Self-localized solitons of the nonlinear wave blocking problem

In this paper, we propose a numerical framework to study the shapes, dynamics, and stabilities of the self-localized solutions of the nonlinear wave blocking problem. To our best knowledge, blocking of the solitons in the marine environment has not been studied before in the existing literature. With this motivation, we use the nonlinear Schrödinger 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 a 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 applicability of the proposed approach.