Collaborative Research RUI: Dynamics of Soliton Interactions and Applications

本研究计划结合解析与数值技术，探讨向量非线性薛定谔系统（包括连续与离散）、短脉冲方程与耦合Maxwell-Bloch方程的解与孤子相互作用。非线性薛定谔方程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，因此可以通过逆散射变换（IST）方法求解。然而，这种方法的矢量NLS在正常色散制度（散焦制度）尚未完全开发。因此，这种方法的完整开发和孤子相互作用的描述是该项目的主要部分。作为原子物理学、激光物理学和光学中的模型而出现的耦合Maxwell-Bloch系统也将通过IST方法的适当调整来研究。此外，IST方法将扩展到最近推导的短脉冲方程，它提供了一个更合适的模型比准单色近似。这些系统的IST理论发展和孤子动力学的阐明将通过数值模拟和构造特殊解的直接方法来指导，由于它们在水波、磁自旋波、光纤、波导和玻色-爱因斯坦凝聚等不同领域的物理适用性，NLS和本项目中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）解的动力学也是非常有趣的，因为矢量孤子的相互作用为设计受控逻辑门和全光计算机、自诱导透明现象以及多级介质中光的偏振切换机制奠定了实验基础。 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的发展将在统一的数学框架中推进对复杂物理现象的基本理解，并提供有关此类系统行为的具体信息。该研究计划的合作性质，无论是主要研究人员之间，并与同事在附近的机构，将有助于加快发展一个相互联系的研究社区访问学生在蒙特克莱尔州立大学和科罗拉多大学在科罗拉多泉（两个本科院校）。值得注意的是，这两个机构的学生群体包括大量代表性不足的群体，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