10.5 Dominant contribution to the asymptotic velocity

It is important to note that, not only does the kinetic energy density converge pointwise, but in the limit, the only term contributing is P 2τ. In fact, the following result holds (see [79Jump To The Next Citation Point, Proposition 1.3, p. 983]):

Proposition 1 Consider a solution to Equations (16View Equation)–(17View Equation) and let 𝜃0 ∈ S1. Then, either Pτ(τ,𝜃0) converges to v∞ (𝜃0) or to − v∞ (𝜃0). If Pτ(τ,𝜃0) → − v∞ (𝜃0), then (Q1, P1) = Inv(Q, P ) has the property that P1τ(τ,𝜃0) → v∞ (𝜃0). Furthermore, if v∞ (𝜃0) > 0, then Q1 (τ,𝜃0) converges to 0.

Similar to what we have already seen for false spikes, see Section 9.3, we see that by applying an inversion we can always obtain a non-negative limit for P τ.

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