  By using holographic methods, the radii of convergence of the hydrodynamic
shear and sound dispersion relations were previously computed in the ${\cal N}
= 4$ supersymmetric Yang-Mills theory at infinite 't Hooft coupling and
infinite number of colours. Here, we extend this analysis to the domain of
large but finite 't Hooft coupling. To leading order in the perturbative
expansion, we find that the radii grow with increasing inverse coupling,
contrary to naive expectations. However, when the equations of motion are
solved using a qualitative non-perturbative resummation, the dependence on the
coupling becomes piecewise continuous and the initial growth is followed by a
decrease. The piecewise nature of the dependence is related to the dynamics of
branch point singularities of the energy-momentum tensor finite-temperature
two-point functions in the complex plane of spatial momentum squared. We repeat
the study using the Einstein-Gauss-Bonnet gravity as a model where the
equations can be solved fully non-perturbatively, and find the expected
decrease of the radii of convergence with the effective inverse coupling which
is also piecewise continuous. Finally, we provide arguments in favour of the
non-perturbative approach and show that the presence of non-perturbative modes
in the quasinormal spectrum can be indirectly inferred from the analysis of
perturbative critical points.
