The equation describing surface waves in a one-dimensional magnetohydrodynamic discontinuity involves the Hilbert transform and is therefore nonlocal. It is found that by taking the analytic function whose real part is the first order perturbation one obtains a local equation. This equation yields several integral equalities which may be studied with the help of certain classical results of the theory of Hardy spaces. In particular this method provides information on the relation between the smoothness of the solution and its possible blow up in time.

中文翻译：

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磁更年期中的表面波：规律性和时间演化

一维磁流体动力不连续面中描述表面波的方程涉及希尔伯特变换，因此是非局部的。发现通过采用其实部是一阶扰动的解析函数，可以得到一个局部方程。这个方程产生了几个积分等式，可以在哈代空间理论的某些经典结果的帮助下进行研究。特别地，该方法提供了关于溶液的平滑度与其可能的时间膨胀之间的关系的信息。