Existence of multi-pulses of the regularized short-pulse and Ostrovsky equations

The existence of multi-pulse solutions near orbit-flip bifurcations of a primary single-humped pulse is shown in reversible, conservative, singularly perturbed vector fields. Similar to the non-singular case, the sign of a geometric condition that involves the first integral decides whether multi-pulses exist or not. The proof utilizes a combination of geometric singular perturbation theory and Lyapunov--Schmidt reduction through Lin's method. The motivation for considering orbit flips in singularly perturbed systems comes from the regularized short-pulse equation and the Ostrovsky equation, which both fit into this framework and are shown here to support multi-pulses.