Nanoscale pattern formation on the surface of a solid that is bombarded with a broad ion beam is studied for angles of ion incidence, $\theta$, just above the threshold angle for ripple formation, ${\theta }_c$. We carry out a systematic expansion in powers of the small parameter $\epsilon \equiv {(\theta -{\theta }_c)}^{1/2}$ and retain all terms up to a given order in $\epsilon$. In the case of two diametrically opposed, obliquely incident beams, the equation of motion close to threshold and at sufficiently long times is rigorously shown to be a particular version of the anisotropic Kuramoto-Sivashinsky equation. We also determine the long-time, near-threshold scaling behavior of the rippled surface's wavelength, amplitude, and transverse correlation length for this case. When the surface is bombarded with a single obliquely incident beam, linear dispersion plays a crucial role close to threshold and dramatically alters the behavior: highly ordered ripples can emerge at sufficiently long times and solitons can propagate over the solid surface. A generalized crater function formalism that rests on a firm mathematical footing is developed and is used in our derivations of the equations of motion for the single and dual beam cases.

中文翻译：

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离子轰击在图案形成阈值附近产生的纳米级波纹形貌理论。

研究了用宽离子束轰击的固体表面的纳米级图案形成，以了解离子的入射角， $\theta$，刚好高于形成波纹的阈值角， ${\theta }_C$。我们对小参数的幂进行系统的扩展$\epsilon \equiv {（\theta -{\theta }_C）}^{\mathrm{1个}/2}$ 并保留给定订单中的所有条款 $\epsilon$。在两个完全相反的，倾斜入射的光束的情况下，严格地表明，运动方程接近阈值并且在足够长的时间内是各向异性的Kuramoto-Sivashinsky方程的特定形式。在这种情况下，我们还确定了波纹表面的波长，幅度和横向相关长度的长期，接近阈值的缩放行为。当用单个倾斜入射光束轰击表面时，线性色散在接近阈值的过程中起着至关重要的作用，并显着改变其行为：在足够长的时间内会出现高度有序的波纹，孤子可以在固态表面上传播。