# What is Soliton: Definition and 25 Discussions

In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.
The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation".

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1. ### Soliton model of the action potential

Has anyone else come across the soliton model of the action potential? https://en.wikipedia.org/wiki/Soliton_model_in_neuroscience It seems extremely non-mainstream, especially given that it presented as an alternative to the Hodgkin-Huxley model, which is undoubtedly the most successful...
2. ### A Help with the Derrick scaling argument and topological solitons

I have been reading Manton & Sutcliffe for some time now and can't quite wrap my head around something. If you take the Hopf invariant N of a topological soliton ϕ then its Skyrme-Faddeev energy (which I hope I've gotten right up to some constants) E=∫∂iϕ⋅∂iϕ+(∂iϕ×∂jϕ)⋅(∂iϕ×∂jϕ) d3x satisfies...
3. ### I Understanding the definition of a soliton

I'm learning about solitons from a book called Solitons and Instantons by R. Rajaraman. He defines (page 14-15) a soliton as a solution to a (possibly non-linear) PDE where the energy density of the system is of the form ##\epsilon (x,t) = \sum_i \epsilon_0(x-a_i-u_i t-\delta_i)##, as ##t...
4. ### I Is there a relation between a soliton and a Goldstone boson?

I am currently reading this notes by t'Hooft, and I realized that a soliton and a Goldstone boson behave in a similar way: Both of them interpolate between the vacua. Keeping in mind Soliton is described classically in the notes(atleast until first few sections in chap1), Is there a relation...
5. ### A Do Falaco solitons also occur in water on a microscopic scale

Falaco solitons are unique in their longevity, reportedly up to 15 minutes on the surface of a body of still water. For details and the math see {{ http://www22.pair.com/csdc/pdf/pdf/falsol.pdf }} In addition, this persistence occurs at relatively macroscopic sizes in the range of 10 to 40 cm...
6. ### A Bright, dark soliton for cubic-quintic nonlinear Schrodinger

For a given stationary cubic-quintic nonlinear Schrodinger equation, EU=-U_XX+G1|U|^2U+G2 |U|^4 U, where X=X(t,x). There are bright and dark solitons. In many references, it is found that there is typo or mistake in dark soliton by substituting their soliton solution to this above eqaution. The...
7. ### What is electron? Is it a perfect point? What does it mean?

Electron is usually imagined as a simple point charge, but in fact it is a very complex entity: https://dl.dropboxusercontent.com/u/12405967/electron.png - being electric charge itself means singular(-like?) configuration of electric field - E behaves like 1/r^2, - it is also magnetic dipole...
8. ### Generic Soliton Solution Periodicity: Restrictions & Examples

Hello everyone, I have a question regarding the possible periodicity of time in a generic metric. Suppose that for some reason I have a solution to Einstein's equations of the kind (in Euclidean time): $$ds^2_E=+f(r)dt_E^2+\frac{dr^2}{g(r)}+r^2(dx^2+dy^2).$$ Am I always allowed to assign...
9. ### Why is the solution of the phi^6 potential not a soliton?

Homework Statement Consider a theory with a \phi^6-scalar potential: \mathcal{L} = \frac{1}{2}(\partial_\mu\phi)^2-\phi^2(\phi^2-1)^2. Why is the solution to the equation of motion not a soliton? Homework Equations \phi''=\frac{\partial V}{\partial\phi} The Attempt at a Solution...
10. ### Finding Constants to Solution Given 2nd Order, Nonlinear DE

Homework Statement Okay, here's the deal:I have been given a second order nonlinear differential equation, and I have also been given the general solution with constants A and B. I am supposed to find the constants A and B. The solution represents a fermion at rest, since the solution does not...
11. ### Good explanation for Soliton and Skyrmion?

I am looking for a Book in which Solitons and Skyrmions are easy explained. I want to understand them for Solid State Physics. Thank U Abby
12. ### What are Soliton Particle Models?

I see there are mainly discussed here very abstract approaches like string theory. I would like to suggest a general discussion about much less abstract models: to get not exactly beyond, but rather behind the standard model - ask about the internal structure of particles (behind abstract...
13. ### Interaction tetween two waves soliton

Hello if i have two waves soliton y=A \sech^2 (k(x \pm ct)), both solution of KdV differential equation how i find a equation for the interaction between the right and left waves \pm c i think on supperposition waves, y=A \sech^2 (k(x + ct))+B \sech^2 (k(x - ct)), but i don't understand if...
14. ### Soliton Solutions in Wave Theory

Hello. I'm not sure the template applies here, as this isn't a textbook style question. I tried to read over the rules and I hope this is the relevant place to put the following query: I am trying to design a research project in my 1st Engineering Physics class, in which we have full freedom...
15. ### KdV Equation - Modelling Soliton

Hi all, I am attempting to model soliton formation numerically. The solitons will be formed by moving a body of some sort through a shallow channel of water with the free surface subject to atmospheric pressure. My goal would be to numerically predict wave amplitudes, wavelengths...
16. ### Why is the behaviour at infinity important in classifying gauge theory states?

I don't understand the link from soliton solution of QFT to the homotopy group. The argument goes like following: Consider the field configuration such that the action is finite, therefore we must require the field vanishes at spacetime infinity, hence, we defined a map from the...
17. ### Can a Non-Linear Differential Equation Have a Soliton Solution?

I have a non-linear differential equation and I wonder whether it has a soliton solution or not. How can I approach to the problem? So far I have never dealt with non-linear differential equations, hence, any suggestion is appreciated.
18. ### Speed of a soliton and its energy

Hi All, A one dimensional soliton will have the mathematical form f(x - ct) where c is the speed of propagation and f represents some envelope function. If we suppose that c1 is a speed and c2 another one where c2>c1. If E1 is the energy needed to maintain the speed and shape of the...
19. ### Stability? nonlinear mode? soliton?

Can someone explain to me what it means by nonlinear mode? I heard people saying that soliton is a nonlinear mode of the nonlinear schrondinger equation and therefore perturbed pulses tend to reshape to the soliton shape. In the reshaping proces, the energy dispersed is known as continuous...
20. ### Infinitely many integrable/conserved quantities? Soliton?

Hi all. I would like to know what's so special about those "integratable systems"? I heard that KdV and NLS models belong to these systems and so they have soliton solution? But why? What's the importance of this? And what's the significance of many conserved quantities? I know, say, KdV has...
21. ### Understanding the Box-Ball Soliton System and Its Soliton Collisions

Well, I don't have a clue on where to put this, but I'll go with this because solitons are a physical phenomenom, too. But I'll guess this is a wrong place for this anyway, so I apologize in advance. Anyway, could somebody help me understand the box-ball soliton system (soliton, specifically...
22. ### Why Do Simulated and Theoretical Soliton Collisions Differ?

Hi all. I am studying the collision of two solitons of the NLS equation. Actually I just want to plot out the exact solutions given in R.S.Johnson's book. (page 321, figure 4.4). I have used MATLAB to do this and produced the figure using exactly the same values for the parameters. However...
23. ### What Is a Soliton? Exploring the Difference from Normal Waves

Hi all. Can somone explain me the difference between "soliton" and those "normal waves" i learn in high school physics? I can't really distinguish between them. While a soliton travel at a constant shape and velocity, doesn't the same apply to normal waves? That two solitons collide and...
24. ### Is This Equation a Soliton? - Exploring the Properties of f_{\mu}

f_{\mu}=\frac{\beta_{\mu}\psi\xi_{_{g}}}{\cosh^{2}\psi\xi} ,\hspace{2em} \xi=\xi_{_{T}}x^{0}+\xi_{_{S}}\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}} ,\hspace{2em} \beta_{\mu},\:\xi_{_{g}}},\:\xi_{_{T}},\:\xi_{_{S}}\rightarrow constant Is this f_{\mu} soliton ?
25. ### Can Solitons Predict the Future of Traveling Waves?

Hello all, I very interested by biological solitons (travelling waves) but my mathematical/physics knowledge is largely out to date. I post a picture of a biologic soliton below. Solitons involve maths/physics but I post the subject here? My asking is the following 1/ if a soliton is...