Nonlinear Waves Seminar
Nonlinear Waves
The Nonlinear Waves seminar is motivated by wave phenomena in various systems. However, more broadly, we are interested in computational and applied mathematics. All are welcome to attend. All meetings are in-person, unless specified otherwise.
If you would like to give a talk, email Justin Cole (jcole13@uccs.edu).
Wednesdays 4:00pm-5:00pm, OSB B138
Abstracts Date: Wednesday March, 6 Abstract: Certain well-known dispersive partial differential equations that model the propagation of waves are known to be integrable - that is, exactly solvable. Recently, a peculiar class of integrable systems with spatial and/or temporal nonlocality has been introduced. This talk will give an overview of this class of nonlocal equations, as well as a discussion of some technical details that separate these systems from their local counterparts.
Date: Wednesday March, 13 Abstract: Electromagnetic wave propagation in a wave guide can be described by the onedimensional time independent Schrodinger equation, which, in the deep lattice limit, can be sufficiently approximated by an SSH type model through constant coefficients interaction terms. A numerical program has been developed to compute the interaction coefficients based on user input potential or spectral band data. A nonlinear least squares approximation method (Levenberg-Marquardt) is utilized to minimize an objective function. The discrete models generated by the algorithm are capable of reproducing the expected physical and topological properties of a given system.
Date: Wednesday March, 19 Abstract: Motivated by the work of J.K. Jang et al., Nat. Commun. {\bf 6}, 7370 (2015), where the authors experimentally tweeze cavity solitons in a passive loop of optical fiber, we study the amenability to tweezing of cavity solitons as the properties of a localized tweezer are varied. The system is modeled by the Lugiato-Lefever equation, a variant of the complex Ginzburg-Landau equation. We produce an effective, localized, trapping tweezer potential by assuming a Gaussian phase-modulation of the holding beam. The potential for tweezing is then assessed as the total (temporal) displacement and speed of the tweezer are varied, and corresponding phase diagrams are presented. As the relative speed of the tweezer is increased we find two possible dynamical scenarios: successful tweezing and release of the cavity soliton. We also deploy a non-conservative variational approximation (NCVA) based on a Lagrangian description which reduces the original dissipative partial differential equation to a set of coupled ordinary differential equations for the cavity soliton parameters. We illustrate the ability of the NCVA to accurately predict the separatrix between successful and failed tweezing. This showcases the versatility of the NCVA to provide a low-dimensional description of the experimental realization of the temporal tweezing. The above is a joint work with Julia Rossi, Ricardo Carretero-González and Panayotis Kevrekidis. Date: Wednesday April, 3 Abstract: Optical Thermodynamics represents a novel statistical theory capable of describing the equilibrium state of weakly nonlinear multimode arrangements. In this talk we will introduce the principles of Optical Thermodynamics by drawing comparisons with classical statistical mechanics to understand its distinct features. Furthermore, we will present the results of various numerical simulations conducted on different discrete nonlinear lattices and their direct comparison with the theoretical predictions.
Date: Wednesday April, 10 Abstract: Approximating rational solutions to partial differential equations is often difficult using traditional numerical methods such as spectral Fourier methods, due to slow (algebraic) decay of the functions. This talk will introduce the Malmquist-Takenaka (MT) functions as a suitable basis for representing rational functions. The MT functions are set of orthogonal rational functions that, importantly, can be related to the discrete Fourier transform and computed via a modified fast Fourier transform. Many examples illustrating the effectiveness of this approach will be given.
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