| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | June 14, 2013 |
| Latest Amendment Date: | May 6, 2015 |
| Award Number: | 1313107 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Victor Roytburd
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 15, 2013 |
| End Date: | June 30, 2017 (Estimated) |
| Total Intended Award Amount: | $184,655.00 |
| Total Awarded Amount to Date: | $184,655.00 |
| Funds Obligated to Date: |
FY 2014 = $66,488.00 FY 2015 = $56,256.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
2385 IRVING HILL RD LAWRENCE KS US 66045-7563 (785)864-3441 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
1460 Jayhawk Boulevard Lawrence KS US 66045-7594 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | APPLIED MATHEMATICS |
| Primary Program Source: |
01001415DB NSF RESEARCH & RELATED ACTIVIT 01001516DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The main theme of the proposed research will be the linear and asymptotic stability of special solutions (waves) of a wide class of partial differential equations. In a series of recent works, the PI and collaborators have characterized the linear stability of travelling and standing waves for second order in time equations and systems. Such results provide necessary information for the asymptotic stability of the same waves. The PI and collaborators will build on their previous work to show asymptotic stability of standing waves for the following models: the Klein-Gordon equation, the sine Gordon equation and the Dirac equation. A second goal of the proposal is to study the existence and stability of coherent structures, arising in spatially discrete models. More concretely, the project will deal, among other things, with Hertzian granular chains/crystals and the discrete nonlinear Schr\"odinger equation.
The project deals with nonlinear dispersive equations, which model wavelike behavior of important physical processes. Important class of problems under consideration include the propagation of light in optical waveguides, motion of quantum particles, the mechanics of fluids to mention a few. The overarching theme of the investigation will be the stability of coherent configuration - that is, if one is initially close to such coherent structure, does it stay close to it forever? The mathematical formulation of such problems, as well as their analysis and predictions about their long time behavior will greatly enhance our understanding of these and related processes.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
Partial differential equations are mathematical tool that give reliable mathematical models that describe many phenomena in a variety of actual engineering applications, such as optics, fluid dynamics to mention a few. The research objective in this proposal is to study the solution solutions, which represent a single wave or train of waves traveling through the medium. Such objects are instrumental for predicting behavior and designing engineering devices for the wide range of physically unrelated phenomena, such as water wave dynamics, in particular, gigantic ocean waves, magnetization issues, propagation of light in optical fibers or in quantum mechanical dynamics. In particular, understanding the dynamics of these (and nearby) objects provides important clues about the global dynamics/behavior of the system. In addition, a feedback mechansim of sorts applies, as singular behavior signals a breakdown of the model itself or some pathological behavior of the system, which are both important to understand.
The PI, in collaboration with other scientists, has studied the linear and asymptotic stability of solitary waves of a wide class of partial differential equations. Specific results concerning existence and stability, including explicit ranges of parameters), were obtained. Among the notable achievements were the explicit characterizations of the stable waves for the Klein-Gordon-Zakharov system (and its version in periodic waveguides), generalized Ostrovsky models and asymptotic stability results for the Gross-Neveu problem. Most of the results obtained in this direction apply for a wider class of examples, while the methods developed to address these problems will certainly be useful in the study of other outstanding problems in the field. In addition, the PI and collaborators have studied the existence of waves in spatially discrete models, such as the ones appearing in the modeling of granular chains.
Last Modified: 10/16/2017
Modified by: Atanas G Stefanov
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