It is possible to consider baryons as solitons in the large N_c limit. This observation is due to Witten in 1979. There was proposal by Skyrme dating back to 1959 considering  baryons as topological solitons in the nonlinear sigma model for pseudoscalar mesons. These solitons are called skyrmions. Skyrme's  work fits in nicely with the large N_c expectation. See papers

‘Exotic Levels from Topology in the Quantum Chromodynamic Effective Lagrangian,’ A.P. Balachandran, V.P. Nair, S.G. Rajeev and A. Stern, Phys. Rev. Lett. 49, 1124 (1982).

(This shows that skyrmions carry nonzero baryon number.)

E. Witten, Current algebra,baryons and quark confinement, Nucl. Phys. B223, 433 (1983) (This shows that skyrmions in the sigma model are fermions; also demonstrates nonzero baryon number.)

G. Adkins, C. Nappi and E. Witten, Static proprties of nucleons in the Skyrme model,Nucl. Phys. B228, 552 (1983).

There are literally over a 1000 papers elaborating on these early works. Skyrme's work, which was way ahead of its time, can be traced from these papers.

There are models of hadrons as solitons, see for example the paper

"Geometric models of matter" by Atiyah, Manton and Schroers, in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

vol 468 (2012) pp 1252-1279.

but, as far as I'm aware, these are models for the hadrons themselves and do not describe the underlying quarks.

@Brian and Christina: thank you for your answers. I will try to look for Atiyah's paper. Best wishes

The Atiyah, Manton & Schroers paper is the perfect example of "not even wrong." All details are postponed "for future work,"  while all ideas are to be found in previous papers of other authors.

Hadrons are quantum objects, the equations that describe solitons describe classical objects. Therefore the latter can't but be considered as the backgrounds about which to compute the quantum corrections and construct the corresponding quantum fields theories, that would describe the hadrons. There was considerable work done in the 1970s, that is reviewed in the 1975 Erice lectures by S. Coleman, ``Classical lumps and their quantum descendants'', Subnucl.\ Ser.\ {\bf 13} (1977) 297. The approach worked in 1+1 dimensions, whereas hadrons, as relativistic objects, are described in 3+1 dimensions. Also, the scattering of hadrons is different than the scattering of solitons: the former involves particle production, the latter doesn't.  This is the main reason that it was, already then, realized that it could only provide theoretical  hints rather than phenomenological insight. Therefore it was, essentially, abandoned, though the SLAC and MIT bag models were inspired by those efforts. In 4d supersymmetric theories things seem to be under more control, solitons have been found in many cases  and there has been a lot of work in that direction. The problem is, of course, that it isn't known how to break supersymmetry, while keeping the quantitative control in these models-deformations of supersymmetric theories have been developed and are being studied. They are, however, more theoretical laboratories rather than phenomenological arenas.

Klaus Hasselmann had his metron model.  

http://xxx.lanl.gov/abs/hep-th/9810086

Majority of solitaries are not solitons. Let the Lax pair exists at least for one model from mentioned above, then solitons - may be. This is unique, single and unrepeatable proof of their existence.

Yes and no.  Probably the closest thing to a solitonic model of a hadron is the so-called Skyrmion, also called a "chiral soliton."  The modern (last 30 years or so) interest was sparked by a paper of Witten:

http://inspirehep.net/record/13235?ln=en

The basic idea is to start with a "chiral Lagrangian" (a truncation of the effective field theory of mesons) with the addition of a Wess-Zumino term.  Witten showed that this theory has a soliton, the "Skyrmion," which one can interpret as a baryon.  Subsequent work has shown that versions of this model can reproduce many hadronic properties, typically with 20-30% accuracy, with a relatively small number of parameters.  There are hundreds of papers on this, one of the earliest being this one by Adkins, Nappi, and Witten:

http://inspirehep.net/record/190174?ln=en

You can't really expect to do better than about 30% accuracy with the Skyrme model, because it is not QCD in any controlled approximation.  It is believed by many that the relative success of the Skyrme model is due to the existence of quantities which are more-or-less model independent.

One of the practical problems with these chiral soliton models is that when you try to make them more realistic, they become awkward very quickly.  For example, one approach to adding strangeness to the Skyrme model is the Callan-Klebanov bound state approach:

http://inspirehep.net/record/285161?ln=en

which is already quite complicated.

Indeed, but, once more, the Skyrmion (which was, also, discussed in Cokeman's lectures) is a classical configuration-its Lagrangian cannot be given a definition as a quantum field theory. The only solitons that make sense as quantum configurations in four spacetime dimensions are found in certain supersymmetric theories.

I think it's in the limit of large number of colors, not flavors-but these mesons and baryons are not solitons, because there is particle production in the large N_c limit in these models-they're not, necessarily, integrable, whereas solitons scatter elastically; if they don't, the notion of a soliton isn't useful. Incidentally, cf. S.~R.~Coleman,``1/N,'', Erice summer school 1979,SLAC-PUB-2484. The expansion in 1/N_colors does seem to capture properties of mesons-for baryons the picture is much murkier. 

(There is another limit, that does involve the ratio N_flavors/N_colors, but neither does this have anything to do, necessarily,  with solitons.) All these approximations simply try to construct the hadrons as collective excitations of effective degrees of freedom, that are weakly interacting at large distances, whereas the quarks are the weakly interacting degrees of freedom at short distances.  One shouldn't focus on solitons, but just recall that the aim is computational control, which solitons do provide, but they aren't the only description.

Yes, I tried changing "flavors" to "colors" and model-dependent to model-independent after posting my comment, but couldn't figure out how, so I deleted it. I simply wanted to point out that there is a limit in which the soliton model makes a connection to QCD, and that it can be quantized to some degree. The duality between the descriptions in terms of quark degrees of freedom and meson degrees of freedom is intriguing in terms of its connection to string theory, which can also be motivated through a large N_c description. It is probably not an accident that the early 80's interest in Skyrmions comes just before the rebirth of string theory.

Indeed, and it's now become the AdS/QCD correspondence, which, still, is work in progress. But solitons don't play a particularly prominent  role there (unless it's the extremal black holes, on the gravity side).  The duality is gauge/gravity, rather than quarks/mesons (which doesn't ring any particular bell.i think there were attempts in that direction, but they don't seem to have to been followed up further ). 

@all contributors: thank you for sharing all your answers here. Correct me if i am wrong, i got an impression that the prospect for soliton model of quarks (and hadron masses) is quite hopeless. Thanks 

I think it's unfair to say that solitons are "hopeless" for understanding hadrons.  Just like every other model of hadrons which is not QCD (e.g. constituent quarks, hadronic strings, bags, AdS/QCD, etc.), the soliton model describes some things well and some things poorly.  If your goal is to reproduce all the numbers in the PDG book, then of course you need to study the real theory.

I would say, though, that probably the easy things to understand using chiral solitons were worked out back in the 80's.  For example, a nice paper on the baryon mass spectrum is this one by Guadagnini:

http://inspirehep.net/record/14287?ln=en

who used chiral solitons to derive a generalization of the Gell-Mann-Okubo formula (which works.)

Incidentally, I agree with Drs Nicolis and Yost that the quantization of chiral solitons is problematic (maybe not impossible, but certainly not understood) except perhaps in the large-N limit or for special supersymmetric theories.

The connection with gauge/gravity duality is interesting.  The baryons should be related to D-branes, which are in some sense solitons of string theory.  There is a lot to be understood there, although the relevance to experiment may be tenuous.

@Peter: thank you for your answer. Best wishes

It is possible to consider baryons as solitons in the large N_c limit. This observation is due to Witten in 1979. There was proposal by Skyrme dating back to 1959 considering  baryons as topological solitons in the nonlinear sigma model for pseudoscalar mesons. These solitons are called skyrmions. Skyrme's  work fits in nicely with the large N_c expectation. See papers

‘Exotic Levels from Topology in the Quantum Chromodynamic Effective Lagrangian,’ A.P. Balachandran, V.P. Nair, S.G. Rajeev and A. Stern, Phys. Rev. Lett. 49, 1124 (1982).

(This shows that skyrmions carry nonzero baryon number.)

E. Witten, Current algebra,baryons and quark confinement, Nucl. Phys. B223, 433 (1983) (This shows that skyrmions in the sigma model are fermions; also demonstrates nonzero baryon number.)

G. Adkins, C. Nappi and E. Witten, Static proprties of nucleons in the Skyrme model,Nucl. Phys. B228, 552 (1983).

There are literally over a 1000 papers elaborating on these early works. Skyrme's work, which was way ahead of its time, can be traced from these papers.

@Dr. V.P. Nair and Luiz Botelho: thank you for your answers. Best wishes

What's going on?

Where is colored soliton?

My solly outside, while yours within;

Though the friendship must won,

Because of hadron,

A review of recent mathematical work on solitonic models of hadrons is

Atiyah, Manton and Schroers' paper:

Geometric Models of Matter

Michael Atiyah, Nicholas S. Manton, Bernd J. Schroers

(https://arxiv.org/abs/1108.5151)

In fact, all known integrable non-linear evolution equations are a reduction of the self-dual Yang-Mills equation.  The sine-Gordon equation is better suited than the Korteweg-de Vries one because it is relativistically covariant.  The Skyrme model is one of its possible generalisations.  Actually, Skyrme first studied the sine-Gordon equation.