arXiv subjects

Agissilaos G. Athanassoulis

Publications and source records attributed to Agissilaos G. Athanassoulis.

5 recordsLinked to original sources

arXiv2023

Modulation instability and convergence of the random phase approximation for stochastic sea states

The nonlinear Schrödinger equation is widely used as an approximate model for the evolution in time of the water wave envelope. In the context of simulating ocean waves, initial conditions are typically generated from a measured power spectrum using the random phase approximation, and periodized on an interval of length $L$. It is known that most realistic ocean waves power spectra do not exhibit modulation instability, but the most severe ones do; it is thus a natural question to ask whether the periodized random phase approximation has the correct stability properties. In this work we specify a random phase approximation scaling so that, in the limit of $L\to\infty,$ the stability properties of the periodized problem are identical to those of the continuous power spectrum on the infinite line. Moreover, it is seen through concrete examples that using a too short computational domain can completely suppress the modulation instability.

math.AP↗

arXiv2020

Strong solutions for the Alber equation and stability of unidirectional wave spectra

The Alber equation is a moment equation for the nonlinear Schrödinger equation, formally used in ocean engineering to investigate the stability of stationary and homogeneous sea states in terms of their power spectra. In this work we present the first well-posedness theory for the Alber equation with the help of an appropriate equivalent reformulation. Moreover, we show linear Landau damping in the sense that, under a stability condition on the homogeneous background, any inhomogeneities disperse and decay in time. The proof exploits novel $L^2$ space-time estimates to control the inhomogeneity and our result applies to any regular initial data (without a mean-zero restriction). Finally, the sufficient condition for stability is resolved, and the physical implications for ocean waves are discussed. Using a standard reference dataset (the "North Atlantic Scatter Diagram") it is found that the vast majority of sea states are stable, but modulationally unstable sea states do appear, with likelihood $O(1/1000);$ these would be the prime breeding ground for rogue waves.

math-ph↗

arXiv2008

Coarse-scale representations and smoothed Wigner transforms

Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and / or semiclassical limits of wave equations. We derive explicit, closed formulations for the coarse-scale representation of the action of pseudodifferential operators. The resulting ``smoothed operators'' are in general of infinite order. The formulation of an appropriate framework, resembling the Gelfand-Shilov spaces, is necessary. Similarly we treat the ``smoothed Wigner calculus''. In particular this allows us to reformulate any linear equation, as well as certain nonlinear ones (e.g. Hartree and cubic non-linear Schrödinger), as coarse-scale phase-space equations (e.g. smoothed Vlasov), with spatial and spectral resolutions controlled by two free parameters. Finally, it is seen that the smoothed Wigner calculus can be approximated, uniformly on phase-space, by differential operators in the semiclassical regime. This improves the respective weak-topology approximation result for the Wigner calculus.

math.AP↗

arXiv2007

Smoothed Wigner transforms in the numerical simulation of semiclassical (high-frequency) wave propagation

The numerical simulation of wave propagation in semiclassical (high-frequency) problems is well known to pose a formidable challenge. In this work, a new phase-space approach for the numerical simulation of semiclassical wave propagation, making use of the smoothed Wigner Transform (SWT), is proposed. There are numerous works which use the Wigner Transform (WT) in the study of a variety of wave propagation problems including high-frequency limits for linear, nonlinear and/or random waves. The WT however is well known to present significant difficulties in the formulation of numerical schemes. Working with concrete examples for the semiclassical linear Schrodinger equation it is seen that the SWT approach is indeed significantly faster (in a well-defined sense) to work with than the WT and than full numerical solutions of the original equation in the semiclassical regime. Comparisons with exact and numerical solutions are used to keep track of numerical errors.

math.AP↗

arXiv2007

Exact equations for smoothed Wigner transforms and homogenization of wave propagation

The Wigner Transform (WT) has been extensively used in the formulation of phase-space models for a variety of wave propagation problems including high-frequency limits, nonlinear and random waves. It is well known that the WT features counterintuitive 'interference terms', which often make computation impractical. In this connection, we propose the use of the smoothed Wigner Transform (SWT), and derive new, exact equations for it, covering a broad class of wave propagation problems. Equations for spectrograms are included as a special case. The 'taming' of the interference terms by the SWT is illustrated, and an asymptotic model for the Schroedinger equation is constructed and numerically verified.

math.AP↗