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
