Soliton

Solitons are travelling wave solutions of completely integrable equations. The term soliton is also used to describe travelling wave or solitary wave solutions of non-integrable equations. For example, assuming $u(t,x)=f(x-ct)$ is a solution of the KdV equation leads to an ODE of the form Failed to parse (unknown function "\math"): {\displaystyle -c f + {\frac{d}{dx}}^2 f + f^2 = 0<\math>. This equation can be explicitly solved in terms of <math>\sech} . Other equations, such as focusing nonlinear Schrodinger equations also have Solitons are remarkably robust. For certain equations, Solitons have been shown to be orbitally stable and asymptotically stable.