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 $-cf+{\frac {d^{2}}{dx^{2}}}f+f^{2}=0$. This equation can be explicitly solved in terms of the hyperbolic secant function. 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.