This work is devoted to the construction of weakly nonlinear, highly oscillating, current vortex sheet solutions to the incompressible magnetohydrodynamics equations. Current vortex sheets are piecewise smooth solutions to the incompressible magnetohydrodynamics equations that satisfy suitable jump conditions for the velocity and magnetic field on the (free) discontinuity surface. In this work, we complete an earlier work by Ali and Hunter and construct approximate solutions at any arbitrarily large order of accuracy to the free boundary problem in three space dimensions when the initial discontinuity displays high frequency oscillations. As evidenced in earlier works, high frequency oscillations of the current vortex sheet give rise to `surface waves' on either side of the sheet. Such waves decay exponentially in the normal direction to the current vortex sheet and, in the weakly nonlinear regime that we consider here, their leading amplitude is governed by a nonlocal Hamilton-Jacobitype equation known as the `HIZ equation' (standing for Hamilton-Il'insky-Zabolotskaya) in the context of Rayleigh waves in elastodynamics. The main achievement of our work is to develop a systematic approach for constructing arbitrarily many correctors to the leading amplitude. Based on a suitable duality formula, we exhibit necessary and sufficient solvability conditions for the corrector equations that need to be solved iteratively. Theverification of these solvability conditions is based on a combination of mere algebra and arguments of combinatorial analysis. The construction of arbitrarily many correctors enables us to produce infinitely accurate approximate solutions to the free boundary problem. Eventually, we show that the rectification phenomenon exhibited by Marcou in the context of Rayleigh waves does not arise in the same way for the current vortex sheet problem.

中文翻译：

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磁流体动力学中的弱非线性表面波。一世

这项工作致力于构建不可压缩磁流体动力学方程的弱非线性、高振荡、电流涡流片解。当前的涡流片是不可压缩磁流体动力学方程的分段平滑解，该方程满足（自由）不连续面上的速度和磁场的合适跳跃条件。在这项工作中，我们完成了 Ali 和 Hunter 的早期工作，并在初始不连续性显示高频振荡时，以任意大的精度构建了三个空间维度上的自由边界问题的近似解。正如早期工作所证明的那样，当前涡旋片的高频振荡会在涡旋片的任一侧产生“表面波”。这样的波在当前涡旋片的法线方向上呈指数衰减，并且在我们在这里考虑的弱非线性区域中，它们的超前振幅由称为“HIZ 方程”的非局部 Hamilton-Jacobitype 方程控制（代表 Hamilton-Il 'insky-Zabolotskaya）在弹性动力学中的瑞利波的背景下。我们工作的主要成就是开发了一种系统方法，可以构建任意多个超前幅度的校正器。基于合适的对偶公式，我们展示了需要迭代求解的校正方程的必要和充分的可解性条件。这些可解性条件的验证基于单纯的代数和组合分析参数的组合。任意多个校正器的构造使我们能够对自由边界问题产生无限精确的近似解。最终，我们表明 Marcou 在瑞利波的背景下表现出的整流现象不会以与当前涡旋片问题相同的方式出现。