  In the framework of a one-dimensional model with a tightly localized
self-attractive nonlinearity, we study the formation and transfer (dragging) of
a trapped mode by "nonlinear tweezers", as well as the scattering of coherent
linear wave packets on the stationary localized nonlinearity. The use of the
nonlinear trap for the dragging allows one to pick up and transfer the relevant
structures without grabbing surrounding "garbage". A stability border for the
dragged modes is identified by means of of analytical estimates and systematic
simulations. In the framework of the scattering problem, the shares of trapped,
reflected, and transmitted wave fields are found. Quasi-Airy stationary modes
with a divergent norm, that may be dragged by the nonlinear trap moving at a
constant acceleration, are briefly considered too.
