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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.

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    $\begingroup$ 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>. $\endgroup$
    – Sophie Swett
    Commented Mar 5, 2012 at 19:05
  • $\begingroup$ @TannerL Swett, so oxford definition matches with Wikipedia... Thanks for the commment. $\endgroup$
    – zapkm
    Commented Mar 5, 2012 at 19:26

3 Answers 3

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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.

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    $\begingroup$ 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? $\endgroup$
    – Qfwfq
    Commented Sep 2, 2018 at 22:22
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    $\begingroup$ @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. $\endgroup$
    – Ben McKay
    Commented Sep 3, 2018 at 9:26
  • $\begingroup$ 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). $\endgroup$
    – Qfwfq
    Commented Sep 3, 2018 at 9:44
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Dear Pradip
There is the historical and theoretical survey "The Symmetry of Solitons" by Richard Palais.
I have liked it very much, so I hope you can find it useful.

Bye.

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    $\begingroup$ Also in the Bulletin is Terry Tao's "Why are solitons stable" ams.org/journals/bull/2009-46-01/S0273-0979-08-01228-7/… $\endgroup$
    – Willie Wong
    Commented Mar 6, 2012 at 7:58
  • $\begingroup$ The interesting material starts on p. 17 of the pdf, and on p. 23 is a wonderful historical anecdote. $\endgroup$
    – user21349
    Commented Aug 20, 2016 at 22:03
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Leaving the Ricci solitons aside, there is a number of different (and not equivalent) definitions. It is mostly agreed that solitons in nonlinear systems are solitary waves which balance dispersion and nonlinearity and maintain their shape, even after elastic interactions like collisions, see e.g. the Scholarpedia article and the discussion at Physics.SX.

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