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?