Evaluating the robustness of rogue waves under perturbations

Highlights: • Developed numerical method to obtain the Peregrine rogue wave in the nonlinear Schrödinger equation. • Applied the method to variants of the NLS equation. • Obtained converged rogue wave solutions in nonintegrable models for the first time. • Checked the accuracy of the method by comparing the obtained solutions with those from a numerical time integration method. -- Abstract: 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 this methodology for different perturbations of nonlinear Schrödinger models. In particular, in addition to benchmarking known solutions (and confirming their numerical propagation under controllable error) this 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.