This research project combines analytical and numerical techniques to investigate the solutions and soliton interactions for vector nonlinear Schrodinger systems (both continuous and discrete), the short-pulse equation and coupled Maxwell-Bloch equations. The nonlinear Schrodinger equation (NLS) and its vector generalization are universal asymptotic models for quasi-monochromatic waves in nonlinear media. Both NLS and vector NLS are completely integrable and can therefore be solved by the Inverse Scattering Transform (IST) method. However, this method for vector NLS in the normal dispersion regime (defocusing regime) is not yet fully developed. Hence, the complete development of this method and description of the soliton interactions is a principal part of the project. The coupled Maxwell-Bloch system, which arises as a model in atomic physics, laser physics and optics, will also be investigated by a suitable adaptation of the IST method. Further, the IST method will be extended to the recently derived short-pulse equation, which provides a more appropriate model than the quasi-monochromatic approximation. Theoretical development of the IST for these systems and elucidation of the soliton dynamics will be guided by numerical simulation and direct methods for constructing special solutions.
Because of their physical applicability in such diverse fields as water waves, magnetic spin waves, optical fibers, waveguides and Bose-Einstein condensates, NLS and other systems investigated in this project are of wide scientific relevance. Dynamics of their solitary wave (soliton) solutions is also of keen interest from the point of view of applications, as interaction of vector solitons sets forth the experimental foundations for designing controlled logic gates and all-optical computers, of the phenomenon of self-induced transparency, as well as a mechanism for polarization switching of light in multi-level media. The equations investigated in this project are not only models for phenomena at the frontier of physical science, but also have intrinsic mathematical value. The development of a modern IST for these equations will advance the fundamental understanding of complex physical phenomena in a unified mathematical framework and provide concrete information about the behavior of such systems. The collaborative nature of the research program, both between the principal investigators and with colleagues at nearby institutions, will serve to accelerate the development of an interconnected research community accessible to students at Montclair State and at University of Colorado at Colorado Springs (both undergraduate institutions). Significantly, the student population at both institutions includes a substantial number of members of underrepresented groups and the collaboration leverages existing programs directed to these students
This project concerned both theoretical and applied investigations of ubiquitous nonlinear wave phenomena, arising in a wide range of physical applications. In particular, the nonlinear Schrodinger systems that have been studied model many important natural phenomena such as: water waves, optical fibers, lasers and Bose-Einstein condensates, etc. Over the past fifty years, a large body of knowledge has been accumulated on these systems, which continue to be extensively studied worldwide. Nonetheless, problems in which the boundary conditions play a crucial role still pose significant mathematical challenges, and a comprehensive picture is currently elusive. The achievements of this research project, which has focused on the investigation of the model equations within the framework of the Inverse Scattering Transform (IST), will contribute pushing the discipline toward frontier research directions, thanks to the combination of analytical, asymptotic and numerical methods that have been developed and used in order to obtain solutions and understand their behavior. Results obtained from the investigations have been disseminated through publications in peer-reviewed journals, as well as via research presentations in conferences and workshops in the PI's own and related disciplines, both in the US and abroad. The research outcomes of this project have several concrete scientific and technological applications. Therefore the results of this project will not only affect the mathematical community, but they will also provide practical information that will help scientists and engineers, thus potentially benefiting society at large. Moreover, the strong track record of international collaboration, conference participation and organization that has been continued and enhanced through this project by the PI contributes to fostering vital research connections between the US and other countries. The research activities under this project have been carried out at a primarily undergraduate institution which fosters undergraduate research. The project has contributed to the promotion of teaching and learning at the PI's institution on a broad scale. The PI has collaborated in developing advanced level courses and thesis projects for both undergraduate and master's students. Moreover, the PI is chair of the colloquium committee of the Math Department at UCCS, and she also organizes the annual Distinguished Lecture of the Mathematics Department. In the three years of NSF support, two undergraduate students and two graduate students were involved in this project. The students also gained valuable experience by participating in conferences and presenting the work carried out under the PI's guidance. The PI was also very active in communicating the general theme of her research in Applied Mathematics to a broader audience and the community at large.
