Do solitons in QFT really exist?

Question

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:

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?

Answer 1

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.