  We study the non-equilibrium dynamics obtained by an abrupt change (a {\em
quench}) in the parameters of an integrable classical field theory, the
nonlinear Schr\"odinger equation. We first consider explicit one-soliton
examples, which we fully describe by solving the direct part of the inverse
scattering problem. We then develop some aspects of the general theory using
elements of the inverse scattering method. For this purpose, we introduce the
{\em quench map} which acts on the space of scattering data and represents the
change of parameter with fixed field configuration (initial condition). We
describe some of its analytic properties by implementing a higher level version
of the inverse scattering method, and we discuss the applications of
Darboux-B\"acklund transformations, Gelfand-Levitan-Marchenko equations and the
Rosales series solution to a related, dual quench problem. Finally, we comment
on the interplay between quantum and classical tools around the theme of
quenches and on the usefulness of the quantization of our classical approach to
the quantum quench problem.
