Current Search: variational solitons (x)
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- Title
- SOLITON SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS USING VARIATIONAL APPROXIMATIONS AND INVERSE SCATTERING TECHNIQUES.
- Creator
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Vogel, Thomas, Kaup, David, University of Central Florida
- Abstract / Description
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Throughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their limited reach in solving large classes of NPDE. This body of work will study the analysis of NPDE using two of the most ubiquitous techniques developed in the last century. In this body of work, the analysis and techniques herein are applied to unsolved physical problems in both the fields of...
Show moreThroughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their limited reach in solving large classes of NPDE. This body of work will study the analysis of NPDE using two of the most ubiquitous techniques developed in the last century. In this body of work, the analysis and techniques herein are applied to unsolved physical problems in both the fields of variational approximations and inverse scattering transform. Additionally, a new technique for estimating the error of a variational approximation is established. Note that the material in chapter 2, "Quantitative Measurements of Variational Approximations" has recently been published. Variational problems have long been used to mathematically model physical systems. Their advantage has been the simplicity of the model as well as the ability to deduce information concerning the functional dependence of the system on various parameters embedded in the variational trial functions. However, the only method in use for estimating the error in a variational approximation has been to compare the variational result to the exact solution. In this work, it is demonstrated that one can computationally obtain estimates of the errors in a one-dimensional variational approximation, without any a priori knowledge of the exact solution. Additionally, this analysis can be done by using only linear techniques. The extension of this method to multidimensional problems is clearly possible, although one could expect that additional difficulties would arise. One condition for the existence of a localized soliton is that the propagation constant does not fall into the continuous spectrum of radiation modes. For a higher order dispersive systems, the linear dispersion relation exhibits a multiple branch structure. It could be the case that in a certain parameter region for which one of the components of the solution has oscillations (i.e., is in the continuous spectrum), there exists a discrete value of the propagation constant, k(ES), for which the oscillations have zero amplitude. The associated solution is referred to as an embedded soliton (ES). This work examines the ES solutions in a CHI(2):CHI(3), type II system. The method employed in searching for the ES solutions is a variational method recently developed by Kaup and Malomed [Phys. D 184, 153-61 (2003)] to locate ES solutions in a SHG system. The variational results are validated by numerical integration of the governing system. A model used for the 1-D longitudinal wave propagation in microstructured solids is a KdV-type equation with third and fifth order dispersions as well as first and third order nonlinearities. Recent work by Ilison and Salupere (2004) has identified certain types of soliton solutions in the aforementioned model. The present work expands the known family of soliton solutions in the model to include embedded solitons. The existence of embedded solitons with respect to the dispersion parameters is determined by a variational approximation. The variational results are validated with selected numerical solutions.
Show less - Date Issued
- 2007
- Identifier
- CFE0001800, ucf:47379
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0001800
- Title
- VARIATIONAL EMBEDDED SOLITONS, AND TRAVELING WAVETRAINS GENERATED BY GENERALIZED HOPF BIFURCATIONS, IN SOME NLPDE SYSTEMS.
- Creator
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Smith, Todd, Choudhury, Roy, University of Central Florida
- Abstract / Description
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In this Ph.D. thesis, we study regular and embedded solitons and generalized and degenerate Hopf bifurcations. These two areas of work are seperate and independent from each other. First, variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochhammer PDE and a generalized microstructure PDE that are currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations...
Show moreIn this Ph.D. thesis, we study regular and embedded solitons and generalized and degenerate Hopf bifurcations. These two areas of work are seperate and independent from each other. First, variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochhammer PDE and a generalized microstructure PDE that are currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the family of the trial functions). Thus, the residual is calculated, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that only the parameter regimes for the existence of solitary waves had previously been analyzed for the microstructure PDE considered here, the results obtained here are both new and timely. Second, we consider generalized and degenerate Hopf bifurcations in three different models: i. a predator-prey model with general predator death rate and prey birth rate terms, ii. a laser-diode system, and iii. traveling-wave solutions of twospecies predator-prey/reaction-diusion equations with arbitrary nonlinear/reaction terms. For speci c choices of the nonlinear terms, the quasi-periodic orbit in the post-bifurcation regime is constructed for each system using the method of multiple scales, and its stability is analyzed via the corresponding normal form obtained by reducing the system down to the center manifold. The resulting predictions for the post-bifurcation dynamics provide an organizing framework for the variety of possible behaviors. These predictions are veri ed and supplemented by numerical simulations, including the computation of power spectra, autocorrelation functions, and fractal dimensions as appropriate for the periodic and quasiperiodic attractors, attractors at in nity, as well as bounded chaotic attractors obtained in various cases. The dynamics obtained in the three systems is contrasted and explained on the basis of the bifurcations occurring in each. For instance, while the two predator-prey models yield a variety of behaviors in the post-bifurcation regime, the laser-diode evinces extremely stable quasiperiodic solutions over a wide range of parameters, which is very desirable for robust operation of the system in oscillator mode.
Show less - Date Issued
- 2011
- Identifier
- CFE0003634, ucf:48887
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0003634
- Title
- Comparing the Variational Approximation and Exact Solutions of the Straight Unstaggered and Twisted Staggered Discrete Solitons.
- Creator
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Marulanda, Daniel, Kaup, David, Moore, Brian, Vajravelu, Kuppalapalle, University of Central Florida
- Abstract / Description
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Discrete nonlinear Schr(&)#246;dinger equations (DNSL) have been used to provide models of a variety of physical settings. An application of DNSL equations is provided by Bose-Einstein condensates which are trapped in deep optical-lattice potentials. These potentials effectively splits the condensate into a set of droplets held in local potential wells, which are linearly coupled across the potential barriers between them [3]. In previous works, DNLS systems have also been used for symmetric...
Show moreDiscrete nonlinear Schr(&)#246;dinger equations (DNSL) have been used to provide models of a variety of physical settings. An application of DNSL equations is provided by Bose-Einstein condensates which are trapped in deep optical-lattice potentials. These potentials effectively splits the condensate into a set of droplets held in local potential wells, which are linearly coupled across the potential barriers between them [3]. In previous works, DNLS systems have also been used for symmetric on-site-centered solitons [11]. A few works have constructed different discrete solitons via the variational approximation (VA) and have explored their regions for their solutions [11, 12]. Exact solutions for straight unstaggered-twisted staggered (SUTS) discrete solitons have been found using the shooting method [12].In this work, we will use Newton's method, which converges to the exact solutions of SUTS discrete solitons. The VA has been used to create starting points. There are two distinct types of solutions for the soliton's waveform: SUTS discrete solitons and straight unstaggered discrete solitons, where the twisted component is zero in the latter soliton. We determine the range of parameters for which each type of solution exists. We also compare the regions for the VA solutions and the exact solutions in certain selected cases. Then, we graphically and numerically compare examples of the VA solutions with their corresponding exact solutions. We also find that the VA provides reasonable approximations to the exact solutions.
Show less - Date Issued
- 2016
- Identifier
- CFE0006350, ucf:51570
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0006350
