The goal of the research that will be supported by this award is to contribute to the next generation of mathematical and computational tools for studying ultrashort solitary waves in optical media. In particular, this research will help characterize the impact of perturbations. For this purpose, the awardee will study three models: the classic cubic nonlinear Schr¨odinger equation with higher order terms and two recently developed models of ultra-short pulses in nonlinear media that both possess solitary wave solutions, namely the short-pulse equation (SPE) derived by Sch¨afer and Wayne in 2004 and the nonlocal short-pulse equation (NSPE) derived by Chung and Sch¨afer in 2007. In the first part of the project, the stability of the solitary waves with respect to perturbations of the initial conditions will be studied. The second part will focus on extensions of these models to more complicated linear and nonlinear response functions. The third part of the project will be devoted to the characterization of the soliton?s response to stochastic variations of the media. As a part of this work, methods to coarse-grain noise in systems with multiple time scales will be developed. All three parts will require a combination of analytical and numerical techniques. A C++ based computational library will be developed to implement the new methods and will be made available freely on the Internet. More broadly, the research will be important not just for optics, but for a variety of scientific areas in which nonlinearity, nonlocality, and randomness meet. As part of the project, undergraduate students will participate in the research, and course material for a new class on the mathematics of optical communications will be developed.
In recent years, experimental success in the creation and detection of ultra-fast optical pulses has opened the door to a new range of optical phenomena that take place on very small scales and hence are extremely fast. Current optical technology allows to design optical devices whose structures are more complex than standard optical fibers. These new devices exhibit remarkable phenomena never seen in standard optical fibers. High bit-rate telecommunications, laser surgery and ultra-broadband generation will benefit from these advances. These potential new applications generate the need for novel mathematical models that describe the such ultra-fast phenomena correctly in a variety of situations. The focus of this research is the question whether such light pulses will remain stable as they propagate through non-perfect wave guides. The research will use a class of mathematical models that was developed by the awardee and his collaborators in 2004.
