论文标题
关于卡瓦哈拉方程的分散量化和分化
On dispersive quantization and fractalization for the Kawahara equation
论文作者
论文摘要
在本文中，我们研究了溶液对卡瓦哈拉方程的二分法行为，具有有界变化的初始数据，类似于塔尔伯特效应。具体而言，我们观察到该解决方案是在合理时间进行量化的，而在非理性时间，它是一个没有分形曲线的连续可区分函数。然而，尚未探索这种现象的kawahara方程，这是第五阶KDV型方程。为了实现这一目标，我们得出了非线性Duhamel溶液的平滑估计值，当与线性解决方案上的已知结果结合使用时，它提供了对Talbot效应的数学描述。
In this paper, we investigate the dichotomous behavior of solutions to the Kawahara equation with bounded variation initial data, analogous to the Talbot effect. Specifically, we observe that the solution is quantized at rational times, whereas at irrational times, it is a nowhere continuous differentiable function with a fractal profile. This phenomenon, however, has not been explored for the Kawahara equation, which is a fifth-order KdV type equation. To achieve this, we derive smoothing estimates for the nonlinear Duhamel solution, which, when combined with the known results on the linear solution, provides a mathematical description of the Talbot effect.
