Abstract
A statistical theory of rogue waves is proposed and tested against experimental data collected in a long water tank where random waves with different degrees of nonlinearity are mechanically generated and free to propagate along the flume. Strong evidence is given that the rogue waves observed in the tank are hydrodynamic instantons, that is, saddle point configurations of the action associated with the stochastic model of the wave system. As shown here, these hydrodynamic instantons are complex spatio-temporal wave field configurations, which can be defined using the mathematical framework of Large Deviation Theory and calculated via tailored numerical methods. These results indicate that the instantons describe equally well rogue waves that originate from a simple linear superposition mechanism (in weakly nonlinear conditions) or from a nonlinear focusing one (in strongly nonlinear conditions), paving the way for the development of a unified explanation to rogue wave formation.
Highlights
A fascinating phenomenon observed in a wide class of nonlinear dispersive systems is the occurrence of rogue waves with abnormally large amplitude; they are found in sea surface gravity waves [1,2], nonlinear fiber optics [3], plasmas [4], and Bose-Einstein condensates
These results indicate that the instantons describe well rogue waves created by simple linear superposition or by nonlinear focusing, paving the way for the development of a unified explanation to rogue wave formation
In the full range of experimental conditions tested, the rogue waves we observe closely resemble hydrodynamic instantons [11,12,13,14,15,16]: these are specific spatio-temporal configurations of the wave field which we define within the framework of large deviation theory (LDT) as the minimizers of an action associated with the random wave model used to describe the system
Summary
A fascinating phenomenon observed in a wide class of nonlinear dispersive systems is the occurrence of rogue waves with abnormally large amplitude; they are found in sea surface gravity waves [1,2], nonlinear fiber optics [3], plasmas [4], and Bose-Einstein condensates. Because the instanton calculus proposed in this paper uses as limiting parameter the maximal wave amplitude itself, without condition on model parameters or regimes in the NLSE, it allows us to assess the validity of the quasideterministic and semiclassical theories by comparing them to the results of our approach in appropriate regimes. We stress that the method proposed here can be generalized to the full two-dimensional setting, as well as other relevant physical systems where an understanding of extreme events is important [27,28] but made challenging by the complexity of the models involved combined with the stochasticity of their evolution and the uncertainty of their parameters [27,29,30,31,32] In this sense our approach adds to other rare events methods [33,34,35,36,37,38,39,40]. VI by discussing the implications of our results in the context of a unified theory of rogue waves
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