Lighthill Whitham Richards Payne交通流方程的一些精确解
Some exact solutions to the Lighthill Whitham Richards Payne traffic flow equations
摘要：
我们找到了一类精确解Lighthill Whitham Richards Payne（LWRP）交通流方程。使用两个连续的拉格朗日变换，实现线性化。接下来，取决于初始密度，我们要么应用（同样是两个）朗伯函数，并获得汽车密度和速度对x和t的依赖关系的精确公式，否则，失败的是，在参数表示中得到相同的结果。计算总是涉及一致性条件的两种可能的因式分解。两者都必须考虑。从物理学的角度来说，这个星群通常以不同的速度分成两个分支。每一个速度很快就变得均匀。这个结果在许多方面类似于Korteweg-de Vries方程的两个孤子解。我们检查一般的保护要求。尽管自LWRP以来交通流研究已经取得了巨大的发展，但这种精确的计算可能为解决类似问题（如气体动力学或河流中的水流）打开大门。考虑到这种可能性，我们在最后详细介绍了这个过程。
英文摘要：
We find a class of exact solutions to the Lighthill Whitham Richards Payne (LWRP) traffic flow equations. Using two consecutive lagrangian transformations, a linearization is achieved. Next, depending on the initial density, we either apply (again two) Lambert functions and obtain exact formulas for the dependence of the car density and velocity on x and t, or else, failing that, the same result in a parametric representation. The calculation always involves two possible factorizations of a consistency condition. Both must be considered. In physical terms, the lineup usually separates into two offshoots at different velocities. Each velocity soon becomes uniform. This outcome in many ways resembles the two soliton solution to the Korteweg-de Vries equation. We check general conservation requirements. Although traffic flow research has developed tremendously since LWRP, this calculation, being exact, may open the door to solving similar problems, such as gas dynamics or water flow in rivers. With this possibility in mind, we outline the procedure in some detail at the end.
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