In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a solitary wave can refer to

• The solitary wave (water waves) or wave of translation, as observed by John Scott Russell
  John Scott Russell
  John Scott Russell was a Scottish naval engineer who built the Great Eastern in collaboration with Isambard Kingdom Brunel, and made the discovery that gave birth to the modern study of solitons.-Personal life:John Scott Russell was born John Russell on 9 May 1808 in Parkhead, Glasgow, the son of...
  
   in the Union Canal, near Edinburgh in 1834. It is the prototype for a soliton.
• A soliton
  Soliton
  In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
  
  , a generalization of the wave of translation to general systems of partial differential equation
  Partial differential equation
  In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
  
  s.
• A topological defect
  Topological defect
  In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of...
  
  , a generalization of the idea of a soliton to any system which is stable against decay due to homotopy theory.

The name soliton

Soliton

In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...

 appears to have been coined by Norman Zabusky

Norman Zabusky

Norman J. Zabusky is an American physicist, who is noted for the discovery of the soliton in the Korteweg–de Vries equation, in work completed with Martin Kruskal. This result early in his career was followed by an extensive body of work in computational fluid dynamics, which led him more...

 and Martin Kruskal

Martin Kruskal

Martin David Kruskal was an American mathematician and physicist. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and from nonlinear analysis to asymptotic analysis...

. However, the name solitary wave—as used in the propagation of non-dispersive energy bundles through discrete and continuous media—is more general, irrespective of whether the Korteweg–de Vries equation

Korteweg–de Vries equation

In mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified...

 (KdV equation), sine-Gordon equation

Sine-Gordon equation

The sine–Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative...

, non-linear Schrödinger equation, Toda field theory

Toda field theory

In the study of field theory and partial differential equations, a Toda field theory is derived from the following Lagrangian:...

or some other equation which allows for solitary wave solutions is used.