Localized magnetostatic wave pulses propagating in a magnetic thin film can be modeled by a nonlinear Schroedinger equation, which has stable envelope soliton solutions. When the wave velocity depends also on the wave amplitude it is necessary to add a self-steepening term to the evolution equation. However, when this is done the original envelope will evolve from a symmetric structure into an asymmetric structure as a direct result of the self-steepening term. An approximate time envelope shape is analytically calculated to obtain the time dependence of the asymmetry, and numerical simulations also indicate similar temporal development. Finally, the analytical calculation is compared with previous experimental pulse shapes obtained during propagation in yttrium iron garnet thin films.