Identify the asymptotic-branch root without continuation

Determine whether a filtering procedure applied to the candidate roots s of the scalar frame-selection function μ_l(s; θ, x_s) = ⟨Φ_l(x − s; s) | X(x − θ; x_s)⟩ in the linear quenching theory for excitable media can uniquely discriminate the root that lies on the asymptotic branch (the branch connecting to the x_s → ∞ solution) without computing the asymptotic solution via continuation.

Background

In the linear quenching framework, selecting the reference frame for comparing a perturbed stable pulse to an unstable pulse is posed as finding roots of μ_l(s; θ, x_s) = ⟨Φ_l(x − s; s) | X(x − θ; x_s)⟩, where Φ_l are heuristic shift selectors and X is the perturbation envelope. For fixed perturbation width x_s and position θ, μ_l may have many roots in s, making the choice of reference frame ambiguous.

To obtain a unique reference frame in practice, the authors continue solutions from the asymptotic regime x_s → ∞, where a unique root is assumed to exist, to finite x_s. While effective, this continuation can be computationally inefficient for single parameter pairs and does not directly address whether one can pick the correct root at a given x_s without computing the asymptotic case.

The open question is whether a root-filtering criterion can be devised that reliably selects the asymptotic-branch root directly from the set of roots of μ_l(s; θ, x_s), eliminating the need for continuation from x_s → ∞.