# What is soliton

I am new to this word.. This is not research level problem and it is soft question in nature. Just for curiosity, i am asking..

In literature, i am finding following words:(Wikipedia+ others).

Soliton is a self-reinforcing solitary wave

Solition is a phenomenon.

Solition is a property

Solitonic solution

As wikipedia also says, single definition is difficult to find. Can somebody explain this term according to you... It will be better if you give the idea of Soliton more mathematically rather only intuitively.

More precisely my question is WHAT IS SOLITON.

• For what it's worth, here's the Oxford English Dictionary definition: "a travelling, non-dissipative wave which is neither preceded nor followed by another such disturbance." Citation: "solitary, adj.". OED Online. December 2011. Oxford University Press. 5 March 2012 <oed.com/view/Entry/184311>. – Tanner Swett Mar 5 '12 at 19:05
• @TannerL Swett, so oxford definition matches with Wikipedia... Thanks for the commment. – zapkm Mar 5 '12 at 19:26

A soliton (at least in my field) is a 'self-similar solution' to a PDE. For instance a solution $(g_t)$ to the Ricci flow equation $$\frac{\partial g }{ \partial t} = - 2 \mathrm{Ric}(g(t))$$ is a Ricci soliton if it takes the form $g(t)= \alpha (t) \phi_t^* (g(0))$ where the $\alpha(t)$'s are scalars and the $\phi_t$'s are diffeomorphisms, i.e. the metric at time $t$ differs from the initial metric by the action of diffeomorphisms and/or dilation.

• Could you [or anybody else.. I realize this answer is from 2012] make an elementary example of a family $g(t)$ of metrics which is not of the form $\alpha(t)\cdot \phi_t^*(g(0))$ for $\alpha(t)$ a (time dependent) scalar and $\phi_t$ a family of diffeomorphisms? – Qfwfq Sep 2 '18 at 22:22
• @Qfwfq: take any metric. Its exponential map identifies a ball with a ball in its tangent space (radius=injectivity radius). The family $g_t=(1-t)g_0+tg_1$ of metrics in that ball starts at the original metric $g_0$ and ends at the induced Euclidean metric $g_1$ on the tangent space. In some smaller ball, this is a metric for all $t$. If the original metric is not conformally flat, for example, taking the Fubini-Study metric on the complex projective plane, then you have a family which contains a confomally non-flat and a flat, so is not a diffeomorphism followed by a conformal rescaling. – Ben McKay Sep 3 '18 at 9:26
• Thank you. In reading your (very apt) example I was confused at first, because $g_0$ and $\phi_t^*(g_0)$ are not conformally equivalent, in general. Still, diffeomorphisms preserve intrinsic properties of metrics (such as being conformally flat, indeed). – Qfwfq Sep 3 '18 at 9:44