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Full-Text Articles in Physical Sciences and Mathematics
Fourth Order Dispersion In Nonlinear Media, Georgios Tsolias
Fourth Order Dispersion In Nonlinear Media, Georgios Tsolias
Doctoral Dissertations
In recent years, there has been an explosion of interest in media bearing quartic
dispersion. After the experimental realization of so-called pure-quartic solitons, a
significant number of studies followed both for bright and for dark solitonic struc-
tures exploring the properties of not only quartic, but also setic, octic, decic etc.
dispersion, but also examining the competition between, e.g., quadratic and quartic
dispersion among others.
In the first chapter of this Thesis, we consider the interaction of solitary waves in
a model involving the well-known φ4 Klein-Gordon theory, bearing both Laplacian and biharmonic terms with different prefactors. As a …
Coherent Control Of Dispersive Waves, Jimmie Adriazola
Coherent Control Of Dispersive Waves, Jimmie Adriazola
Dissertations
This dissertation addresses some of the various issues which can arise when posing and solving optimization problems constrained by dispersive physics. Considered here are four technologically relevant experiments, each having their own unique challenges and physical settings including ultra-cold quantum fluids trapped by an external field, paraxial light propagation through a gradient index of refraction, light propagation in periodic photonic crystals, and surface gravity water waves over shallow and variable seabeds. In each of these settings, the physics can be modeled by dispersive wave equations, and the technological objective is to design the external trapping fields or propagation media such …
On The Intermediate Long Wave Propagation For Internal Waves In The Presence Of Currents, Joseph Cullen, Rossen Ivanov
On The Intermediate Long Wave Propagation For Internal Waves In The Presence Of Currents, Joseph Cullen, Rossen Ivanov
Articles
A model for the wave motion of an internal wave in the presence of current in the case of intermediate long wave approximation is studied. The lower layer is considerably deeper, with a higher density than the upper layer. The flat surface approximation is assumed. The fluids are incompressible and inviscid. The model equations are obtained from the Hamiltonian formulation of the dynamics in the presence of a depth-varying current. It is shown that an appropriate scaling leads to the integrable Intermediate Long Wave Equation (ILWE). Two limits of the ILWE leading to the integrable Benjamin-Ono and KdV equations are …
Camassa-Holm Cuspons, Solitons And Their Interactions Via The Dressing Method, Rossen Ivanov, Tony Lyons, Nigel Orr
Camassa-Holm Cuspons, Solitons And Their Interactions Via The Dressing Method, Rossen Ivanov, Tony Lyons, Nigel Orr
Articles
A dressing method is applied to a matrix Lax pair for the Camassa–Holm equation, thereby allowing for the construction of several global solutions of the system. In particular, solutions of system of soliton and cuspon type are constructed explicitly. The interactions between soliton and cuspon solutions of the system are investigated. The geometric aspects of the Camassa–Holm equation are re-examined in terms of quantities which can be explicitly constructed via the inverse scattering method.
Asymptotic And Numerical Analysis Of Coherent Structures In Nonlinear Schrodinger-Type Equations, Cory Ward
Asymptotic And Numerical Analysis Of Coherent Structures In Nonlinear Schrodinger-Type Equations, Cory Ward
Doctoral Dissertations
This dissertation concerns itself with coherent structures found in nonlinear Schrödinger-type equations and can be roughly split into three parts. In the first part we study a deformation of the defocusing nonlinear Schrödinger (NLS) equation, the defocusing Camassa-Holm NLS (CH-NLS) equation in both one and two space dimensions. We use asymptotic multiscale expansion methods to reduce this model to a Boussinesq-like equation, which is then subsequently used to obtain approximate solitary wave solutions for both the 1D and 2D CH-NLS equations. We then use direct numerical simulations to investigate the validity of these approximate solutions, their evolution, and their head-on …
Riemann-Hilbert Problem, Integrability And Reductions, Vladimir Gerdjikov, Rossen Ivanov, Aleksander Stefanov
Riemann-Hilbert Problem, Integrability And Reductions, Vladimir Gerdjikov, Rossen Ivanov, Aleksander Stefanov
Articles
Abstract. The present paper is dedicated to integrable models with Mikhailov reduction groups GR ≃ Dh. Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the GR-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with Dh symmetries are presented.
Surface Waves Over Currents And Uneven Bottom, Alan Compelli, Rossen Ivanov, Calin I. Martin, Michail D. Todorov
Surface Waves Over Currents And Uneven Bottom, Alan Compelli, Rossen Ivanov, Calin I. Martin, Michail D. Todorov
Articles
The propagation of surface water waves interacting with a current and an uneven bottom is studied. Such a situation is typical for ocean waves where the winds generate currents in the top layer of the ocean. The role of the bottom topography is taken into account since it also influences the local wave and current patterns. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types. The arising KdV equation with variable coefficients, dependent on the bottom topography, is studied numerically when the initial condition is in the form of the one soliton solution …
Equatorial Wave–Current Interactions, Adrian Constantin, Rossen Ivanov
Equatorial Wave–Current Interactions, Adrian Constantin, Rossen Ivanov
Articles
We study the nonlinear equations of motion for equatorial wave–current interactions in the physically realistic setting of azimuthal two-dimensional inviscid flows with piecewise constant vorticity in a two-layer fluid with a flat bed and a free surface. We derive a Hamiltonian formulation for the nonlinear governing equations that is adequate for structure-preserving perturbations, at the linear and at the nonlinear level. Linear theory reveals some important features of the dynamics, highlighting differences between the short- and long-wave regimes. The fact that ocean energy is concentrated in the long-wave propagation modes motivates the pursuit of in-depth nonlinear analysis in the long-wave …
Global Well-Posedness For The Derivative Nonlinear Schrödinger Equation Through Inverse Scattering, Jiaqi Liu
Global Well-Posedness For The Derivative Nonlinear Schrödinger Equation Through Inverse Scattering, Jiaqi Liu
Theses and Dissertations--Mathematics
We study the Cauchy problem of the derivative nonlinear Schrodinger equation in one space dimension. Using the method of inverse scattering, we prove global well-posedness of the derivative nonlinear Schrodinger equation for initial conditions in a dense and open subset of weighted Sobolev space that can support bright solitons.
Dark Bright Solitons In Coupled Nonlinear Schrodinger Equations With Unequal Dispersion Coefficients, E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskaki, B. A. Malomed
Dark Bright Solitons In Coupled Nonlinear Schrodinger Equations With Unequal Dispersion Coefficients, E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskaki, B. A. Malomed
Efstathios Charalampidis
We study a two component nonlinear Schrodinger system with equal, repulsive cubic interactions and different dispersion coefficients in the two components. We consider states that have a dark solitary wave in one component. Treating it as a frozen one, we explore the possibility of the formation of bright solitonic structures in the other component. We identify bifurcation points at which such states emerge in the bright component in the linear limit and explore their continuation into the nonlinear regime. An additional analytically tractable limit is found to be that of vanishing dispersion of the bright component. We numerically identify regimes …
Vector Rogue Waves And Dark Bright Boomeronic Solitons In Autonomous And Non Autonomous Settings, R. Babu Mareeswaran, E. G. Charalampidis, T. Kanna, P. G. Kevrekidis, D. J. Frantzeskakis
Vector Rogue Waves And Dark Bright Boomeronic Solitons In Autonomous And Non Autonomous Settings, R. Babu Mareeswaran, E. G. Charalampidis, T. Kanna, P. G. Kevrekidis, D. J. Frantzeskakis
Efstathios Charalampidis
In this work, we consider the dynamics of vector rogue waves and ark bright solitons in two component nonlinear Schrodinger equations with various physically motivated time dependent non linearity coefficients, as well as spatio temporally dependent potentials. A similarity transformation is utilized to convert the system into the integrable Manakov system and subsequently the vector rogue and dark bright boomeron like soliton solutions of the latter are converted back into ones of the original non autonomous model. Using direct numerical simulations we find that, in most cases, the rogue waves formation is rapidly followed by a modulational instability that leads …
On The N-Wave Equations With Pt-Symmetry, Vladimir Gerdjikov, Georgi Grahovski, Rossen Ivanov
On The N-Wave Equations With Pt-Symmetry, Vladimir Gerdjikov, Georgi Grahovski, Rossen Ivanov
Articles
We study extensions of N-wave systems with PT-symmetry. 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 3-wave (related to the algebra sl(3,C)) and 4-wave (related to the algebra so(5,C)) systems, the properties of different types of 1- and 2-soliton solutions are discussed. It is shown that the PT symmetric 3-wave equations may have regular multi-soliton solutions for some specific choices of their parameters.
Exciting And Harvesting Vibrational States In Harmonically Driven Granular Chains, Efstathios G. Charalampidis, Christopher Chong, Eunho Kim, Heetae Kim, F. Li, Panayotis G. Kevrekidis, J. Lydon, Chiara Daraio, Jianke Yang
Exciting And Harvesting Vibrational States In Harmonically Driven Granular Chains, Efstathios G. Charalampidis, Christopher Chong, Eunho Kim, Heetae Kim, F. Li, Panayotis G. Kevrekidis, J. Lydon, Chiara Daraio, Jianke Yang
Mathematics and Statistics Department Faculty Publication Series
This article explores the excitation of different vibrational states in a spatially extended dynamical system through theory and experiment. As a prototypical example, we consider a one-dimensional packing of spherical particles (a so-called granular chain) that is subject to harmonic boundary excitation. The combination of the multi-modal nature of the system and the strong coupling between the particles due to the nonlinear Hertzian contact force leads to broad regions in frequency where different vibrational states are possible. In certain parametric regions, we demonstrate that the Nonlinear Schr¨odinger (NLS) equation predicts the corresponding modes fairly well. We propose that nonlinear multi-modal …
Dark Bright Solitons In Coupled Nonlinear Schrodinger Equations With Unequal Dispersion Coefficients, E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskaki, B. A. Malomed
Dark Bright Solitons In Coupled Nonlinear Schrodinger Equations With Unequal Dispersion Coefficients, E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskaki, B. A. Malomed
Mathematics and Statistics Department Faculty Publication Series
We study a two component nonlinear Schrodinger system with equal, repulsive cubic interactions and different dispersion coefficients in the two components. We consider states that have a dark solitary wave in one component. Treating it as a frozen one, we explore the possibility of the formation of bright solitonic structures in the other component. We identify bifurcation points at which such states emerge in the bright component in the linear limit and explore their continuation into the nonlinear regime. An additional analytically tractable limit is found to be that of vanishing dispersion of the bright component. We numerically identify regimes …
Pulses And Snakes In Ginzburg-Landau Equation, S.C. Mancas, Roy S. Choudhury
Pulses And Snakes In Ginzburg-Landau Equation, S.C. Mancas, Roy S. Choudhury
Publications
Using a variational formulation for partial differential equations combined with numerical simulations on ordinary differential equations (ODEs), we find two categories (pulses and snakes) of dissipative solitons, and analyze the dependence of both their shape and stability on the physical parameters of the cubic-quintic Ginzburg–Landau equation (CGLE). In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. Numerical simulations reveal very interesting bifurcations sequences as the parameters of the CGLE …
Vector Rogue Waves And Dark Bright Boomeronic Solitons In Autonomous And Non Autonomous Settings, R. Babu Mareeswaran, E. G. Charalampidis, T. Kanna, P. G. Kevrekidis, D. J. Frantzeskakis
Vector Rogue Waves And Dark Bright Boomeronic Solitons In Autonomous And Non Autonomous Settings, R. Babu Mareeswaran, E. G. Charalampidis, T. Kanna, P. G. Kevrekidis, D. J. Frantzeskakis
Mathematics and Statistics Department Faculty Publication Series
In this work, we consider the dynamics of vector rogue waves and ark bright solitons in two component nonlinear Schrodinger equations with various physically motivated time dependent non linearity coefficients, as well as spatio temporally dependent potentials. A similarity transformation is utilized to convert the system into the integrable Manakov system and subsequently the vector rogue and dark bright boomeron like soliton solutions of the latter are converted back into ones of the original non autonomous model. Using direct numerical simulations we find that, in most cases, the rogue waves formation is rapidly followed by a modulational instability that leads …
Lattice Three Dimensional Skyrmions Revisited, E. G. Charalampidis, T. A. I, P. G. Kevrekidis
Lattice Three Dimensional Skyrmions Revisited, E. G. Charalampidis, T. A. I, P. G. Kevrekidis
Mathematics and Statistics Department Faculty Publication Series
In the continuum a skyrmion is a topological nontrivial map between Riemannian manifolds, an a stationary point of a particular energy functional. This paper describes lattice analogues of the aforementioned skyrmions, namely a natural way of using the topological properties of the three dimensional continuum Skyrme model to achieve topological stability on the lattice. In particular, using fixed point iterations, numerically exact lattice skyrmions are constructed: and their stability under small perturbation sis explored by means of linear stability analysis. While stable branches of such solutions are identified, it is also shown that they possess a particularly delicate bifurcation structure, …
The Krein Matrix: General Theory And Concrete Applications In Atomic Bose-Einstein Condensates, Todd Kapitula, Panayotis G. Kevrekidis, Dong Yan
The Krein Matrix: General Theory And Concrete Applications In Atomic Bose-Einstein Condensates, Todd Kapitula, Panayotis G. Kevrekidis, Dong Yan
University Faculty Publications and Creative Works
When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems it is especially important to locate not only the unstable eigenvalues (i.e., those with positive real part) but also those which are purely imaginary but have negative Krein signature. These latter eigenvalues have the property that they can become unstable upon collision with other purely imaginary eigenvalues; i.e., they are a necessary building block in the mechanism leading to the so-called Hamiltonian-Hopf bifurcation. In this paper we review a general theory for constructing a meromorphic matrix-valued function, the so-called Krein matrix, which has the property of not only locating the unstable …
The Krein Matrix: General Theory And Concrete Applications In Atomic Bose-Einstein Condensates, Todd Kapitula, Panayotis G. Kevrekidis, Dong Yan
The Krein Matrix: General Theory And Concrete Applications In Atomic Bose-Einstein Condensates, Todd Kapitula, Panayotis G. Kevrekidis, Dong Yan
University Faculty Publications and Creative Works
When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems it is especially important to locate not only the unstable eigenvalues (i.e., those with positive real part) but also those which are purely imaginary but have negative Krein signature. These latter eigenvalues have the property that they can become unstable upon collision with other purely imaginary eigenvalues; i.e., they are a necessary building block in the mechanism leading to the so-called Hamiltonian-Hopf bifurcation. In this paper we review a general theory for constructing a meromorphic matrix-valued function, the so-called Krein matrix, which has the property of not only locating the unstable …
On The Quadratic Bundles Related To Hermitian Symmetric Spaces, Tihomir Valchev
On The Quadratic Bundles Related To Hermitian Symmetric Spaces, Tihomir Valchev
Articles
We develop the direct scattering problem for quadratic bundles associated to Hermitian symmetric spaces. We adapt the dressing method for quadratic bundles which allows us to find special solutions to multicomponent derivative Schrodinger equation for instance. The latter is an infinite dimensional Hamiltonian system possessing infinite number of integrals of motion. We demonstrate how one can derive them by block diagonalization of the corresponding Lax pair.
Remarks On Quadratic Bundles Related To Hermitian Symmetric Spaces, Tihomir Valchev
Remarks On Quadratic Bundles Related To Hermitian Symmetric Spaces, Tihomir Valchev
Conference papers
We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m)\times U(n)). We discuss the spectral properties of scattering operator, develop the direct scattering problem associated with it and stress on the effect of reduction on these. By applying a modification of Zakharov-Shabat's dressing procedure we demonstrate how one can obtain reflectionless potentials. That way one is able to generate soliton solutions to the nonlinear evolution equations belonging to the integrable hierarchy associated with quadratic bundles under study.
On The Peakon And Soliton Solutions Of An Integrable Pde With Cubic Nonlinearities, Rossen Ivanov, Tony Lyons
On The Peakon And Soliton Solutions Of An Integrable Pde With Cubic Nonlinearities, Rossen Ivanov, Tony Lyons
Conference papers
The interest in the singular solutions (peakons) has been inspired by the Camassa-Holm (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.
G-Strands, Darryl Holm, Rossen Ivanov, James Percival
G-Strands, Darryl Holm, Rossen Ivanov, James Percival
Articles
A G-strand is a map g(t,s): RxR --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3)K-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincar'e system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3) …
Nonlinear Waves And Solitons On Contours And Closed Surfaces, Andrei Ludu
Nonlinear Waves And Solitons On Contours And Closed Surfaces, Andrei Ludu
Andrei Ludu
No abstract provided.
Smooth And Peaked Solitons Of The Camassa-Holm Equation And Applications, Darryl Holm, Rossen Ivanov
Smooth And Peaked Solitons Of The Camassa-Holm Equation And Applications, Darryl Holm, Rossen Ivanov
Articles
The relations between smooth and peaked soliton solutions are reviewed for the Camassa- Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann- Hilbert problem. The momentum map from the action-angle scattering variables T∗(TN) to the flow momentum (X∗) provides the Eulerian representation of the N-soliton solution of CH in terms of the scattering data and squared eigenfunctions of its isospectral eigenvalue problem. The …
Two Component Integrable Systems Modelling Shallow Water Waves, Rossen Ivanov
Two Component Integrable Systems Modelling Shallow Water Waves, Rossen Ivanov
Conference papers
Our aim is to describe the derivation of shallow water model equations for the constant vorticity case and to demonstrate how these equations can be related to two integrable systems: a two component integrable generalization of the Camassa-Holm equation and the Kaup - Boussinesq system.
Equations Of The Camassa-Holm Hierarchy, Rossen Ivanov
Equations Of The Camassa-Holm Hierarchy, Rossen Ivanov
Articles
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 …
Generalised Fourier Transform For The Camassa-Holm Hierarchy, Adrian Constantin, Vladimir Gerdjikov, Rossen Ivanov
Generalised Fourier Transform For The Camassa-Holm Hierarchy, Adrian Constantin, Vladimir Gerdjikov, Rossen Ivanov
Articles
The squared eigenfunctions of the spectral problem associated to the Camassa-Holm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform for the Camassa-Holm 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 Camassa-Holm spectral problem. We show that all the fundamental properties of the Camassa-Holm 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 …
Inverse Scattering Transform For The Camassa-Holm Equation, Adrian Constantin, Vladimir Gerdjikov, Rossen Ivanov
Inverse Scattering Transform For The Camassa-Holm Equation, Adrian Constantin, Vladimir Gerdjikov, Rossen Ivanov
Articles
An Inverse Scattering Method is developed for the Camassa-Holm 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.
Lattice Quantum Algorithm For The Schrodinger Wave Equation In 2+1 Dimensions With A Demonstration By Modeling Soliton Instabilities, Jeffrey Yepez, George Vahala, Linda L. Vahala
Lattice Quantum Algorithm For The Schrodinger Wave Equation In 2+1 Dimensions With A Demonstration By Modeling Soliton Instabilities, Jeffrey Yepez, George Vahala, Linda L. Vahala
Electrical & Computer Engineering Faculty Publications
A lattice-based quantum algorithm is presented to model the non-linear Schrödinger-like equations in 2 + 1 dimensions. In this lattice-based model, using only 2 qubits per node, a sequence of unitary collide (qubit-qubit interaction) and stream (qubit translation) operators locally evolve a discrete field of probability amplitudes that in the long-wavelength limit accurately approximates a non-relativistic scalar wave function. The collision operator locally entangles pairs of qubits followed by a streaming operator that spreads the entanglement throughout the two dimensional lattice. The quantum algorithmic scheme employs a non-linear potential that is proportional to the moduli square of the wave function. …