The 3-Wave Resonant Interaction Model: Spectra and Instabilities of Plane Waves

Romano, Marzia (2019) The 3-Wave Resonant Interaction Model: Spectra and Instabilities of Plane Waves. Post-Doctoral thesis, Northumbria University.

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Abstract

The aim of this thesis is the analysis of the spectral stability of plane wave solutions of the 3-wave resonant interaction (3WRI) model, when such solutions undergo localised perturbations.
For the first time, we provide a comprehensive topological classification of the spatial stability spectra with respect to the parameters space and the gain functions associated to any stability spectrum. We find that all the stability spectra of the coupled nonlinear Schrödinger (CNLS) system are enclosed in those of the 3WRI system. The topological features of the CNLS stability spectra are gaps on the real axis (solutions not bounded in space), and branches and loops off the real axis (solutions bounded in space which can be linearly unstable in time). New topological components exist in the stability spectra of the 3WRI model: we name such components twisted loops. They are associated with explosive instability (the corresponding solutions blow up in a finite time) and their gain function is non-zero in a whole neighbourhood of the origin. We observe that the gain function associated to the branches is non-zero at low wave numbers, symmetrically located with respect to the zero-value of the wave number, but it is zero at the origin of the plot (linear instability of baseband-type). The gain function associated to the loops is non-zero only away from the origin (linear instability of passband-type).
We show that the plane wave solutions of the 3WRI model are linearly unstable in time for any choice of the physical parameters, including those ones associated to the solutions that are explosive. Thus, there is linear instability of the plane wave for any choice of the physical parameters corresponding to a positive gain-function.
Finally, we conjecture that the existence of branches in the stability spectra is a necessary condition for the onset of rogue waves ascribable to rational or semi-rational solutions obtained by Darboux Dressing Transformation. Indeed, we observe numerically linear in- stability of plane waves with the subsequent generation of localised structures whose onset, as a result of the perturbation of plane waves, must be investigated further due to the dispersionless nature of the 3WRI system.

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