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Displaying Results 1  25 of 31 on page 1 of 2
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A Generalized Nonlinear Model for the Evolution of Low Frequency Freak Waves
(2011)
Blackledge, Jonathan
A Generalized Nonlinear Model for the Evolution of Low Frequency Freak Waves
(2011)
Blackledge, Jonathan
Abstract:
This paper presents a generalized model for simulating wave fields associated with the sea surface. This includes the case when `freak waves' may occur through an effect compounded in the nonlinear (cubic) Schrodinger equation. After providing brief introductions to linear sea wave models, `freak waves' and the linear and nonlinear Schrodinger equations, we present a unified model that provides for a piecewise continuous transition from a linear to a nonlinear state. This is based on introducing a fractional time derivative to develop a fractional nonlinear partial differential equation with a stochastic source function. In order to explore the characteristics of this equation, we consider a separation of variables approach in order to derive governing equations for the spatial and temporal behaviour. Models for the source function (which, in physical terms, describes the conversion of wind energy into wave energy) are also considered on a separable basis. With regard to t...
https://arrow.dit.ie/engscheleart2/41
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Algebraic Discretization of the CamassaHolm and HunterSaxton Equations
(2008)
Ivanov, Rossen
Algebraic Discretization of the CamassaHolm and HunterSaxton Equations
(2008)
Ivanov, Rossen
Abstract:
The CamassaHolm (CH) and HunterSaxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the H1 and H.1 rightinvariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a rightinvariant metric on the infinitedimensional group of diffeomorphisms preserving the volume element of the domain of fluid flow and to the Euler equations of rigid body whith a fixed point, describing geodesics for a leftinvariant metric on SO(3). The CH and HS equations are integrable bihamiltonian equations and one of their Hamiltonian structures is associated to the Virasoro algebra. The parallel with the integrable SO(3) top is made explicit by a discretization of both equation based on Fourier modes expansion. The obtained equations represent integrable tops with infinitely many momentum components. An emphasis is given on the structure of the phase space of these equations, ...
https://arrow.dit.ie/scschmatart/68
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An explicit mapping between the frequency domain and the time domain representations of nonlinear systems
(2005)
Condon, Marissa; Ivanov, Rossen
An explicit mapping between the frequency domain and the time domain representations of nonlinear systems
(2005)
Condon, Marissa; Ivanov, Rossen
Abstract:
Explicit expressions are presented that describe the inputoutput behaviour of a nonlinear system in both the frequency and the time domain. The expressions are based on a set of coefficients that do not depend on the input to the system and are universal for a given system. The anharmonic oscillator is chosen as an example and is discussed for different choices of its physical parameters. It is shown that the typical approach for the determination of the Volterra Series representation is not valid for the important case when the nonlinear system exhibits oscillatory behaviour and the input has a pole at the origin (in the frequency domain), e.g. the unitstep function. For this case, resonant effects arise and the analysis requires additional care.
https://arrow.dit.ie/scschmatart/71
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Conformal and geometric properties of the CamassaHolm hierarchy
(2007)
Ivanov, Rossen
Conformal and geometric properties of the CamassaHolm hierarchy
(2007)
Ivanov, Rossen
Abstract:
Integrable equations with second order Lax pair like KdV and CamassaHolm (CH) exhibit interesting conformal properties and can be written in terms of the socalled conformal invariants (Schwarz form). These properties for the CH hierarchy are discussed in this ontribution. The squared eigenfunctions of the spectral problem, associated to the CamassaHolm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform (IST) for the CamassaHolm hierarchy as a Generalised Fourier Transform (GFT). Using GFT we describe explicitly some members of the CH hierarchy, including integrable deformations for the CH equation. Also we show that solutions of some 2+1dimensional generalizations of CH can be constructed via the IST for the CH hierarchy.
https://arrow.dit.ie/scschmatart/99
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Conformal Properties and Baecklund Transform for the Associated CamassaHolm Equation
(2005)
Ivanov, Rossen
Conformal Properties and Baecklund Transform for the Associated CamassaHolm Equation
(2005)
Ivanov, Rossen
Abstract:
Integrable equations exhibit interesting conformal properties and can be written in terms of the socalled conformal invariants. The most basic and important example is the KdV equation and the corresponding SchwarzKdV equation. Other examples, including the CamassaHolm equation and the associated CamassaHolm equation are investigated in this paper. It is shown that the B¨acklund transform is related to the conformal properties of these equations. Some particular solutions of the Associated CamassaHolm Equation are discussed also.
https://arrow.dit.ie/scschmatart/73
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Cryptography Using Steganography: New Algorithms and Applications
(2011)
Blackledge, Jonathan
Cryptography Using Steganography: New Algorithms and Applications
(2011)
Blackledge, Jonathan
Abstract:
Developing methods for ensuring the secure exchange of information is one of the oldest occupations in history. With the revolution in Information Technology, the need for securing information and the variety of methods that have been developed to do it has expanded rapidly. Much of the technology that forms the basis for many of the techniques used today was originally conceived for use in military communications and has since found a place in a wide range of industrial and commercial sectors. This has led to the development of certain industry standards that are compounded in specific data processing algorithms together with the protocols and procedures that are adopted in order to implement them. These standards are of course continually scrutinized for their effectiveness and undergo improvements and/or changes as required.
https://arrow.dit.ie/engscheleart2/40
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Currency Trading using the Fractal Market Hypothesis
(2011)
Blackledge, Jonathan; Murphy, Kieran
Currency Trading using the Fractal Market Hypothesis
(2011)
Blackledge, Jonathan; Murphy, Kieran
Abstract:
We report on a research and development programme in financial modelling and economic security undertaken in the Information and Communications Security Research Group (ICSRG, 2011) which has led to the launch of a new company  Currency Traders Ireland Limited  funded by Enterprise Ireland. Currency Traders Ireland Limited (CTI, 2011) has a fifty year exclusive license to develop a new set of indicators for analysing currency exchange rates (Forex trading). We consider the background to the approach taken and present examples of the results obtained to date.
https://arrow.dit.ie/engscheleart2/39
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Electromagnetic Scattering Solutions for Digital Signal Processing
(2009)
Blackledge, Jonathan
Electromagnetic Scattering Solutions for Digital Signal Processing
(2009)
Blackledge, Jonathan
Abstract:
Electromagnetic scattering theory is fundamental to understanding the interaction between electromagnetic waves and inhomogeneous dielectric materials. The theory unpins the engineering of electromagnetic imaging systems over a broad range of frequencies, from optics to radio and microwave imaging, for example. Developing accurate scattering models is particularly important in the field of image understanding and the interpretation of electromagnetic signals generated by scattering events. To this end there are a number of approaches that can be taken. For relatively simple geometric configurations, approximation methods are used to develop a transformation from the object plane (where scattering events take place) to the image plane (where a record of some measure of the scattered field is taken). The most common approximation is the weak scattering approximation which ignores the effect of multiple scattering interactions and the first part of this thesis investigates the use of t...
https://arrow.dit.ie/engschelecon/3
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Equations of the CamassaHolm Hierarchy
(2009)
Ivanov, Rossen
Equations of the CamassaHolm Hierarchy
(2009)
Ivanov, Rossen
Abstract:
The squared eigenfunctions of the spectral problem associated with the CamassaHolm (CH) equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the CH hierarchy as a generalized Fourier transform (GFT). All the fundamental properties of the CH equation, such as the integrals of motion, the description of the equations of the whole hierarchy, and their Hamiltonian structures, can be naturally expressed using the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions. Using the GFT, we explicitly describe some members of the CH hierarchy, including integrable deformations for the CH equation. We also show that solutions of some (1 + 2)  dimensional members of the CH hierarchy can be constructed using results for the inverse scattering transform for the CH equation. We give an example of the peakon solution of one such equation.
https://arrow.dit.ie/scschmatart/63
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EulerPoincar´e equations for GStrands
(2014)
Holm, Darryl; Ivanov, Rossen
EulerPoincar´e equations for GStrands
(2014)
Holm, Darryl; Ivanov, Rossen
Abstract:
The Gstrand equations for a map R×R into a Lie group G are associated to a Ginvariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The Gstrand itself is the map g(t,s):R×R→G, where t and s are the independent variables of the Gstrand equations. The EulerPoincar'e reduction of the variational principle leads to a formulation where the dependent variables of the Gstrand equations take values in the corresponding Lie algebra g and its coalgebra, g∗ with respect to the pairing provided by the variational derivatives of the Lagrangian. We review examples of different Gstrand constructions, including matrix Lie groups and diffeomorphism group. In some cases the Gstrand equations are completely integrable 1+1 Hamiltonian systems that admit soliton solutions.
https://arrow.dit.ie/scschmatcon/16
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Examples of GStrand Equations
(2014)
Holm, Darryl; Ivanov, Rossen
Examples of GStrand Equations
(2014)
Holm, Darryl; Ivanov, Rossen
Abstract:
The Gstrand equations for a map R×R into a Lie group G are associated to a Ginvariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The Gstrand itself is the map g(t,s):R×R→G, where t and s are the independent variables of the Gstrand equations. The EulerPoincare´ reduction of the variational principle leads to a formulation where the dependent variables of the Gstrand equations take values in the corresponding Lie algebra g and its coalgebra, g∗ with respect to the pairing provided by the variational derivatives of the Lagrangian. We review examples of Gstrand constructions, including matrix Lie groups of low ranks, and the Diffeomorphism group. In some cases the arising Gstrand equations are completely integrable 1+1 Hamiltonian systems that admit soliton solutions.Our presentation is aimed to illustrate the Gstrand construction with several simple but instructive examples: (i) SO(3)strand integrable equations for Lax operators, ...
https://arrow.dit.ie/scschmatcon/17
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GStrands and Peakon Collisions on Diff(R)
(2013)
Holm, Darryl; Ivanov, Rossen
GStrands and Peakon Collisions on Diff(R)
(2013)
Holm, Darryl; Ivanov, Rossen
Abstract:
A Gstrand is a map g : R x R > G for a Lie group G that follows from Hamilton's principle for a certain class of Ginvariant Lagrangians. Some Gstrands on finitedimensional groups satisfy 1+1 spacetime evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that Gstrands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of Gstrands when G = Diff(R) is the group of diffeomorphisms of the real line R, for which the group product is composition of smooth invertible functions. In the case of peakonantipeakon collisions, the solution reduces to solving either Laplace's equation or the wave equation (depending on a sign in the Lagrangian) and is written in terms of their solu...
https://arrow.dit.ie/scschmatart/136
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Generalised Fourier Transform and Perturbations to Soliton Equations
(2009)
Grahovski, Georgi; Ivanov, Rossen
Generalised Fourier Transform and Perturbations to Soliton Equations
(2009)
Grahovski, Georgi; Ivanov, Rossen
Abstract:
A brief survey of the theory of soliton perturbations is presented. The focus is on the usefulness of the socalled Generalised Fourier Transform (GFT). This is a method that involves expansions over the complete basis of “squared solutions” of the spectral problem, associated to the soliton equation. The Inverse Scattering Transform for the corresponding hierarchy of soliton equations can be viewed as a GFT where the expansions of the solutions have generalised Fourier coefficients given by the scattering data. The GFT provides a natural setting for the analysis of small perturbations to an integrable equation: starting from a purely soliton solution one can ’modify’ the soliton parameters such as to incorporate the changes caused by the perturbation. As illustrative examples the perturbed equations of the KdV hierarchy, in particular the Ostrovsky equation, followed by the perturbation theory for the Camassa Holm hierarchy are presented.
https://arrow.dit.ie/scschmatart/64
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Generalised Fourier transform for the CamassaHolm hierarchy
(2007)
Constantin, Adrian; Gerdjikov, Vladimir; Ivanov, Rossen
Generalised Fourier transform for the CamassaHolm hierarchy
(2007)
Constantin, Adrian; Gerdjikov, Vladimir; Ivanov, Rossen
Abstract:
The squared eigenfunctions of the spectral problem associated to the CamassaHolm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform for the CamassaHolm hierarchy as a Generalised Fourier transform. The main result of this work is the derivation of the completeness relation for the squared solutions of the CamassaHolm spectral problem. We show that all the fundamental properties of the CamassaHolm equation such as the integrals of motion, the description of the equations of the whole hierarchy and their Hamiltonian structures can be naturally expressed making use of the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions.
https://arrow.dit.ie/scschmatart/97
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Hamiltonian formulation and integrability of a complex symmetric nonlinear system
(2006)
Ivanov, Rossen
Hamiltonian formulation and integrability of a complex symmetric nonlinear system
(2006)
Ivanov, Rossen
Abstract:
The integrability of a complex generalisation of the ’elegant’ system, proposed by D. Fairlie and its relation to the Nahm equation and the Manakov top is discussed.
https://arrow.dit.ie/scschmatart/67
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Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation
(2007)
Choudhury, S. Roy; Ivanov, Rossen; Liu, Yue
Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation
(2007)
Choudhury, S. Roy; Ivanov, Rossen; Liu, Yue
Abstract:
The Ostrovsky equation is a model for gravity waves propagating down a channel under the influence of Coriolis force. This equation is a modification of the famous Kortewegde Vries equation and is also Hamiltonian. However the Ostrovsky equation is not integrable and in this contribution we prove its nonintegrability. We also study local bifurcations of its solitary waves.
https://arrow.dit.ie/scschmatart/98
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Inverse Scattering Transform for the CamassaHolm equation
(2006)
Constantin, Adrian; Gerdjikov, Vladimir; Ivanov, Rossen
Inverse Scattering Transform for the CamassaHolm equation
(2006)
Constantin, Adrian; Gerdjikov, Vladimir; Ivanov, Rossen
Abstract:
An Inverse Scattering Method is developed for the CamassaHolm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data. The main difference with respect to the standard Inverse Scattering Transform lies in the fact that we have a weighted spectral problem. We therefore have to develop different asymptotic expansions.
https://arrow.dit.ie/scschmatart/91
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Issues in the design of switched linear control systems: A benchmark study
(2003)
Leith, Douglas J.; Shorten, Robert N.; Leithead, William; Mason, Oliver; Curran, Paul
Issues in the design of switched linear control systems: A benchmark study
(2003)
Leith, Douglas J.; Shorten, Robert N.; Leithead, William; Mason, Oliver; Curran, Paul
Abstract:
In this paper we present a tutorial overview of some of the issues that arise in the design of switched linear control systems. Particular emphasis is given to issues relating to stability and control system realisation. A benchmark regulation problem is then presented. This problem is most naturally solved by means of a switched control design. The challenge to the community is to design a control system that meets the required performance specifications and permits the application of rigorous analysis techniques. A simple design solution is presented and the limitations of currently available analysis techniques are illustrated with reference to this example.
http://mural.maynoothuniversity.ie/1864/
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Multi–component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory
(2010)
Gerdjikov, Vladimir; Grahovski, Georgi
Multi–component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory
(2010)
Gerdjikov, Vladimir; Grahovski, Georgi
Abstract:
The algebraic structure and the spectral properties of a special class of multicomponent NLS equations, related to the symmetric spaces of BD.Itype are analyzed. The focus of the study is on the spectral theory of the associated Lax operator to these nonlinear evolutionary equations for different fundamental representations of the underlying simple Lie algebra g. Special attention is paid to the spinor representation of the orthogonal Lie algebras of B type.
https://arrow.dit.ie/scschmatart/54
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Nonlinear Behaviour of Sea Surface Waves Based on LowGradient PhaseOnly Scattering Effects
(2011)
Blackledge, Jonathan; Coyle, Eugene; Kearney, Derek
Nonlinear Behaviour of Sea Surface Waves Based on LowGradient PhaseOnly Scattering Effects
(2011)
Blackledge, Jonathan; Coyle, Eugene; Kearney, Derek
Abstract:
Nonlinear sea waves generated by the wind, including freak waves, are considered to be phenomena that can be modelled using the nonlinear (cubic) Schrodinger equation, for example. However, there is a problem with this approach which is that sea surface waves, driven by wind speeds of varying strength, must be considered to be composed of two distinct types, namely, linear waves and nonlinear waves. In this paper, we consider a different approach to modelling ‘nonlinear’ waves that is based on a solution to the linear wave equation under a lowgradient, phaseonly condition. This approach is entirely compatible with the fluid equations of motion (the NavierStokes equations) and is thereby not based on a phenomenological model such as the nonlinear Schr¨odinger equation.
https://arrow.dit.ie/engscheleart/149
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On Soliton Interactions for a Hierarchy of Generalized Heisenberg Ferromagnetic Models on SU(3)/S(U(1) $\times$ U(2)) Symmetric Space
(2012)
Gerdjikov, Vladimir; Grahovski, Georgi; Mikhailov, Alexander; VAlchev, Tihomir
On Soliton Interactions for a Hierarchy of Generalized Heisenberg Ferromagnetic Models on SU(3)/S(U(1) $\times$ U(2)) Symmetric Space
(2012)
Gerdjikov, Vladimir; Grahovski, Georgi; Mikhailov, Alexander; VAlchev, Tihomir
Abstract:
We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator L. The Lax representation is Z2 \times Z2 reduced and is naturally associated with the symmetric space SU(3)/S(U(1) \times U(2)). The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the Nsoliton solutions for an arbitrary member of the hierarchy by using the ZakharovShabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The onesoliton solutions of NLEEs with even and odd dispersion laws have different properties. In particular, the onesoliton solutions for NLEEs with even dispersion laws are not traveling waves; their velocities and their amplitudes are time dependent. Calculating the asymptotics of the Nsoliton solutions for t \rightarrow \pm \infty we analyze the interactions of quadruplet solitons.
https://arrow.dit.ie/scschmatart/129
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On the (Non)Integrability of the Perturbed KdV Hierarchy with Generic Selfconsistent Sources
(2011)
Gerdjikov, Vladimir; Grahovski, Georgi; Ivanov, Rossen
On the (Non)Integrability of the Perturbed KdV Hierarchy with Generic Selfconsistent Sources
(2011)
Gerdjikov, Vladimir; Grahovski, Georgi; Ivanov, Rossen
https://arrow.dit.ie/scschmatcon/8
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On the Nwave Equations with PTsymmetry
(2016)
Gerdjikov, Vladimir; Grahovski, Georgi; Ivanov, Rossen
On the Nwave Equations with PTsymmetry
(2016)
Gerdjikov, Vladimir; Grahovski, Georgi; Ivanov, Rossen
Abstract:
We study extensions of Nwave systems with PTsymmetry. The types of (nonlocal) reductions leading to integrable equations invariant with respect to P (spatial reflection) and T (time reversal) symmetries is described. The corresponding constraints on the fundamental analytic solutions and the scattering data are derived. Based on examples of 3wave (related to the algebra sl(3,C)) and 4wave (related to the algebra so(5,C)) systems, the properties of different types of 1 and 2soliton solutions are discussed. It is shown that the PT symmetric 3wave equations may have regular multisoliton solutions for some specific choices of their parameters.
https://arrow.dit.ie/scschmatart/219
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On the Peakon and Soliton Solutions of an Integrable PDE with Cubic Nonlinearities
(2012)
Ivanov, Rossen; Lyons, Tony
On the Peakon and Soliton Solutions of an Integrable PDE with Cubic Nonlinearities
(2012)
Ivanov, Rossen; Lyons, Tony
Abstract:
The interest in the singular solutions (peakons) has been inspired by the CamassaHolm (CH) equation and its peakons. An integrable peakon equation with cubic nonlinearities was first discovered by Qiao. Another integrable equation with cubic nonlinearities was introduced by V. Novikov . We investigate the peakon and soliton solutions of the Qiao equation.
https://arrow.dit.ie/scschmatcon/15
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Rational Bundles and Recursion Operators for Integrable Equations on A.IIItype Symmetric Spaces
(2011)
Gerdjikov, Vladimir; Grahovski, Georgi; Mikhailov, Alexander; Valtchev, Tihomir
Rational Bundles and Recursion Operators for Integrable Equations on A.IIItype Symmetric Spaces
(2011)
Gerdjikov, Vladimir; Grahovski, Georgi; Mikhailov, Alexander; Valtchev, Tihomir
Abstract:
We analyze and compare the methods of construction of the recursion operators for a special class of integrable nonlinear differential equations related to A.IIItype symmetric spaces in Cartan’s classification and having additional reductions.
https://arrow.dit.ie/scschmatart/115
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