# Self-Localized Solitons of the Nonlinear Wave Blocking Problem

## TL;DR

This paper introduces a spectral renormalization method to find and analyze the stability of self-localized solitons in the nonlinear wave blocking problem modeled by the NLSE, revealing stability depends on the current gradient profile.

## Contribution

The paper presents a novel numerical framework combining spectral renormalization and Runge-Kutta methods to study soliton shapes, dynamics, and stability in the nonlinear wave blocking context.

## Key findings

- Self-localized solitons exist for constant, linear, and sinusoidal current gradients.
- Stable solitons are found for constant and linear gradients.
- Solitons are unstable under sinusoidal current gradient.

## Abstract

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.