表示交通阻塞的扰动mKdV方程的行波解
Travelling wave solutions of the perturbed mKdV equation that represent traffic congestion
摘要：
一个著名的最佳速度（OV）模型描述车辆沿单车道的道路，这减少到一个扰动的修改Korteweg-de Vries（mKdV）方程的不稳定政权。这个方程的定常行波解，然后推导出与多尺度扰动技术。层次结构中的一阶解是用慢变量和快变量来表示的。在下面的顺序，一个系统的微分方程突出显示，管理缓慢演变的性质的领先的解决方案。然后，它表明，该系统的临界点表示行波没有慢变化。其结果是，一个家庭的稳定波具有恒定的振幅和周期。当满足周期边界条件时，这些解的参数与出现在OV模型中的驾驶员灵敏度$\hat{a}$相关联。对于某些给定的$\hat{a}$，存在向上和向下形式的解，其中向下类型对应于交通拥塞。数值模拟被用来验证渐近分析，并检查我们的解决方案的长期行为。
英文摘要：
A well-known optimal velocity (OV) model describes vehicle motion along a single lane road, which reduces to a perturbed modified Korteweg-de Vries (mKdV) equation within the unstable regime. Steady travelling wave solutions to this equation are then derived with a multi-scale perturbation technique. The first order solution in the hierarchy is written in terms of slow and fast variables. At the following order, a system of differential equations are highlighted that govern the slowly evolving properties of the leading solution. Then, it is shown that the critical points of this system signify travelling waves without slow variation. As a result, a family of steady waves with constant amplitude and period are identified. When periodic boundary conditions are satisfied, these solutions' parameters are associated with the driver's sensitivity, $\hat{a}$, which appears in the OV model. For some given $\hat{a}$, solutions of both an upward and downward form exist, with the downward type corresponding to traffic congestion. Numerical simulations are used to validate the asymptotic analysis and also to examine the long-time behaviour of our solutions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
