54 research outputs found
Strong and weak semiclassical limits for some rough Hamiltonians
We present several results concerning the semiclassical limit of the time
dependent Schr\"odinger equation with potentials whose regularity doesn't
guarantee the uniqueness of the underlying classical flow. Different topologies
for the limit are considered and the situation where two bicharateristics can
be obtained out of the same initial point is emphasized
Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves
In this paper, we model Rogue Waves as localized instabilities emerging from homogeneous and stationary background wavefields, under NLS dynamics. This is achieved in two steps: given any background Fourier spectrum P(k), we use the Wigner transform and Penrose’s method to recover spatially periodic unstable modes, which we call unstable Penrose modes. These can be seen as generalized Benjamin–Feir modes, and their parameters are obtained by resolving the Penrose condition, a system of nonlinear equations involving P(k). Moreover, we show how the superposition of unstable Penrose modes can result in the appearance of localized unstable modes. By interpreting the appearance of an unstable mode localized in an area not larger than a reference wavelength λ0 as the emergence of a Rogue Wave, a criterion for the emergence of Rogue Waves is formulated. Our methodology is applied to δ spectra, where the standard Benjamin–Feir instability is recovered, and to more general spectra. In that context, we present a scheme for the numerical resolution of the Penrose condition and estimate the sharpest possible localization of unstable modes. Keywords: Rogue Waves; Wigner equation; Nonlinear Schrodinger equation; Penrose modes; Penrose conditio
A FAST CONVERGENT MODAL-EXPANSION OF THE WAVE POTENTIAL WITH APPLICATION TO THE HYDRODYNAMIC AND HYDROELASTIC ANALYSIS OF FLOATING BODIES IN GENERAL BATHYMETRY
ABSTRACT Numerical examples are presented, showing that useful results can be obtained for the analysis of large floating elastic bodies or structures very efficiently by keeping only a few terms in the expansion. Ideas for extending our approach to 3D are also discussed. A non-linear coupled-mode system of horizontal equations has been derived with the aid of Luke's (1967) variational principle, modelling the evolution of nonlinear water waves in intermediate depth and over a general bathymetr
A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions
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