Predicting effective quenching of stable pulses in slow-fast excitable media

We develop a linear theory for the prediction of excitation wave quenching -- the construction of minimal perturbations which return stable excitations to quiescence -- for localized pulse solutions in models of excitable media. The theory accounts for an additional equivariance compared to the homogeneous ignition problem, and thus requires a reconsideration of heuristics for choosing optimal reference states from their group representation. We compare predictions made with the linear theory to direct numerical simulations across a family of perturbations and assess their accuracy for several models with distinct stable excitation structures. We find that the theory achieves qualitative predictive power with only the effort of continuing a scalar root, and achieves quantitative predictive power in many circumstances. Finally, we compare the computational cost of our prediction technique to other numerical methods for the determination of transitions in extended excitable systems.