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 integrability society, it is usually called non-linear 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).

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).