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In general, solitons are single-crest waves which travel at constant speed and don't loose their shape (due to their non-dispersivity), and there are many examples of them in the real world.

Now in QFT a soliton can be defined as a single crest which travels through a potential that (when we consider two dimensions) has the form of a sinusoid in the x-direction and has a value in the y direction that is the same as the corresponding point in the x-direction, more or less like the surface of a frozen sea with sinusoid waves.

Now one part of the soliton lies between two crests of the sinusoid, while the other part lies between the next two crests of the sinusoid, and it's stretched and moves in the y-direction. The y-direction extends to infinity on both sides. For those interested in the math take a look at Sine Gordon equation.

A soliton in QFT can be represented by this photograph:

enter image description here

Both sides of the rod about which the little rods rotate extend to infinity.

Now does a soliton in QFT really exist, or is it a mathematical construction? Or more concrete, does the described potential really exist? And IF they exist, how do they make themselves detectable?

Deschele Schilder
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    "Now does a soliton in QFT really exist, or is it a mathematical construction?". This is a meaningless question until you define what it means for a soliton to exist. What does it mean for an object to exist? At what point does a concept become just math as opposed to physically real? – AccidentalFourierTransform May 24 '17 at 10:13
  • There are many real, material solitons, but in the context of QFT a very strange (and in my eyes unreal) form of a potential energy is demanded to literally construct them, without any correspondence to the material world. – Deschele Schilder May 25 '17 at 10:57
  • "A soliton in QFT can be represented by this figure" - this claim is somewhere between very unclear and very wrong - what are you even trying to say here? What QFT are you talking about? A scalar theory? A gauge theory? What does "existence" mean here, given that a quantum field does not actually have a well-defined value at all point, so no configuration of the field "exists" in the classical sense. – ACuriousMind May 25 '17 at 11:20
  • I'm interested if the potentials which give rise to solitons really exist. The Mexican hat potential is after all also one that is invented. Only for the purpose to let the breaking of some symmetry happen. The excitations of the accompanying Higgs field suppose to proof the existence of this potential, while in fact, it's a mathematical construct. – Deschele Schilder May 25 '17 at 18:13
  • The well known L. Motl once wrote (see the first linked question):
    "First of all, most of the work on solitons is formal theory - analyses of soliton solutions in QFTs that are interesting theoretically but that are known not to describe the Universe around us." Where it must, of course, be noted that he wrote "most".     – Deschele Schilder May 26 '17 at 10:13

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All solitons are mathematical constructions. By definition, a soliton is a solution of a nonlinear partial differential equation (PDE) in which the dispersion is exactly cancelled by the non-linearity yielding a propagating non-dispersing wave like solution. There are many PDE's with this property and there are many physical systems that may be approximated by such PDE models. These models are never exact, so the "real world" solitons are non-existent. Such models are never-the-less frequently used to gain valuable insights into the behaviors of nonlinear mechanical (see my answer to this question), optical (see here), and fluid (see here) systems.

When the properties of solitons were first discovered, their temporal permanence was recognized as analogous to that of elementary particles and there was considerable study of QFT models with soliton solutions. I am not aware of any of these models that were based upon theories that became part of the Standard Model, but you may be interested in this question and the comments that it elicited.

Lewis Miller
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    These models are never exact, so the "real world" solitons are non-existent. In that case most objects in physics do not exist. And besides all solitons are real world material arrangements of particles (except in the context of QFT) from which we construct a math model.

     – Deschele Schilder May 25 '17 at 11:01
  • Physical objects are what exist in the real wourd. Models are mathematical idealizations that match the properties of objects to a greater or lesser extent. BTW I agree with you that QFT soliton are less connected with reality than the other examples I cited. – Lewis Miller May 25 '17 at 14:27