Broglie in 1927 and revived in the 1950s, is the natural continuation of his ideas developed between 1923 and 1926, which extended the wave–particle duality introduced by Albert Einstein for the light quanta, to all the particles of matter. The observation of accelerating surface gravity water wave soliton using an external hydrodynamic linear potential was demonstrated in 2019. This experiment also demonstrated the ability to excite and measure the phases of ballistic solitons.