  We investigate the non-linear coupling between radial and non-radial
oscillations of static spherically symmetric neutron stars as a possible
mechanism for the generation of gravitational waves that may lead to observable
signatures. In this paper we concentrate on the axial sector of the non-radial
perturbations. By using a multi-parameter perturbative framework we introduce a
complete description of the non-linear coupling between radial and axial
non-radial oscillations; we study the gauge invariant character of the
associated perturbative variables and develop a computational scheme to evolve
the non-linear coupling perturbations in the time domain. We present results of
simulations corresponding to different physical situations and discuss the
dynamical behaviour of this non-linear coupling. Of particular interest is the
occurrence of signal amplifications in the form of resonance phenomena when a
frequency associated with the radial pulsations is close to a frequency
associated with one of the axial w-modes of the star. Finally, we mention
possible extensions of this work and improvements towards more astrophysically
motivated scenarios.
