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 Failed to parse (unknown function "\sech"): {\displaystyle \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.