Hilbert Transforms

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 λ 0 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.B A. G. Athanassoulis

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“…by the standard results on the Hilbert transform (King 2009). In fact, since F ζ,σ (k) has zero mean, the Hilbert transform p.v.…”

Section: Proof Of the Claimmentioning

confidence: 93%

“…Thus, we recover a kind of Sokhotsky-Plemelj formula (King 2009, Section 3.7) for our problem, namely…”

Section: On the Solvability Of The Penrose Conditionmentioning

confidence: 99%

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

Athanassoulis

Sapsis

2017

J. Ocean Eng. Mar. Energy

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“…of (3.12), (3.13), and use the following relationship between the Fourier and the Hilbert transform [8]:…”

Section: Reduction Of the Problem To One Weakly Singular Integral Equmentioning

confidence: 99%

A Rigid Stamp Indentation into a Semiplane with a Curvature-Dependent Surface Tension on the Boundary

Walton¹,

Zemlyanova²

2016

SIAM J. Appl. Math.

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Abstract. It has been shown that taking into account surface mechanics is extremely important for accurate modeling of many physical phenomena such as those arising in nanoscience, fracture propagation, and contact mechanics. This paper is dedicated to a contact problem of a rigid stamp indentation into an elastic isotropic semiplane with curvature-dependent surface tension acting on the boundary of the semiplane. Cases of both frictionless and adhesive contact of the stamp with the boundary of the semiplane are considered. Using the method of integral transforms, each problem is reduced to a system of singular integro-differential equations, which is further reduced to one or two weakly singular integral equations. It has been shown that the introduction of the curvaturedependent surface tension eliminates the classical singularities of the order 1/2 of the stresses and strains at the end-points of the contact interval. The numerical solution of the problem is obtained by approximation of unknown functions with Taylor polynomials.

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“…A more convenient representation of the WH factors can be found using the Hilbert transform and the Plemelj-Sokhotsky relations [37]. The Hilbert transform of a function f (ξ) is defined as…”

Section: Spitzer Identity and Wiener-hopf Factorizationmentioning

confidence: 99%

Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options

Fusai

Germano

Marazzina

2016

European Journal of Operational Research

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The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Lévy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Lévy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and * Corresponding author Email addresses: gianluca.fusai@unipmn.it, gianluca.fusai.1@city.ac.uk (Gianluca Fusai), g.germano@ucl.ac.uk, g.germano@lse.ac.uk (Guido Germano), daniele.marazzina@polimi.it (Daniele Marazzina) Preprint submitted to ElsevierNovember 27, 2015the error decays exponentially with the number of grid points.

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Hilbert Transforms

Cited by 204 publications

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Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

A Rigid Stamp Indentation into a Semiplane with a Curvature-Dependent Surface Tension on the Boundary

Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options

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