In our recent work we found a surprising breakdown of symmetry conservation: using standard numerical discretization with very high precision the computed numerical solutions corresponding to very nice initial data may converge to completely incorrect steady states due to the gradual accumulation of machine round-off error. We solved this issue by introducing a new Fourier filter technique for solutions with certain band gap properties. To further investigate the attracting basin of steady states we classify in this work all possible bounded nontrivial steady states for the Allen-Cahn equation. We characterize sharp dependence of nontrivial steady states on the diffusion coefficient and prove strict monotonicity of the associated energy. In particular, we establish a certain self-replicating property amongst the hierarchy of steady states and give a full classification of their energies and profiles. We develop a new modulation theory and prove sharp convergence to the steady state with explicit rates and profiles.

中文翻译：

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急剧收敛到 Allen-Cahn 的稳态

在我们最近的工作中，我们发现了对称守恒的一个令人惊讶的崩溃：使用非常高精度的标准数值离散化，由于机器舍入误差的逐渐积累，对应于非常好的初始数据的计算数值解可能会收敛到完全不正确的稳态。我们通过为具有某些带隙特性的解决方案引入新的傅立叶滤波器技术来解决这个问题。为了进一步研究稳态的吸引盆，我们在这项工作中对 Allen-Cahn 方程的所有可能的有界非平凡稳态进行了分类。我们描述了非平凡稳态对扩散系数的强烈依赖性，并证明了相关能量的严格单调性。特别是，我们在稳态层次结构中建立了某种自我复制的属性，并对其能量和轮廓进行了完整分类。我们开发了一种新的调制理论，并证明了对具有明确速率和轮廓的稳态的急剧收敛。