We numerically study the dynamics and interactions of vortex rings in the nonlinear Schrödinger equation (NLSE). Single ring dynamics for both bright and dark vortex rings are explored including their traverse velocity, stability, and perturbations resulting in quadrupole oscillations. Multi-ring dynamics of dark vortex rings are investigated, including scattering and merging of two colliding rings, leapfrogging interactions of co-traveling rings, as well as co-moving steady-state multi-ring ensembles. Simulations of choreographed multi-ring setups are also performed, leading to intriguing interaction dynamics.
Due to the inherent lack of a close form solution for vortex rings and the dimensionality where they live, efficient numerical methods to integrate the NLSE have to be developed in order to perform the extensive number of required simulations. To facilitate this, compact high-order numerical schemes for the spatial derivatives are developed which include a new semi-compact modulus-squared Dirichlet boundary condition. The schemes are combined with a fourth-order Runge-Kutta time-stepping scheme in order to keep the overall method fully explicit. To ensure efficient use of the schemes, a stability analysis is performed to find bounds on the largest usable time step-size as a function of the spatial step-size.
The numerical methods are implemented into codes which are run on NVIDIA graphic processing unit (GPU) parallel architectures. The codes running on the GPU are shown to be many times faster than their serial counterparts. The codes are developed with future usability in mind, and therefore are written to interface with MATLAB utilizing custom GPU-enabled C codes with a MEX-compiler interface. Reproducibility of results is achieved by combining the codes into a code package called NLSEmagic which is freely distributed on a dedicated website.
","abstract_html":"<p>We numerically study the dynamics and interactions of vortex rings in the nonlinear Schrödinger equation (NLSE). Single ring dynamics for both bright and dark vortex rings are explored including their traverse velocity, stability, and perturbations resulting in quadrupole oscillations. Multi-ring dynamics of dark vortex rings are investigated, including scattering and merging of two colliding rings, leapfrogging interactions of co-traveling rings, as well as co-moving steady-state multi-ring ensembles. Simulations of choreographed multi-ring setups are also performed, leading to intriguing interaction dynamics.</p> <p>Due to the inherent lack of a close form solution for vortex rings and the dimensionality where they live, efficient numerical methods to integrate the NLSE have to be developed in order to perform the extensive number of required simulations. To facilitate this, compact high-order numerical schemes for the spatial derivatives are developed which include a new semi-compact modulus-squared Dirichlet boundary condition. The schemes are combined with a fourth-order Runge-Kutta time-stepping scheme in order to keep the overall method fully explicit. To ensure efficient use of the schemes, a stability analysis is performed to find bounds on the largest usable time step-size as a function of the spatial step-size.</p> <p>The numerical methods are implemented into codes which are run on NVIDIA graphic processing unit (GPU) parallel architectures. The codes running on the GPU are shown to be many times faster than their serial counterparts. The codes are developed with future usability in mind, and therefore are written to interface with MATLAB utilizing custom GPU-enabled C codes with a MEX-compiler interface. Reproducibility of results is achieved by combining the codes into a code package called NLSEmagic which is freely distributed on a dedicated website.</p>","abstract_has_math":false,"creators":["Caplan, Ronald Meyer"],"institution":null,"degree_name":"Computational Science Joint PhD with San Diego State University, PhD","degree_level":"Open Access Dissertation","degree_discipline":"School of Mathematical Sciences","degree_department":null,"school":null,"contributors":["Peter Blomgren","Michael Bromley","Ali Nadim"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T01:39:44Z","subjects":["Bose-Einstein condensates","GPU","high-order compact schemes nonlinear dynamics nonlinear Schrödinger equation quantized vortex rings","Applied Mathematics","Computer Sciences","Condensed Matter Physics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarship.claremont.edu/cgu_etd/52","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Peter Blomgren","Michael Bromley","Ali Nadim"]},{"key":"dc:creator","label":"Author","values":["Caplan, Ronald Meyer"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2012-06-01T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["School of Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Computational Science Joint PhD with San Diego State University, PhD"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Bose-Einstein condensates","GPU","high-order compact schemes nonlinear dynamics nonlinear Schrödinger equation quantized vortex rings","Applied Mathematics","Computer Sciences","Condensed Matter Physics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarship.claremont.edu/cgu_etd/52"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We numerically study the dynamics and interactions of vortex rings in the nonlinear Schrödinger equation (NLSE). Single ring dynamics for both bright and dark vortex rings are explored including their traverse velocity, stability, and perturbations resulting in quadrupole oscillations. Multi-ring dynamics of dark vortex rings are investigated, including scattering and merging of two colliding rings, leapfrogging interactions of co-traveling rings, as well as co-moving steady-state multi-ring ensembles. Simulations of choreographed multi-ring setups are also performed, leading to intriguing interaction dynamics.
Due to the inherent lack of a close form solution for vortex rings and the dimensionality where they live, efficient numerical methods to integrate the NLSE have to be developed in order to perform the extensive number of required simulations. To facilitate this, compact high-order numerical schemes for the spatial derivatives are developed which include a new semi-compact modulus-squared Dirichlet boundary condition. The schemes are combined with a fourth-order Runge-Kutta time-stepping scheme in order to keep the overall method fully explicit. To ensure efficient use of the schemes, a stability analysis is performed to find bounds on the largest usable time step-size as a function of the spatial step-size.
The numerical methods are implemented into codes which are run on NVIDIA graphic processing unit (GPU) parallel architectures. The codes running on the GPU are shown to be many times faster than their serial counterparts. The codes are developed with future usability in mind, and therefore are written to interface with MATLAB utilizing custom GPU-enabled C codes with a MEX-compiler interface. Reproducibility of results is achieved by combining the codes into a code package called NLSEmagic which is freely distributed on a dedicated website.
"]},{"key":"dc:title","label":"Title","values":["Study of Vortex Ring Dynamics in the Nonlinear Schrödinger Equation Utilizing GPU-Accelerated High-Order Compact Numerical Integrators"]}]}],"canonical_facts":{"dc:contributor":["Peter Blomgren","Michael Bromley","Ali Nadim"],"dc:creator":["Caplan, Ronald Meyer"],"dc:date.available":["2012-06-01T07:00:00Z"],"dc:description.abstract":["We numerically study the dynamics and interactions of vortex rings in the nonlinear Schrödinger equation (NLSE). Single ring dynamics for both bright and dark vortex rings are explored including their traverse velocity, stability, and perturbations resulting in quadrupole oscillations. Multi-ring dynamics of dark vortex rings are investigated, including scattering and merging of two colliding rings, leapfrogging interactions of co-traveling rings, as well as co-moving steady-state multi-ring ensembles. Simulations of choreographed multi-ring setups are also performed, leading to intriguing interaction dynamics.
Due to the inherent lack of a close form solution for vortex rings and the dimensionality where they live, efficient numerical methods to integrate the NLSE have to be developed in order to perform the extensive number of required simulations. To facilitate this, compact high-order numerical schemes for the spatial derivatives are developed which include a new semi-compact modulus-squared Dirichlet boundary condition. The schemes are combined with a fourth-order Runge-Kutta time-stepping scheme in order to keep the overall method fully explicit. To ensure efficient use of the schemes, a stability analysis is performed to find bounds on the largest usable time step-size as a function of the spatial step-size.
The numerical methods are implemented into codes which are run on NVIDIA graphic processing unit (GPU) parallel architectures. The codes running on the GPU are shown to be many times faster than their serial counterparts. The codes are developed with future usability in mind, and therefore are written to interface with MATLAB utilizing custom GPU-enabled C codes with a MEX-compiler interface. Reproducibility of results is achieved by combining the codes into a code package called NLSEmagic which is freely distributed on a dedicated website.
"],"dc:identifier":["https://scholarship.claremont.edu/cgu_etd/52"],"dc:subject":["Bose-Einstein condensates","GPU","high-order compact schemes nonlinear dynamics nonlinear Schrödinger equation quantized vortex rings","Applied Mathematics","Computer Sciences","Condensed Matter Physics"],"dc:title":["Study of Vortex Ring Dynamics in the Nonlinear Schrödinger Equation Utilizing GPU-Accelerated High-Order Compact Numerical Integrators"],"thesis:degree_discipline":["School of Mathematical Sciences"],"thesis:degree_level":["Open Access Dissertation"],"thesis:degree_name":["Computational Science Joint PhD with San Diego State University, PhD"]},"updated_at":"2026-07-24T01:39:44Z"}