Traveling waves for nonlocal models of traffic flow

Figure 1.  Typical profiles $P(\cdot)$ with $v(\rho) = 1-\rho, h = 0.2, \ell = 0.01$ and various $[\rho^-, \rho^+]$ values given in the legends. In the left we use the weight function $w(x) = \frac{2}{h}-\frac{2x}{h^2}$ on $(0, h)$ where $w'<0$, while in right plot we use $w(x) = \frac{2x}{h^2}$ on $(0, h)$ where $w'>0$

Figure 2.  Typical solutions of the FtLs model $(z_i(t), \rho_i(t))$ at various time $t$, with oscillatory initial condition. Above: $w(x) = \frac{2}{h}-\frac{2x}{h^2}$ and the solution approaches some profile as $t$ grows. Below: $w(x) = \frac{2x}{h^2}$ and the solution oscillates more as $t$ grows

Figure 3.  Left: Graphs of $Q(x)$ and $\widehat Q(x)$ on $[x_1, x_1+h]$. Right: Graphs of shifted functions $Q(s+x_1)$ and $\widehat Q(s+x_2)$

Figure 4.  Typical profiles $Q(\cdot)$ with $v(\rho) = 1-\rho, h = 0.2$, and various asymptotic values $[\rho^-, \rho^+]$ given in the legends. For the left plot we use $w(x) = \frac{2}{h}-\frac{2x}{h^2}$ for $x\in(0, h)$, and for right plot we use $w(x) = \frac{2x}{h^2}$ for $x\in(0, h)$

Figure 5.  Solution for the nonlocal conservation law (1.1) with oscillatory initial condition, at $t = 0, 0.4, 0.8$. Top: With $w(x) = \frac{2}{h}-\frac{2x}{h^2}$, i.e., $w'<0$, the solution quickly approaches the traveling wave profile $Q(x)$ as $t$ grows. Bottom: With $w(x) = \frac{2x}{h^2}$, i.e., $w'>0$, the solution becomes more oscillatory as $t$ grows