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 soliton solutions.

Solitons are remarkably robust. For certain equations, solitons have been shown to be orbitally stable and asymptotically stable.

Many completely integrable solutions also admit multisoliton solutions, which are nonlinear superpositions of soliton solutions (typically of different amplitude and speed). A remarkable consequence of complete integrability is that, while the evolution is nonlinear, the soliton components of a multisoliton can pass through each other with almost no interaction other than a phase shift or a spatial translation. This is not the case for non-integrable equations, nevertheless in many cases one can still construct asymptotic multisoliton solutions for such equations, and it is suspected (but unproven for any focusing non-integrable equation) that solutions for such equations, if they do not blow up, should generically resolve into such an asymptotic multisoliton state, plus a radiation term.