Extreme wave solutions: Parametric studies and wavelet analysis

• A GPU accelerated map related to periodic theta function NLSE solutions is presented.
• Several rogue waves and their associated wave fields are profiled.
• Wavelet analysis shows the dispersion relation in classic and other rogue waves.
• Time–frequency analyses of energy transfer associated with a rogue wave formation are also provided.

Wave fields for near homoclinic, single mode rogue-wave solutions of the periodic nonlinear Schrödinger equation are presented. Parameters of candidate solutions are estimated and refined through an eigenvalue solution procedure. An overview of the estimation and refining procedure used by the authors is provided. Solutions are scaled to facilitate experimental implementation. The continuous wavelet transform is used to carry out time–frequency analyses and the results obtained are demonstrative of the dispersion relation as well as the time varying side band energy transfer associated with the Benjamin–Feir instability. The analysis framework and approach used are validated with the Peregrine solution. Other extreme wave solutions are analyzed as well. The framework presented here could serve as a basis for experimental investigations into single mode rogue waves as well as other localizations in wave fields.