Current Search: Schober, Constance (x)
View All Items
 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 KleinGordon and the sineGordon equations. For those equations we develop two multisymplectic (MS) integrators and compare their performance to other popular symplectic and nonsymplectic 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 KleinGordon and the sineGordon equations. For those equations we develop two multisymplectic (MS) integrators and compare their performance to other popular symplectic and nonsymplectic 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 nonMS 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 wavetrain with the high modes traveling faster (MS schemes), or slower (nonMS 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 nonMS ones. Moreover, observations of the convergence patterns in the wave profile obtained for the sineGordon equation for the initial data corresponding to the doublepole 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.
Show less  Date Issued
 2007
 Identifier
 CFE0001817, ucf:47344
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0001817
 Title
 PHASE SYNCHRONIZATION IN THREEDIMENSIONAL 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 wellknown Rossler system as a prototypical model, phase synchronization in one oscillator with periodic external forcing and in twocoupled nonidentical oscillators was explored at first. The study was further extended to consider threedimensional 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 wellknown Rossler system as a prototypical model, phase synchronization in one oscillator with periodic external forcing and in twocoupled nonidentical oscillators was explored at first. The study was further extended to consider threedimensional lattices and globally coupled populations of nonidentical oscillators, in which the mathematical formulation that represents phase synchronization in the generalized Ncoupled 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.
Show less  Date Issued
 2005
 Identifier
 CFE0000776, ucf:46559
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0000776
 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
 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
 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 wellknown St(&)#246;rmerVerlet 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 wellknown St(&)#246;rmerVerlet 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 stepsize 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 RungeKutta method that does not preserve conformal properties. Numerical experiments will include the damped harmonic oscillator and the damped nonlinear pendulum.
Show less  Date Issued
 2014
 Identifier
 CFE0005793, ucf:50051
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005793
 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