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

Self-Localized Solitons of the Nonlinear Wave Blocking Problem