Link between integrability and soliton solutions

Question

I have been doing some research on the properties and dynamics of solitons (in particular, solitons in superfluids) and several works and papers mention the link between solitonic solutions and integrability of the non-linear differential equation describing the physical system. However, I found that the explanation of what exactly it entails for a system to "be integrable" and what this has to do with solitonic solutions is often either quite vague or explained in very technical and mathematical terms, with little physical content. Therefore, my question is twofold:

a) Is it possible to explain in physical terms what it means for a differential equation and its underlying system to be integrable, e.g. the 1D Gross-Pitaevskii equation (describing a 1D Bose-Einstein condensate) is said to be integrable.

b) What is the link between integrability of a non-linear differential equation and solitonic solutions of this equation? I have come across non-integrable differential equations which also permit solitary wave solutions, but maybe these are not "true solitons" in the strictest sense of the word?

Answer 1

a) Usually when a physicist refers a differential equation is integrable (classical integrability), it means the nonlinear differential equation can be mapped into an auxiliary linear problem (inverse scattering method). In this case, 1D Gross-Pitaevskii equation (in some scenarios, it is usually called non-linear Schrödinger equation) can be mapped in an auxiliary linear problem, characterized by transfer matrix $T$ with a spectral parameter $\lambda$. The time evolution of the transfer matrix is relatively easy to obtain. And the dynamics in real space can be obtained by mapping the time-evolved transfer matrix back to the real space(usually not an easy task). So the defining property of integrable systems would be that one can obtain the time evolution of any initial profile. Of course, this hints that for integrable systems there are infinitely many conserved charges, which can be derived from the property of the transfer matrix $T$.

b) Soliton is a type of wave that does not disperse while propagating and localizes. If you define like that, some non-integrable models can also host solitons, e.g. kinks in $\varphi^4$ field. But a more rigorous definition is that the scattering between solitons is elastic, ruling out all the "solitons" in non-integrable models. Solitons in integrable models are localized solutions that do not disperse and scatter with each other elastically. One remark is that not all integrable models have solitonic solutions. For instance, sinh-Gordon model doesn't have solitonic solution, only radiative modes (which are dispersive).

Answer 2

In the context of physics we say that a (Hamiltonian) system is integrable if it can, in principle, be solved by quadratures such as a completely separable Hamilton-Jacobi equation. It can be shown that a system with $N$ degrees of freedom is integrable if and only if it has $N$ independent conserved quantities in involution.

For a field theory, which has infinite number of degrees of freedom one needs an infinite number of conserved quantities in order to have integrability. This of course means that the system is infinitely restricted and a soliton precisely have such property. Roughly speaking, in order to retain its shape, (non topological) solitons need a restriction for each point in space or equivalently for each of its degrees of freedom. A classic example which illustrate this is the KdV equation which models non linear waves in shallow water. This is non linear field theory which admits soliton solutions. At same time it is integrable and it was explicitly shown that it possesses infinitely many symmetries giving rise to infinitely many conserved quantities.

Answer 3

There is plenty of integrable partial differential systems which are dispersionless a.k.a. hydrodynamic-type (i.e., can be written as first-order homogeneous quasilinear systems; apparently in the case of more than three independent variables the dispersionless systems form an overwhelming majority of all known integrable systems, see e.g. this article and references therein) and do not necessarily have multisoliton solutions but have some analogs thereof, multiphase solutions, cf.e.g. here.