Nonlinear dynamics of a dispersive field line resonance (FLR) is investigated in a dipolar magnetic geometry using a new finite element code TOPO. Time dependent dispersion and steepening of the perpendicular Alfv¿n velocity profile lead to the acceleration of dispersive effects and to localization of the FLR within an ionospheric density cavity. Consequently, shear Alfv¿n waves are trapped between turning points inside nonlinear density perturbations. This results in a complicated nonlinear FLR structure and an associated overlap of timescales for dispersion and nonlinearity. We show that nonlinear effects lead to fine scale structuring produced by the interplay of nonlinear and dispersive effects. We discuss the relevance of our results to observations of FLR's in the auroral zone.