Traveling Waves for a Microscopic Model of Traffic Flow

We consider the follow-the-leader model for traffic flow. The position of each car $z_i(t)$ satisfies an ordinary differential equation, whose speed depends only on the relative position $z_{i+1}(t)$ of the car ahead. Each car perceives a local density $\rho_i(t)$. We study a discrete traveling wave profile $W(x)$ along which the trajectory $(\rho_i(t),z_i(t))$ traces such that $W(z_i(t))=\rho_i(t)$ for all $i$ and $t>0$; see definition 2.2. We derive a delay differential equation satisfied by such profiles. Existence and uniqueness of solutions are proved, for the two-point boundary value problem where the car densities at $x\to\pm\infty$ are given. Furthermore, we show that such profiles are locally stable, attracting nearby monotone solutions of the follow-the-leader model.


Introduction and Preliminaries
We consider a microscopic model for traffic flow. Let be the length of all the cars, and let z i (t) be the position of ith car at time t. We order the indices for the cars such that for every i ∈ Z. (1.1) For a car with index i, we define the local density Note that if ρ i = 1, then the two cars with indices i and i + 1 will be bumper-to-bumper. Thus 0 ≤ ρ i ≤ 1 for all i.
Note that (1.3) can be rewritten as a system of ODEs for the discrete density functions ρ i (t),ρ If one uses (1.5), then (1.6) becomeṡ Given the initial positions of the cars z i (0) and the speed of the leader as i → ∞, the existence of solution for the ODE system (1.3) is established in the literature [8,14]. We can define a piecewise constant function ρ (t, x) from the discrete densities {ρ i } as ρ (t, x)= ρ i (t), for x ∈ [z i (t), z i+1 (t)). (1.8) As → 0 and the number of the cars tends to ∞, under suitable assumptions one has the convergence ρ (t, x) → ρ(t, x), where the limit function ρ(t, x) provides a weak solution for the scalar conservation law See [8] for a proof using direct properties of the solutions of (1.3), and some more recent works [14,15] where the same results are achieved utilizing a Lagrangian formulation and the properties of monotone numerical schemes. For other related works including model derivations, analysis, and treatment of various conditions, we refer to [1-7, 11, 19] and the references therein.
In this work we seek a "discrete traveling wave profile" for the FtL model, as a corresponding approximation to the shock waves for the conservation law (1.9). To fix the idea, we start with a monotone stationary profile W (x) such that the position of the point (z i (t), ρ i (t)) traces along the graph of the function W (x) as time t evolves. To be precise, we require We remark that for general traveling waves with speed σ = 0, the profile will be stationary in the shifted coordinate x → ξ = x − σt, see the discussion in Section 5.
Differentiating both sides of (1.12) in t, and using (1.3) and (1.6), one gets Note that Since z i is randomly chosen, we write x for z i , and obtain the following equation: If W (x) = 0 for some x, then we set Once the "initial data" is given on an interval [x, ∞) for anyx, the DDE (1.13) can be solved backwards in x, and the profile W (x) can be obtained for all x ≤x. This is in agreement with the following-the-leader principle.
In this paper we study in detail the DDE (1.13). In particular, we study the "two-pointboundary-value" problem. To be specific, we seek solutions of (1.13) that satisfies the boundary conditions at the infinities: (1.14) In the case of the stationary profile W (x), ρ ± must further satisfy Note that any horizontal shift of the profile W (x) is again a profile. Thus, a unique profile can be achieved by requiring a "location-fixing" condition at x = 0, say We show that, for any given ρ ± satisfying (1.15), there exists a profile W (x), unique up to horizontal shifts. Furthermore, such traveling waves are local attractors for the solutions of the FtL model (1.3).
In the literature, solutions for the conservation law (1.9) are approximated by various approaches. These include the viscous equations, kinetic models with relaxation terms, and various numerical approximations. For many of the approximate solutions, the study of traveling wave profiles is one of the key techniques in the analysis. In this paper, we consider the microscopic "particle" model and its traveling waves, filling a missing piece in the literature.
We mention also a study on traveling waves for a non-standard integro-differential equation modeling slow erosion [12], where uniqueness and local stability are achieved.
The rest of the paper is organized as follows. For stationary profiles W (x), in Section 2 we prove several technical Lemmas. These results are utilized in Section 3 where we prove the existence and uniqueness of the profile. Furthermore, such profiles are local attractors for solutions of the FtL model (1.3), proved in Section 4. Extension to general traveling waves with non-zeros speed is outlined in Section 5. Finally, concluding remarks and further open problems are discussed in Section 6.

Technical Lemmas; Properties of the Stationary Profile
We consider the stationary profile W (x), satisfying the DDE (1.13) with boundary conditions (1.14)-(1.15).
We first provide a formal argument which makes connection between the profile W (x) and the viscous shock for the conservation law (1.9). Assuming that /W (x) > 0 is very small, by Taylor expansion we have Dropping the higher order terms, the DDE (1.13) is approximated by This equation can be manipulated into: and then We conclude Now we consider the viscous conservation law Stationary viscous shock wavesρ(x) must satisfy the ODE We observe that the ODEs (2.2) and (2.3) are connected through the relation: For any given profile W (x), one can generate a distribution of car positions {z i }, and vise versa. We make the following definitions.
Definition 2.1. Let the function x → W ∈ (0, 1] be given for x ∈ R. We call a sequence of car positions {z i } a distribution generated by W (x), if If one imposes z 0 = 0, then the distribution is unique.
The following Lemma is an immediate consequence of these definitions. Our first theorem states existence and uniqueness of monotone solutions of (1.13) as an initial value problem, under suitable assumptions on the initial data.
Theorem 2.1. Fix anx. Let ψ(x) ∈ (0, 1) be a continuous monotone function defined on the interval x ≥x such that ψ (x) > 0 for all x ≥x. Let W (x) be the solution of the DDE (1.13) on x <x, solved backwards in x, with initial data ψ(x) given on x ≥x. Then, there exists a unique positive solution W (x), which is monotone increasing such that Proof. The existence and uniqueness of the solution W (x) for the initial value problem follows from an iteration argument. It is understood that the derivative W (x) in (1.13) is the left derivative. We clearly have We claim that, if W (x) exists on I 1 , then W (x) ≥ 0. Indeed, the lower bound W (x) ≥ 0 is clear since 0 is a critical point. Assuming that W (x) becomes negative on some subset of I 1 , then there exists a point x 0 ∈ I 1 such that But this is not possible because by (1.13) we have We further claim that, if W (x) exists on I 1 , then it is monotonically increasing. We prove by contradiction. Assume that W (x) is not monotone on I 1 . Then there exists a valuex, with x <x, where W changes sign, such that However, this would imply a contradiction to (2.6).
Thus, we deduce that which means, Then, the equation (1.13) reduces to an ODE of the form Since F is Lipschitz in both arguments, ψ is continuous and Lipschitz for W > 0, by standard ODE theory, the solution W (x) exists and is unique on I 1 .
Remark 2.1. For general references on standard theory for delay differential equations, see [9,10]. We remark that our equation (1.13) does not fall into the standard setting, therefore we provide a simple proof for existence and uniqueness of solutions. We further note that, if the initial condition shall be monotonically decreasing, such that ψ (x) < 0 for x >x, the global existence of solution W (x) on x ∈ (−∞,x] fails. One simply observes that W (x) < 0, so W (x) increases as x decreases, and W (x) blows up to infinity as W (x) approaches 1.
Next Lemma describes the asymptotic behavior at the limits as x → ±∞.
Then, we have the followings.
• As x → ∞, W (x) can approach ρ + at an exponential rate only if ρ + > ρ * . The exponential rate λ + satisfies the estimate The exponential rate λ − satisfies the estimates Step 1. Consider the asymptotic behavior as x → +∞. By assumption we have where η(x) is the first order perturbation. Plugging this into (1.13), and neglecting the higher order terms, we obtain the following linearized equation for η(x): Denoting the positive constants as This is a linear delay differential equation, which can be solved explicitly using the characteristic equation. Seeking solution of the form where M is an arbitrary constant (which could be both negative or positive), the rate λ satisfies the characteristic equation We locate all the zeros for the function G(λ), in particular the positive ones. We observe Thus, G(·) is a convex function which goes through the origin. Typical graphs of G(λ) for different values of b are illustrated in Figure 1. We have: Figure 1: Typical graphs of G(λ) and location of the zeros.
• If ρ + < ρ * , then the other rate λ − < 0 indicates exponential growth of η(x), which is not valid. The only possible solution is the trivial one.
• If ρ + > ρ * , then the other zero λ + > 0 indicates exponential decay of η(x) in the limit as x → ∞. This is the valid case.
Step 2. A similar computation can be carried out for x → −∞. We write where ζ(x) is a small perturbation. The linearized equation for ζ(x) becomes Denoting the positive constantŝ and seeking solutions of the form we arrive at the characteristic equation To seek positive zeros of H(·), we first observe that H(0) = 0. Furthermore, we have Thus, positive rate λ = λ − exists only for the case whenb < 1, i.e., when ρ − < ρ * .
A similar computation as for (2.7) leads to an estimate with both upper and lower bounds: where −c 0 is an upper bound for f , see (1.10).

22)
On the other hand, if ρ − is close to 0, we have a different estimate On the other end, by estimate (2.23), as ρ − → 0, we have λ − → ∞. Thus, if ρ − = 0, we must have W (x) = 0 for x <x, for somex.
One concludes that, if ρ − = 0, ρ + = 1, the only possible profile is a step function with the jump located at somex. This represents the scenario where cars are bumper-to-bumper on x >x, and the road is empty for x <x.
The next lemma is most interesting. It shows that, if W (x) is a stationary monotone profile such that the solutions {z i (t)} of (1.3) traces along, then the distribution {z i (t)} demonstrates a "periodic" pattern. (E2) The solutions {z i (t)} exhibit the following periodic behavior. There exist a "period" t p , such that after the period each car takes over the initial position of its leader, i.e., Proof.
Step 1. We first prove that (E2) ⇒ (E1). Without loss of generality, we consider a car initially located at z 0 (0) = x for some x, and its leader, initially located at Thus, the evolution of z 0 (t) satisfies the ODE This is a separable equation. (E2) implies the following identity which easily leads to (1.13), proving (E1).
Step 2. To prove the indication (E1) ⇒ (E2), assume that W (x) satisfies the DDE (1.13). For a given time t, let {z i (t)} be a distribution of cars generated by W (x), as in Definition 2.1. We write now Since W (x) solves (1.13), it satisfies (2.26). The time it takes for the ith car to reach the original position of its leader z i+1 is dz.
The next Lemma connects the period t p to the limit values of W (x) at x → ±∞.
Lemma 2.4. Let W (x) and {z i (t)} be given as in the setting of Lemma 2.3. Let ρ − , ρ + be two states that satisfy Then, the following additional properties are equivalent. Proof.
Step 1. We first prove that (E3) ⇒ (E4). Assume (E3) holds, such that W (x) is a monotone profile satisfies (2.29). Let > 0, and consider the limit as x → +∞. There exists an M such that for all x ≥ M we have Since φ (·) < 0, we have, for all x ≥ M , Integrating this inequality over [x, x + /W (x)], one has, for all x ≥ M , dz.

This gives
.
Taking the limit → 0, we get The other limit x → −∞ can be treated in a completely similar way, proving (E4).
Step 2. The implication (E4) ⇒ (E3) follows by contradiction. Assuming that (E4) holds, but lim By the proof in Step 1 we have the contradiction t p = /f = /f .

Approximate Sequence; Existence and Uniqueness of Traveling Wave Profiles
We now construct approximate solutions to W (x) as a two-point-boundary-value problem, and prove their convergence, thus establishing the existence of traveling wave profiles.
there exists a monotone stationary profile W (x) which satisfies the DDE (1.13) and the "boundary" values lim Proof. By Remarks 2.3-2.5, we rule out the trivial cases. For the rest of the proof, we consider The proof takes a few steps.
(1). We first construct the sequence of approximate solutions. Let a sequence {x n } be given such thatx We define the function where λ + is the rate computed in Lemma 2. (2). We derive an estimate for ρ −,n . Given an n such thatx n is sufficiently large, so Consider the solution W n (x), given on x ≤x n . Let {z n i } be a distribution of car positions generated by W n (x), with z n 0 =x n , for i = 0, −1, −2, −3, · · · . Let {z n i (t)} be the solution of the system of ODEs (1.3), for index i < 0, and the leader z n 0 (t) traces along the initial condition ψ(x) on x >x n . By Lemma 2.3, {z n i (t)} (i < 0) demonstrates periodic behavior. We denote the period by t p,n . Note that once t p,n is given, we obtain the unique value of ρ −,n by the relation Thus, it suffices to show that We further observe that, thanks to the periodic behavior of {z n i (t)}, (i < 0), we only need to get an estimate of the time for car located at z n −1 , to reach z n 0 =x n . Here z n −1 is uniquely defined by the implicit relation (3). By the set up, W n (x) is very close to ρ + on the interval [z n −1 , z n 0 ], therefore an estimate on t p,n can be obtained by linearization. By Lemma 2.2, we have the first order approximation of W n (x) on [z n −1 , z n 0 ], denoted as To simplify the notation, we denote the magnitude of the perturbation by The first order approximation for W n (x) can be written as The corresponding distance z n 0 − z n −1 is computed approximately as We can compute t p,n , using a first order approximation in δ n , as Using (3.8), we get where M is a positive constant, depending on the data ρ + , φ, V, , but not on δ n . As n → ∞, δ n → 0, and we conclude (3.6), completing the proof. In Figure 2 we present some numerical simulations of the approximate solutions W n (x). We obverse the convergence asx n → ∞. Furthermore, from (3.9) it holds that t p,n > t p , indicating ρ −,n < ρ − , which is consistent with the simulation results.
Once the existence of the profile W (x) is proved, we establish the uniqueness of the solution for the "two-point-boundary-value-problem" for the DDE (1.13). Proof. We first consider the trivial cases. If ρ − = ρ + = ρ * , the only monotone graph is W (x) ≡ ρ * . If ρ − = 0 and ρ + = 1, then t p → ∞ and nothing moves, so the flux must be 0 everywhere. The only monotone solution is a unit step function. In the rest of the proof, we assume 0 < ρ − < ρ * < ρ + < 1.
We prove by contradiction. Consider the settings of Theorem 3.1, and let W (x),W (x) be two solutions which are different. Assume that, after some horizontal shift, the graphs of W (x) andW (x) intersect at a pointx so that Then, by the periodical property, we have This means that, if the graphs of W andW cross each other atx, then they must cross each other at least one more time on (x,x + /W (x)). Thus, they must cross each other infinitely many times for x ∈ R.
We now freeze the graph of W (x), and shift the graph ofW (x) to the right until they touch each other only atx tangentially, such that Again, by periodicity, we get a contradiction.
We conclude that the graphs of W andW either completely coincide or never cross each other. Then they must be horizontal shifts of each other, proving the uniqueness.
Numerical simulations. Various profiles of W (x) are plotted in Figure 3, for various values of ρ ± . Here we use = 0.1 and φ(ρ) = 1 − ρ, such that ρ * = 0.5. We plot the graphs of W (x) that connect the following pairs of limit values of (ρ − , ρ + )  We make a couple of observations.
(1). For smaller values of (ρ + − ρ − ), the profiles W (x) has a smaller value of W (0). We provide a formal argument. For small , W (x) can be approximated by a linear function W (x) = W (0) + σx, σ= W (0), for x close to 0. Writing out only the first order approximations, we have The periodic property gives .
Working out the integration, we get Multiplying both sides by −σ and then taking the exponential function on both sides, we obtain Moving everything to the right hand side, it gives the equation Recall that f * = V /4 is a constant. The function K(σ) has the following properties: Iff = f * , the only zero for K(σ) is σ = 0. Whenf < f * , then K (0) > 0, and there exists another positive zero σ + for K(σ). One can easily verify that W (0) = σ + decreases asf increases to the value f * , in correspondence to our simulation result.
We remark that this is different from a viscous shock profile, where the diffusion is uniform and the profile is odd symmetric about the location of the shock.

Stability of the discrete traveling waves
We now show that the traveling wave profiles are local attractors for the solution of the FtL model.
Then, there exists a constanth such that Theorem 4.1 implies that as t → ∞, {z i (t), ρ i (t)} approaches asymptotically a distribution generated by the profile W (x −h).
Proof. Since W (x) is monotonically increasing, with lim x→±∞ W (x) = ρ ± , then for any point (z, ρ) with ρ − < ρ < ρ + , there exists a unique value h such that ρ = W (z − h). We define the function Then, for t ≥ 0 and for each i, let Denote Differentiating in t, we getρ Using thatż we getḣ Then, if h i (t) < h i+1 (t), we have Since φ is a monotonically decreasing function, we have φ(W (y i + /ρ i )) < φ(ρ i+1 ) and φ(W (y i )) > φ(W (y i + /ρ i )), It suffices to show that lim Indeed, for any given t ≥ 0, from the previous discussion we have the following.
exists and is non-negative. To show that the limit must be 0, we use contradiction and assume the opposite, such that lim and let {z i ,ρ i } be the asymptotic car distribution, with the corresponding values of {h i }. Now, take {z i ,ρ i } as the initial data and solve the system of ODEs (1.3). There exists an index j whereh j is the maximum withh j >h j+1 . By the previous discussion we have d dth j < 0. If this is the isolated maximum, then d dt h < 0, a contradiction. Ifh j−1 is also a maximum, then after an arbitrarily small amount of time we have d dth j−1 < 0 so d dt h < 0, still a contradiction. Thus, we conclude (4.5), completing the proof.

Extension to general traveling waves
One can extend the analysis to traveling waves with speed different from 0, by a simple coordinate shift. Let V σ be the wave speed and let ξ = x−V σt be the shifted space coordinate. Let ζ i (t) be the position of the ith car in the shifted coordinate, and ρ i the discrete density. We haveζ Since the density is not affected by a horizontal shift, the ODE for ρ i is unchanged.
The corresponding conservation law is The analysis for the stationary traveling wave can be applied here with minimal modifications.

Concluding Remark
In this paper we study traveling wave profiles of a particle model for traffic flow, i.e., the follow-the-leader (FtL) ODE models for car positions. Given any densities ρ ± at x → ±∞, with ρ − < ρ + , we prove that there exists a unique traveling wave profile for the FtL model. Furthermore, such profiles are locally stable which attract nearby solutions of the FtL model. In the limit as → 0, the traveling waves converge to admissible shocks for the solution of the conservation law (1.9). Our results fill a gap in existing theory on traveling waves. The admissible conditions derived from our result are in accordance to the counter part for the viscous equation where stable viscous shocks only exist for upward jumps.
It's interesting and also non-trivial to study the same particle model on a road with rough conditions. For example, let κ(x) denote the speed limit (which reflects the road condition), and assume that it is a piecewise constant function with a jump at x = 0. One would like to seek stationary traveling waves for the FtL model, around x = 0. The corresponding macroscopic model ρ t + f (ρ, κ(x)) x = 0 is a scalar conservation law with discontinuous flux. In existing literature, admissibility conditions on the jump at x = 0 are derived through the viscous model ρ t + f (ρ, κ(x)) x = ερ xx and take the vanishing viscosity limit ε → 0+. However, our preliminary analysis shows a rather different scenario for limits of the FtL model, as → 0, where many of the vanishing viscosity limits are actually not admissible. Details are in a forthcoming work [22].