Collaborative Research RUI: Dynamics of Soliton Interactions and Applications

本研究结合解析和数值方法，研究矢量非线性薛定谔系统(包括连续和离散的)、短脉冲方程和耦合的Maxwell-Bloch方程的解和孤子相互作用。非线性薛定谔方程(NLS)及其矢量推广是非线性介质中准单色波的普遍渐近模型。NLS和矢量NLS都是完全可积的，因此可以用逆散射变换(IST)方法求解。然而，对于正常色散区(散焦区)中的矢量NLS，这种方法还没有完全发展起来。因此，该方法的完整发展和孤子相互作用的描述是该项目的主要部分。作为原子物理、激光物理和光学中的一个模型，耦合的Maxwell-Bloch系统也将被适当地采用IST方法来研究。此外，将IST方法推广到最近导出的短脉冲方程，它提供了比准单色近似更合适的模型。这些系统的IST的理论发展和孤子动力学的阐明将以数值模拟和构造特解的直接方法为指导。由于它们在水波、磁自旋波、光纤、波导和玻色-爱因斯坦凝聚体等不同领域的物理适用性，NLS和本项目所研究的其他系统具有广泛的科学意义。从应用的角度来看，它们的孤波(孤子)解的动力学也引起了人们的极大兴趣，因为矢量孤子的相互作用为设计可控逻辑门和全光计算机、自感透明现象以及多能级介质中光的偏振切换机制提供了实验基础。本项目所研究的方程不仅是物理科学前沿现象的模型，而且具有内在的数学价值。这些方程的现代IST的发展将在一个统一的数学框架中促进对复杂物理现象的基本理解，并提供关于此类系统行为的具体信息。该研究项目的合作性质，无论是在主要研究人员之间，还是与附近机构的同事之间，都将有助于加速发展一个相互联系的研究社区，供蒙特克莱尔州立大学和科罗拉多大学科罗拉多斯普林斯分校(这两所本科院校)的学生使用。值得注意的是，这两所大学的学生群体中都有相当数量的代表不足群体的成员，而且合作利用了针对这些学生的现有项目

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