Existence of traveling wave solutions in continuous OV models

The paper reduces the traffic PDE to the Liénard-type system (1.6) with f(u)=(Ku−V(u)−c)/K, h(u,µ)=f′(u)+µ, g1>0, g2>0, and establishes existence/uniqueness-in-µ of heteroclinic connections (Theorem 1), a complete classification of homoclinic pulses under Condition (C) (Theorem 2), and uniqueness-in-µ plus positivity of µper for periodic orbits (Theorem 3), all via phase-plane shooting and µ-monotonicity of the manifolds w±(u) given by w_u=g1f/w+g2h, with (w±)_µ satisfying a first-order linear ODE; see the system setup and D1/D2 partition and main theorems and the monotonicity lemma . The candidate solution reproduces the same program: system reduction, manifold-graph formulation, strict µ-monotonicity, gap functions for shooting, and an energy identity. Results align item-by-item with Theorems 1–3 (including condition (C) and the role of u0, u1, u2; and nonexistence statements). Minor issues in the model write-up include an incorrect aside that u2 can become a sink (u2 is always a saddle when f′(u2)>0 since det<0) and a heuristic step for µper>0; neither affects the main conclusions, which follow by the same monotonicity/return-map framework used in the paper.