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 Title
 Analytical and Numerical Investigations of the Kudryashov Generalized KdV Equation.
 Creator

Hilton, William, Schober, Constance, Moore, Brian, Choudhury, Sudipto, University of Central Florida
 Abstract / Description

This thesis concerns an analytical and numerical study of the Kudryashov Generalized Kortewegde Vries (KG KdV) equation. Using a refined perturbation expansion of the FermiPastaUlam (FPU) equations of motion, the KG KdV equation, which arises at sixth order, and general higher order KdV equations are derived. Special solutions of the KG KdV equation are derived using the tanh method. A pseudospectral integrator, which can handle stiff equations, is developed for the higher order KdV...
Show moreThis thesis concerns an analytical and numerical study of the Kudryashov Generalized Kortewegde Vries (KG KdV) equation. Using a refined perturbation expansion of the FermiPastaUlam (FPU) equations of motion, the KG KdV equation, which arises at sixth order, and general higher order KdV equations are derived. Special solutions of the KG KdV equation are derived using the tanh method. A pseudospectral integrator, which can handle stiff equations, is developed for the higher order KdV equations. The numerical experiments indicate that although the higher order equations exhibit complex dynamics, they fail to reach energy equipartition on the time scale considered.
Show less  Date Issued
 2018
 Identifier
 CFE0007754, ucf:52395
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007754
 Title
 Paritytime and supersymmetry in optics.
 Creator

Miri, Mohammad, Christodoulides, Demetrios, Abouraddy, Ayman, Likamwa, Patrick, Choudhury, Sudipto, University of Central Florida
 Abstract / Description

Symmetry plays a crucial role in exploring the laws of nature. By exploiting some of the underlying analogies between the mathematical formalism of quantum mechanics and that of electrodynamics, in this dissertation we show that optics can provide a fertile ground for studying, observing, and utilizing some of the peculiar symmetries that are currently out of reach in other areas of physics. In particular, in this work, we investigate two important classes of symmetries, paritytime symmetry ...
Show moreSymmetry plays a crucial role in exploring the laws of nature. By exploiting some of the underlying analogies between the mathematical formalism of quantum mechanics and that of electrodynamics, in this dissertation we show that optics can provide a fertile ground for studying, observing, and utilizing some of the peculiar symmetries that are currently out of reach in other areas of physics. In particular, in this work, we investigate two important classes of symmetries, paritytime symmetry (PT) and supersymmetry (SUSY), within the context of classical optics. The presence of PT symmetry can lead to entirely real spectra in nonHermitian systems. In optics, PTsymmetric structures involving balanced regions of gain and loss exhibit intriguing properties which are otherwise unattainable in traditional Hermitian systems. We show that selective PT symmetry breaking offers a new method for achieving single mode operation in laser cavities. Other interesting phenomena also arise in connection with PT periodic structures. Along these lines, we introduce a new class of optical lattices, the so called mesh lattices. Such arrays provide an ideal platform for observing a range of PTrelated phenomena. We show that defect sates and solitons exist in such periodic environments exhibiting unusual behavior. We also investigate the scattering properties of PTsymmetric particles and we show that such structures can deflect light in a controllable manner. In the second part of this dissertation, we introduce the concept of supersymmetric optics. In this regard, we show that any optical structure can be paired with a superpartner with similar guided wave and scattering properties. As a result, the guided mode spectra of these optical waveguide systems can be judiciously engineered so as to realize new families of mode filters and mode division multiplexers and demultiplexers. We also present the first experimental demonstration of light dynamics in SUSY ladders of photonic lattices. In addition a new type of transformation optics based on supersymmetry is also explored. Finally, using the SUSY formalism in nonHermitian settings, we identify more general families of complex optical potentials with real spectra.
Show less  Date Issued
 2014
 Identifier
 CFE0005844, ucf:50915
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005844
 Title
 Propagation Failure in Discrete Inhomogeneous Media Using a Caricature of the Cubic.
 Creator

Lydon, Elizabeth, Moore, Brian, Choudhury, Sudipto, Kaup, David, University of Central Florida
 Abstract / Description

Spatially discrete Nagumo equations have widespread physical applications, including modeling electrical impulses traveling through a demyelinated axon, an environment typical in multiple scle rosis. We construct steadystate, single front solutions by employing a piecewise linear reaction term. Using a combination of JacobiOperator theory and the ShermanMorrison formula we de rive exact solutions in the cases of homogeneous and inhomogeneous diffusion. Solutions exist only under certain...
Show moreSpatially discrete Nagumo equations have widespread physical applications, including modeling electrical impulses traveling through a demyelinated axon, an environment typical in multiple scle rosis. We construct steadystate, single front solutions by employing a piecewise linear reaction term. Using a combination of JacobiOperator theory and the ShermanMorrison formula we de rive exact solutions in the cases of homogeneous and inhomogeneous diffusion. Solutions exist only under certain conditions outlined in their construction. The range of parameter values that satisfy these conditions constitutes the interval of propagation failure, determining under what circumstances a front becomes pinned in the media. Our exact solutions represent a very specific solution to the spatially discrete Nagumo equation. For example, we only consider inhomogeneous media with one defect present. We created an original script in MATLAB which algorithmically solves more general cases of the equation, including the case for multiple defects. The algorithmic solutions are then compared to known exact solutions to determine their validity.
Show less  Date Issued
 2015
 Identifier
 CFE0005831, ucf:50903
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005831
 Title
 Modeling rogue waves in deep water.
 Creator

Strawn, Maria, Schober, Constance, Moore, Brian, Choudhury, Sudipto, Calini, Annalisa, University of Central Florida
 Abstract / Description

The evolution of surface waves in deep water is governed by the nonlinear Schrodinger (NLS) equation. Spatially periodic breathers (SPBs) and rational solutions of the NLS equation are used as typical models for rogue waves since they exhibit many features of rogue waves. A major component of the dissertation is the stability of solutions of the NLS equation.We address the stability of the rational solutions of the NLS equation used to model rogue waves using squared eigenfunctions of the...
Show moreThe evolution of surface waves in deep water is governed by the nonlinear Schrodinger (NLS) equation. Spatially periodic breathers (SPBs) and rational solutions of the NLS equation are used as typical models for rogue waves since they exhibit many features of rogue waves. A major component of the dissertation is the stability of solutions of the NLS equation.We address the stability of the rational solutions of the NLS equation used to model rogue waves using squared eigenfunctions of the associated Lax Pair. This allows us to contrast to the existing results for SPBs. The stability of the constant amplitude solution of the higher order NLS (HONLS) equation with additional novel perturbations, relevant toour subsequent study on downshifting, is considered next. In addition to the higher order perturbations, we include linear effects and nonlinear damping of the mean flow to the HONLS equation.In addition to stability, we discuss rogue waves and downshifting. Permanent downshifting occurs when energy if permanently transferred from the initially dominant mode to lower modes and is observed in physical experiments and field studies of deep water waves. Although these experimental observations are well documented, neither NLS nor HONLS equations describe this behavior. Nonlinear damping of the mean flow, included in our studies, is shown to model permanent downshifting. We examine the interaction of rogue waves and downshifting in a sea state with both nonlinear and linear effects. We show that there are no rogue waves after permanent downshifting. Analytical and numerical analysis are provided to support the findings.
Show less  Date Issued
 2016
 Identifier
 CFE0006402, ucf:51476
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0006402
 Title
 Buidling Lax Integrable VariableCoefficient Generalizations to Integrable PDEs and Exact Solutions to Nonlinear PDEs.
 Creator

Russo, Matthew, Choudhury, Sudipto, Moore, Brian, Schober, Constance, Christodoulides, Demetrios, University of Central Florida
 Abstract / Description

This dissertation is composed of two parts. In Part I a technique based on extended Lax Pairs isfirst considered to derive variablecoefficient generalizations of various Laxintegrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax or Sintegrable nonlinear partial differential equations (PDEs) with both time and spacedependent coefficients which are thus more general than almost all cases considered earlier via other methods such...
Show moreThis dissertation is composed of two parts. In Part I a technique based on extended Lax Pairs isfirst considered to derive variablecoefficient generalizations of various Laxintegrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax or Sintegrable nonlinear partial differential equations (PDEs) with both time and spacedependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must 'guess' a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we present a generalization to the well known EstabrookWahlquist prolongation technique which provides a systematic procedure for the derivation of the Lax representation. In order to obtain a nontrivial Lax representation we must impose differential constraints on the variable coefficients present in the nlpde. The resulting constraints determine a class of equations which represent generalizations to a previously known integrable constant coefficient nlpde. We demonstrate the effectiveness of this technique by deriving variablecoefficient generalizations to the nonlinear Schrodinger (NLS) equation, derivative NLS equation, PTsymmetric NLS, fifthorder KdV, and three equations in the MKdV hierarchy. In Part II of this dissertation, we introduce three types of singular manifold methods which have been successfully used in the literature to derive exact solutions to many nonlinear PDEs extending over a wide range of applications. The singular manifold methods considered are: truncated Painleve analysis, Invariant Painleve analysis, and a generalized Hirota expansion method. We then consider the KdV and KPII equations as instructive examples before using each method to derive nontrivial solutions to a microstructure PDE and two generalized PochhammerChree equations.
Show less  Date Issued
 2016
 Identifier
 CFE0006173, ucf:51144
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0006173
 Title
 Smooth and NonSmooth Traveling Wave Solutions of Some Generalized CamassaHolm Equations.
 Creator

Rehman, Taslima, Choudhury, Sudipto, Nevai, Andrew, Rollins, David, University of Central Florida
 Abstract / Description

In this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized CamassaHolm (GCH) equations. In the first part, a novel application of phaseplane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible nonsmooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling...
Show moreIn this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized CamassaHolm (GCH) equations. In the first part, a novel application of phaseplane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible nonsmooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling waves. We generalize an existing theorem to establish the existence of peakon solutions of the third GCH equation. This equation is found to also support four segmented, nonsmooth Mwave solutions. While the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes.In the second part of the thesis, smooth traveling waves of the four GCH equations are considered. Here, we use a recent technique to derive convergent multiinfinite series solutions for the homoclinic and heteroclinic orbits of their travelingwave equations, corresponding to pulse and front (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. Of course, the convergence rate is not comparable to typical asymptotic series. However, asymptotic solutions for global behavior along a full homoclinic/heteroclinic orbit are currently not available.
Show less  Date Issued
 2013
 Identifier
 CFE0004918, ucf:49637
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0004918
 Title
 Partially Integrable PTSymmetric Hierarchies of Some Canonical Nonlinear Partial Differential Equations.
 Creator

Pecora, Keri, Choudhury, Sudipto, Schober, Constance, Rollins, David, Christodoulides, Demetrios, University of Central Florida
 Abstract / Description

We generalize the work of Bender and coworkers to derive new partiallyintegrable hierarchies of various PTsymmetric, nonlinear partial differential equations. The possible integrable members are identified employing the Painlev(&)#232; Test, a necessary but not sufficient integrability condition, and are indexed by the integer n, corresponding to the negative of the order of the dominant pole in the singular part of the Painlev(&)#232; expansion for the solution.For the PTsymmetric KdV...
Show moreWe generalize the work of Bender and coworkers to derive new partiallyintegrable hierarchies of various PTsymmetric, nonlinear partial differential equations. The possible integrable members are identified employing the Painlev(&)#232; Test, a necessary but not sufficient integrability condition, and are indexed by the integer n, corresponding to the negative of the order of the dominant pole in the singular part of the Painlev(&)#232; expansion for the solution.For the PTsymmetric KdV equation, as with some other hierarchies, the first or n=1 equation fails the test, the n=2 member corresponds to the regular KdV equation, while the remainder form an entirely new, possibly integrable hierarchy. Integrability properties of the n=3 and n=4 members, typical of partiallyintegrable systems, including B(&)#228;cklund Transformations, a 'nearLax Pair', and analytic solutions are derived. The solutions, or solitary waves, prove to be algebraic in form, and the extended homogeneous balance technique appears to be the most efficient in exposing the Lax Pair.The PTsymmetric Burgers' equation fails the Painlev(&)#232; Test for its n=2 case, but special solutions are nonetheless obtained. Also, PTsymmetric hierarchies of 2+1 Burgers' and KadomtsevPetviashvili equations, which may prove useful in applications, are analyzed. Extensions of the Painlev(&)#232; Test and Invariant Painlev(&)#232; analysis to 2+1 dimensions are utilized, and BTs and special solutions are found for those cases that pass the Painlev(&)#232; Test.
Show less  Date Issued
 2013
 Identifier
 CFE0004736, ucf:49843
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0004736
 Title
 Performance Evaluation of Connectivity and Capacity of Dynamic Spectrum Access Networks.
 Creator

Altameemi, Osama, Chatterjee, Mainak, Bassiouni, Mostafa, Jha, Sumit, Wei, Lei, Choudhury, Sudipto, University of Central Florida
 Abstract / Description

Recent measurements on radio spectrum usage have revealed the abundance of under utilized bands of spectrum that belong to licensed users. This necessitated the paradigm shift from static to dynamic spectrum access (DSA) where secondary networks utilize unused spectrum holes in the licensed bands without causing interference to the licensed user. However, wide scale deployment of these networks have been hindered due to lack of knowledge of expected performance in realistic environments and...
Show moreRecent measurements on radio spectrum usage have revealed the abundance of under utilized bands of spectrum that belong to licensed users. This necessitated the paradigm shift from static to dynamic spectrum access (DSA) where secondary networks utilize unused spectrum holes in the licensed bands without causing interference to the licensed user. However, wide scale deployment of these networks have been hindered due to lack of knowledge of expected performance in realistic environments and lack of costeffective solutions for implementing spectrum database systems. In this dissertation, we address some of the fundamental challenges on how to improve the performance of DSA networks in terms of connectivity and capacity. Apart from showing performance gains via simulation experiments, we designed, implemented, and deployed testbeds that achieve economics of scale. We start by introducing network connectivity models and show that the wellestablished disk model does not hold true for interferencelimited networks. Thus, we characterize connectivity based on signal to interference and noise ratio (SINR) and show that not all the deployed secondary nodes necessarily contribute towards the network's connectivity. We identify such nodes and show that eventhough a node might be communicationvisible it can still be connectivityinvisible. The invisibility of such nodes is modeled using the concept of Poisson thinning. The connectivityvisible nodes are combined with the coverage shrinkage to develop the concept of effective density which is used to characterize the con nectivity. Further, we propose three techniques for connectivity maximization. We also show how traditional flooding techniques are not applicable under the SINR model and analyze the underlying causes for that. Moreover, we propose a modified version of probabilistic flooding that uses lower message overhead while accounting for the node outreach and in terference. Next, we analyze the connectivity of multichannel distributed networks and show how the invisibility that arises among the secondary nodes results in thinning which we characterize as channel abundance. We also capture the thinning that occurs due to the nodes' interference. We study the effects of interference and channel abundance using Poisson thinning on the formation of a communication link between two nodes and also on the overall connectivity of the secondary network. As for the capacity, we derive the bounds on the maximum achievable capacity of a randomly deployed secondary network with finite number of nodes in the presence of primary users since finding the exact capacity involves solving an optimization problem that shows inscalability both in time and search space dimensionality. We speed up the optimization by reducing the optimizer's search space. Next, we characterize the QoS that secondary users can expect. We do so by using vector quantization to partition the QoS space into finite number of regions each of which is represented by one QoS index. We argue that any operating condition of the system can be mapped to one of the precomputed QoS indices using a simple lookup in Olog (N) time thus avoiding any cumbersome computation for QoS evaluation. We implement the QoS space on an 8bit microcontroller and show how the mathematically intensive operations can be computed in a shorter time. To demonstrate that there could be low cost solutions that scale, we present and implement an architecture that enables dynamic spectrum access for any type of network ranging from IoT to cellular. The three main components of this architecture are the RSSI sensing network, the DSA server, and the service engine. We use the concept of modular design in these components which allows transparency between them, scalability, and ease of maintenance and upgrade in a plugnplay manner, without requiring any changes to the other components. Moreover, we provide a blueprint on how to use offtheshelf commercially available software configurable RF chips to build low cost spectrum sensors. Using testbed experiments, we demonstrate the efficiency of the proposed architecture by comparing its performance to that of a legacy system. We show the benefits in terms of resilience to jamming, channel relinquishment on primary arrival, and best channel determination and allocation. We also show the performance gains in terms of frame error rater and spectral efficiency.
Show less  Date Issued
 2016
 Identifier
 CFE0006063, ucf:50980
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0006063
 Title
 Structurepreserving finite difference methods for linearly damped differential equations.
 Creator

Bhatt, Ashish, Moore, Brian, Choudhury, Sudipto, Gurel, Basak, Kauffman, Jeffrey L., University of Central Florida
 Abstract / Description

Differential equations (DEs) model a variety of physical phenomena in science and engineering. Many physical phenomena involve conservative or dissipative forces, which manifest themselves as qualitative properties of DEs that govern these phenomena. Since only a few and simplistic models are known to have exact solutions, approximate solution techniques, such as numerical integration, are used to reveal important insights about solution behavior and properties of these models. Numerical...
Show moreDifferential equations (DEs) model a variety of physical phenomena in science and engineering. Many physical phenomena involve conservative or dissipative forces, which manifest themselves as qualitative properties of DEs that govern these phenomena. Since only a few and simplistic models are known to have exact solutions, approximate solution techniques, such as numerical integration, are used to reveal important insights about solution behavior and properties of these models. Numerical integrators generally result in undesirable quantitative and qualitative errors . Standard numerical integrators aim to reduce quantitative errors, whereas geometric (numerical) integrators aim to reduce or eliminate qualitative errors, as well, in order to improve the accuracy of numerical solutions. It is now widely recognized that geometric (or structurepreserving) integrators are advantageous compared to nongeometric integrators for DEs, especially for long time integration.Geometric integrators for conservative DEs have been proposed, analyzed, and investigated extensively in the literature. The motif of this thesis is to extend the idea of structure preservation to linearly damped DEs. More specifically, we develop, analyze, and implement geometric integrators for linearly damped ordinary and partial differential equations (ODEs and PDEs) that possess conformal invariants, which are qualitative properties that decay exponentially along any solution of the DE as the system evolves over time. In particular, we derive restrictions on the coefficient functions of exponential RungeKutta (ERK) numerical methods for preservation of certain conformal invariants of linearly damped ODEs. An important class of these methods is shown to preserve the damping rate of solutions of damped linear ODEs. Linearly stability and order of accuracy for some specific cases of ERK methods are investigated. Geometric integrators for PDEs are designed using structurepreserving ERK methods in space, time, or both. These integrators for PDEs are also shown to preserve additional structure in certain special cases. Numerical experiments illustrate higher order accuracy and structure preservation properties of various ERK based methods, demonstrating clear advantages over nonstructurepreserving methods, as well as usefulness for solving a wide range of DEs.
Show less  Date Issued
 2016
 Identifier
 CFE0006832, ucf:51763
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0006832