We numerically study the transverse stability of the Peregrine solution, an exact solution to the one dimensional nonlinear Schrodinger (NLS) equation and thus a $y$-independent solution to the 2D NLS. To this end we generalise a previously published approach based on a multi-domain spectral method on the whole real line. We do this in two ways: firstly, a fully explicit 4th order method for the time integration, based on a splitting scheme and an implicit Runge--Kutta method for the linear part, is presented. Secondly, the 1D code is combined with a Fourier spectral method in the transverse variable. It is shown, with several examples, that the Peregrine solution is unstable against all standard perturbations, and that some perturbations can even lead to a blow up.

中文翻译：

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Peregrine溶液横向稳定性的数值研究

我们数值研究了 Peregrine 解的横向稳定性，它是一维非线性薛定谔 (NLS) 方程的精确解，因此是 2D NLS 的 $y$ 独立解。为此，我们在整条实线上推广了先前发布的基于多域谱方法的方法。我们通过两种方式做到这一点：首先，基于分裂方案和线性部分的隐式 Runge--Kutta 方法，提出了一种用于时间积分的完全显式 4 阶方法。其次，将一维码与横向变量中的傅立叶谱方法相结合。几个例子表明，Peregrine 解对所有标准扰动都是不稳定的，有些扰动甚至会导致爆炸。