Abstract
Coherent wave groups are not only characterized by the intrinsic shape of the wave packet, but also by the underlying phase evolution during the propagation. Exact deterministic formulations of hydrodynamic or electromagnetic coherent wave groups can be obtained by solving the nonlinear Schrödinger equation (NLSE). When considering the NLSE, there are two asymptotically equivalent formulations, which can be used to describe the wave dynamics: the time- or space-like NLSE. These differences have been theoretically elaborated upon in the 2016 work of Chabchoub and Grimshaw. In this paper, we address fundamental characteristic differences beyond the shape of wave envelope, which arise in the phase evolution. We use the Peregrine breather as a referenced wave envelope model, whose dynamics is created and tracked in a wave flume using two boundary conditions, namely as defined by the time- and space-like NLSE. It is shown that whichever of the two boundary conditions is used, the corresponding local shape of wave localization is very close and almost identical during the evolution; however, the respective local phase evolution is different. The phase dynamics follows the prediction from the respective NLSE framework adopted in each case.
Highlights
The nonlinear Schrödinger equation (NLSE) is a simple but powerful framework in the description of weakly nonlinear waves
This suggests once again that when using two different boundary conditions to trigger the evolution of the Peregrine breather, the wave envelope may look almost identical in both cases; the characteristic phase-shift dynamics may differ during the propagation
We have experimentally investigated the phase evolution of Peregrine breather hydrodynamics, which have been initiated by two different boundary conditions as defined by the time- and space-like NLSE framework, respectively
Summary
The nonlinear Schrödinger equation (NLSE) is a simple but powerful framework in the description of weakly nonlinear waves. Similar to the Euler equation, the first-derived hydrodynamic NLSE describes the evolution of the wave field in time, implying that initial conditions are required to start the wave motion. This type of NLSE is referred to as space-like NLSE since the spatial wave field is evolved in time. It is shown that despite the water surface elevation being almost identical, there are differences in the phase behavior, especially at the location of maximal breather compression These key features in the phase pattern might be useful for extreme wave detection in random and irregular wave states
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