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I am reading papers about solitons for my small reports, and i could not understand its physical meaning in detail.

I know soliton is solitary wave which behaves like particle. And many text they talk about kink, vortices, strings, monopoles, dyons.......

I read that solutions of Sine-Gordon equation are kink solutions, and the many solitons govern by the non-linear differential equations (such as KdV)

What is a Kink in both physical sense? And what is the difference between kinks, vortices and strings?

From the context of solitons(kinks, vortices, strings, monopoles, dyons ) they are in same section, but i could not understand what are the difference between them.

And lastly, what is the difference between solitons and instanton?

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Are you talking about the self-reinforcing waves called solitons or the topological ones? –  ACuriousMind Aug 24 at 13:45

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A soliton is a localized, non-dispersive solution of a nonlinear theory in Euclidean space. It certainly is a real object: you have a famous story about a certain John Russell who observed soliton-like waves made by a boat on a river (wikipedia knows everything about it!) The so-called morning glory clouds in Australia (http://en.wikipedia.org/wiki/Morning_Glory_cloud) are also a great example. The theoretical physicist Sidney Coleman uses also the term 'lumps' to denote solitons, his book 'Aspects of Symmetry' contains a great chapter on them (and another chapter on instantons, for that matter). The terms 'kinks' and 'anti-kinks' are used for the soliton solutions of the sine-Gordon equation, I do not know whether these terms are used outside that context.

Instantons (the term coined by 't Hooft [1], who did groundbreaking research on them), or pseudo-particles (the name given by Polyakov [2]), are not real, in this sense that they are solutions to the equations of motion of a quantum (field) theory after a Wick rotation, in which time is made imaginary! Therefore, they are not observable as such, but they can be used to calculate and to explain quantum mechanical effects that can be observed, such as tunneling. In QCD, instantons are believed to tunnel between the topologically different color vacua, although I have to admit I don't know the details.

Good sources on these matters are:

[1] 't Hooft, Phys. Rev. D14 (1976) 3432

[2] Polyakov, Nucl. Phys. B120 (1977) 429

Coleman - Aspects of Symmetry - Cambridge University Press (1985)

Vainshtein, Zakharov, Novikov, and Shifman - Abc of Instantons - Sov. Phys. Usp., 25:195 – 215 (1982)

Schulman - Techniques and Applications of Path Integration - John Wiley & Sons (1981)

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I want to add another outstanding book with introduction to solitons, instantons and the like: "Gauge field theories: an introduction with applications" by M. Guidry (1991). This is a personal favourite of mine because Guidry's style of rhetoric clicks well with my internal thought process. Another good reference I know is "Solitons and instantons: An introduction to solitons and instantons in quantum field theory" by R. Rajaraman (1982). –  Wouter Aug 24 at 16:06

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