  In the framework of mapped pseudospectral methods, we introduce a new
polynomial-type mapping function in order to describe accurately the dynamics
of systems developing almost singular structures. Using error criteria related
to the spectral interpolation error, the new polynomial-type mapping is
compared against previously proposed mappings for the study of collapse and
shock wave phenomena. As a physical application, we study the dynamics of two
coupled beams, described by coupled nonlinear Schr\"odinger equations and
modeling beam propagation in an atomic coherent media, whose spatial sizes
differs up to several orders of magnitude. It is demonstrated, also by
numerical simulations, that the accuracy properties of the new polynomial-type
mapping outperforms in orders of magnitude the ones of the other studied
mapping functions.
