  Rogue waves, and their periodic counterparts, have been shown to exist in a
number of integrable models. However, relatively little is known about the
existence of these objects in models where an exact formula is unattainable. In
this work, we develop a novel numerical perspective towards identifying such
states as localized solutions in space-time. Importantly, we illustrate that
this methodology in addition to benchmarking known solutions (and confirming
their numerical propagation under controllable error) enables the continuation
of such solutions over parametric variations to non-integrable models. As a
result, we can answer in the positive the question about the parametric
robustness of Peregrine-like waveforms and even of generalizations thereof on a
cnoidal wave background.
