Probably the earliest introduction of solitary waves was provided by Scott Russell when he described his observations as

"... rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. ... still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height."

In 1895, Korteweg and deVries provided a nonlinear partial differential equation (KdV Equation) that describes traveling solitary waves on shallow water observed by Scott Russell. However, properties of the KdV equation were not well understood until mid twentieth century, when Zebusky and Kruskal applied KdV equations to numerically study the Fermi-Pasta-Ulam system in 1965. Zebusky and Kruskal observed that the KdV equation admits multiple solitary wave solutions that can collide with each other and preserve their shape and speed post collision. Owing to this particle like nature of the waves, Zebusky and Kruskal coined the term Soliton to describe the solitary wave solutions of the KdV equations.

The KdV equation is a partial differential equation of the form

The KdV equation can be solved numerically using a Fourier spectral method on a periodic domain, or a pseudo-spectral method on an arbitrary domain. The left animation below shows two-soliton solution of the KdV equation on a periodic domain (the solution is shown on the polar-coordinates). The two solitons travel at velocities proportional to their amplitude, with the larger soliton overtaking the smaller one. Observe that the solitons preserve their shape and velocity post collision.

The KdV equation have a global attractor and do not show any chaotic behavior. However, when the KdV equation is forced and damped, with the low damping value, the solution shows chaotic behavior. The animation below on left shows solution of a KdV equation forced using a sinusoidal forcing. The solution is initialized with two solitons and the inital behavior is similar to the solution of an unforced KdV equation. However as the forcing is accumulated over time, new waves appear at the boundary. Note that the new waves also show behavior similar to the initial solitons. On the contrary, the damping term damps the soliton, resulting in the complete vanishing of some of the solitons at certain time instances. However, the solitons reappear due to the forcing.

The accompanying animation shows solution of the forced and damped KdV equation with the waves represented as particles. Note the wave-particle duality of the solitons evident from the animations.

The preceding discussion assumed deterministic KdV equations, however, the equations are often uncertain owing to uncertainty in initial conditions, parameters etc. Monte Carlo methods are often used to propagate this uncertainty to the system response. For example in view of uncertain initial condition, the numerical solution can be initialized from ensemble members sampled from the probability distribution of the uncertain initial condition, and each ensemble member can be integrated forward in time. Accompanying animations show solution of forced and damped KdV equation for three ensemble members. Both wave and particle form of the solution is provided in the animations.