  We consider the classical field integrable system whose evolution equation is
the nonlinear Schr\"odinger equation with defocusing non-linearities, which is
the classical limit of the quantum Lieb-Liniger model. We propose a simple
derivation of the relation between two sets of conserved quantities: on the one
hand the trace of the monodromy matrix, parameterized by the spectral parameter
and introduced in the inverse-scattering framework, and on the other hand the
rapidity distribution, a concept imported from the Lieb-Liniger model. To do so
we use the definition of the rapidity distribution as the asymptotic momentum
distribution after an expansion. More precisely we use thought experiments
implementing an expansion and we present two different ways to derive our
result, based on different thought experiments which lead to different
calculations.
