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STABILITY AND PRESERVATION PROPERTIES OF MULTISYMPLECTIC INTEGRATORS
Title
STABILITY AND PRESERVATION PROPERTIES OF MULTISYMPLECTIC INTEGRATORS.
Creator
Wlodarczyk, Tomasz, Schober, Constance, University of Central Florida
Abstract / Description
This dissertation presents results of the study on symplectic and multisymplectic numerical methods for solving linear and nonlinear Hamiltonian wave equations. The emphasis is put on the second order space and time discretizations of the linear wave, the Klein-Gordon and the sine-Gordon equations. For those equations we develop two multisymplectic (MS) integrators and compare their performance to other popular symplectic and non-symplectic numerical methods. Tools used in the linear analysis...
Show moreThis dissertation presents results of the study on symplectic and multisymplectic numerical methods for solving linear and nonlinear Hamiltonian wave equations. The emphasis is put on the second order space and time discretizations of the linear wave, the Klein-Gordon and the sine-Gordon equations. For those equations we develop two multisymplectic (MS) integrators and compare their performance to other popular symplectic and non-symplectic numerical methods. Tools used in the linear analysis are related to the Fourier transform and consist of the dispersion relationship and the power spectrum of the numerical solution. Nonlinear analysis, in turn, is closely connected to the temporal evolution of the total energy (Hamiltonian) and can be viewed from the topological perspective as preservation of the phase space structures. Using both linear and nonlinear diagnostics we find qualitative differences between MS and non-MS methods. The first difference can be noted in simulations of the linear wave equation solved for broad spectrum Gaussian initial data. Initial wave profiles of this type immediately split into an oscillatory wave-train with the high modes traveling faster (MS schemes), or slower (non-MS methods), than the analytic group velocity. This result is confirmed by an analysis of the dispersion relationship, which also indicates improved qualitative agreement of the dispersive curves for MS methods over non-MS ones. Moreover, observations of the convergence patterns in the wave profile obtained for the sine-Gordon equation for the initial data corresponding to the double-pole soliton and the temporal evolution of the Hamiltonian functional computed for solutions obtained from different discretizations suggest a change of the geometry of the phase space. Finally, we present some theoretical considerations concerning wave action. Lagrangian formulation of linear partial differential equations (PDEs) with slowly varying solutions is capable of linking the wave action conservation law with the dispersion relationship thus suggesting the possibility to extend this connection to multisymplectic PDEs.
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Date Issued
2007
Identifier
CFE0001817, ucf:47344
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0001817
PHASE SYNCHRONIZATION IN THREE-DIMENSIONAL LATTICES AND GLOBALLY COUPLED POPULATIONS OF NONIDENTICAL ROSSLER OSCILLATORS
Title
PHASE SYNCHRONIZATION IN THREE-DIMENSIONAL LATTICES AND GLOBALLY COUPLED POPULATIONS OF NONIDENTICAL ROSSLER OSCILLATORS.
Creator
Qi, Limin, Schober, Constance, University of Central Florida
Abstract / Description
A study on phase synchronization in large populations of nonlinear dynamical systems is presented in this thesis. Using the well-known Rossler system as a prototypical model, phase synchronization in one oscillator with periodic external forcing and in two-coupled nonidentical oscillators was explored at first. The study was further extended to consider three-dimensional lattices and globally coupled populations of nonidentical oscillators, in which the mathematical formulation that...
Show moreA study on phase synchronization in large populations of nonlinear dynamical systems is presented in this thesis. Using the well-known Rossler system as a prototypical model, phase synchronization in one oscillator with periodic external forcing and in two-coupled nonidentical oscillators was explored at first. The study was further extended to consider three-dimensional lattices and globally coupled populations of nonidentical oscillators, in which the mathematical formulation that represents phase synchronization in the generalized N-coupled Rossler system was derived and several computer programs that perform numerical simulations were developed. The results show the effects of coupling dimension, coupling strength, population size, and system parameter on phase synchronization of the various Rossler systems, which may be applicable to studying phase synchronization in other nonlinear dynamical systems as well.
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Date Issued
2005
Identifier
CFE0000776, ucf:46559
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0000776
Analytical and Numerical Investigations of the Kudryashov Generalized KdV Equation
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 Korteweg-de Vries (KG KdV) equation. Using a refined perturbation expansion of the Fermi-Pasta-Ulam (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 Korteweg-de Vries (KG KdV) equation. Using a refined perturbation expansion of the Fermi-Pasta-Ulam (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.
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Date Issued
2018
Identifier
CFE0007754, ucf:52395
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007754
Modeling rogue waves in deep water
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.
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Date Issued
2016
Identifier
CFE0006402, ucf:51476
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0006402
Buidling Lax Integrable Variable-Coefficient Generalizations to Integrable PDEs and Exact Solutions to Nonlinear PDEs
Title
Buidling Lax Integrable Variable-Coefficient 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 variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax- or S-integrable nonlinear partial differential equations (PDEs) with both time- and space-dependent 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 variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax- or S-integrable nonlinear partial differential equations (PDEs) with both time- and space-dependent 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 Estabrook-Wahlquist 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 variable-coefficient generalizations to the nonlinear Schrodinger (NLS) equation, derivative NLS equation, PT-symmetric NLS, fifth-order 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 KP-II equations as instructive examples before using each method to derive nontrivial solutions to a microstructure PDE and two generalized Pochhammer-Chree equations.
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Date Issued
2016
Identifier
CFE0006173, ucf:51144
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0006173
Comparison of Second Order Conformal Symplectic Schemes with Linear Stability Analysis
Title
Comparison of Second Order Conformal Symplectic Schemes with Linear Stability Analysis.
Creator
Floyd, Dwayne, Moore, Brian, Schober, Constance, Mohapatra, Ram, University of Central Florida
Abstract / Description
Numerical methods for solving linearly damped Hamiltonian ordinary differential equations are analyzed and compared. The methods are constructed from the well-known St(&)#246;rmer-Verlet and implicit midpoint methods. The structure preservation properties of each method are shown analytically and numerically. Each method is shown to preserve a symplectic form up to a constantand are therefore conformal symplectic integrators, with each method shown to accurately preserve the rate of momentum...
Show moreNumerical methods for solving linearly damped Hamiltonian ordinary differential equations are analyzed and compared. The methods are constructed from the well-known St(&)#246;rmer-Verlet and implicit midpoint methods. The structure preservation properties of each method are shown analytically and numerically. Each method is shown to preserve a symplectic form up to a constantand are therefore conformal symplectic integrators, with each method shown to accurately preserve the rate of momentum dissipation. An analytical linear stability analysis is completed for each method, establishing thresholds between the value of the damping coefficient and the step-size that ensure stability. The methods are all second order and the preservation of the rate of energy dissipation is compared to that of a third order Runge-Kutta method that does not preserve conformal properties. Numerical experiments will include the damped harmonic oscillator and the damped nonlinear pendulum.
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Date Issued
2014
Identifier
CFE0005793, ucf:50051
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005793
Partially Integrable PT-Symmetric Hierarchies of Some Canonical Nonlinear Partial Differential Equations
Title
Partially Integrable PT-Symmetric 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 co-workers to derive new partially-integrable hierarchies of various PT-symmetric, 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 PT-symmetric KdV...
Show moreWe generalize the work of Bender and co-workers to derive new partially-integrable hierarchies of various PT-symmetric, 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 PT-symmetric 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 partially-integrable systems, including B(&)#228;cklund Transformations, a 'near-Lax 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 PT-symmetric Burgers' equation fails the Painlev(&)#232; Test for its n=2 case, but special solutions are nonetheless obtained. Also, PT-symmetric hierarchies of 2+1 Burgers' and Kadomtsev-Petviashvili 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.
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Date Issued
2013
Identifier
CFE0004736, ucf:49843
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004736