  In this paper, we propose a numerical framework to study the shapes, dynamics
and the stabilities of the self-localized solutions of the nonlinear wave
blocking problem. With this motivation, we use the nonlinear Schr\"odinger
equation (NLSE) derived by Smith as a model for the nonlinear wave blocking. We
propose a spectral renormalization method (SRM) to find the self-localized
solitons of this model. We show that for constant, linearly varying or
sinusoidal current gradient, i.e. dU/dx, the self-localized solitons of the
Smith's NLSE do exist. Additionally, we propose a spectral scheme with 4th
order Runge-Kutta time integrator to study the temporal dynamics and
stabilities of such solitons. We observe that self-localized solitons are
stable for the cases of constant or linearly varying current gradient however,
they are unstable for sinusoidal current gradient, at least for the selected
parameters. We comment on our findings and discuss the importance and the
applicability of the proposed approach.
