Solitons are standing waves which may appear in many media, including liquids and light. Their earliest known recording was in 1834, when Scottish engineer John Scott Russell watched a solitary wave in a canal travel over 8 miles without changing shape or amplitude.
Solitons may occur in light beams, magnets, DNA molecules, proteins, and cell membranes. Solitons arise at both the macroscopic and quantum scales, in both matter (fermions) and pure energy (bosons).
Soliton dynamics vary depending upon the medium in which they appear. The mathematical lynchpin of soliton equational solution depends upon their medium.
“Optical soliton propagation in nonlocal media – which include plasmas, nematic liquid crystals and liquid solutions with thermal nonlinearities – is governed by the same model that is used to describe shallow waters, with nonlocality playing the role of surface tension.” ~ Greek mathematician Theodoros P. Horikis
Unlike the seeming simplicity of a standing soliton – in maintaining its coherence – the mathematics of solitons are intricate.
“Equations with soliton solutions have a profound mathematical structure.” ~ English mathematician Mason Porter
Solitons arise via cancellation of nonlinear and dispersive effects in a gas or fluid. Solitons may sustain their standing even when interacting with other solitons. In certain instances, soliton collisions generate complicated wave patterns which mystically maintain themselves.
Beyond characterization, physicists have no explanation for how solitons arise or maintain themselves. Solitons are an instance of coherence flamboyantly pronouncing itself as inherent to Nature.
Theodoros P. Horikis & Dimitrios J. Frantzeskakis, “Patterns of water in light,” Proceedings of the Royal Society A (24 July 2019).
Lisa Zyga, “Patterns typically observed in water can also be found in light,” Phys.org (12 August 2019).
Tarik Yefsah et al, “Heavy solitons in a fermionic superfluid,” Nature 449: 426–430 (25 July 2013).
Christoph Becker, “Dark and heavy,” Nature 449: 413–414 (25 July 2013).
D.Y. Tang et al, “Observation of high-order polarization-locked vector solitons in a fiber laser,” Physical Review Letters 101: 153904 (10 October 2008).
Aleksandr S. Davydov, Solitons in molecular systems. Mathematics and its applications, Kluwer Academic Publishers (1991).
Thomas Heimburg & Andrew D. Jackson, “On soliton propagation in biomembranes and nerves,” PNAS (1 July 2005).
Ludmila V. Yakushevich, Nonlinear Physics of DNA, Wiley (2004).
Zachariah Sinkala, “Soliton/exciton transport in proteins,” Journal of Theoretical Biology 241(4): 919-927 (August 2006).