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 Schrodinger 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.

中文翻译：

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非线性波阻塞问题的自定域孤子

在本文中，我们提出了一个数值框架来研究非线性波阻塞问题的自定域解的形状、动力学和稳定性。出于这个动机，我们使用 Smith 推导出的非线性薛定谔方程 (NLSE) 作为非线性波阻塞的模型。我们提出了一种谱重整化方法（SRM）来寻找该模型的自定位孤子。我们表明，对于恒定的、线性变化的或正弦电流梯度，即 dU/dx，史密斯 NLSE 的自定域孤子确实存在。此外，我们提出了一种具有四阶 Runge-Kutta 时间积分器的光谱方案来研究这种孤子的时间动力学和稳定性。我们观察到自定域孤子在电流梯度恒定或线性变化的情况下是稳定的，然而，它们对于正弦电流梯度是不稳定的，至少对于选定的参数是这样。我们评论我们的发现并讨论所提议方法的重要性和适用性。