Computational Theory and Methods for Finding Multiple Solutions to Differential Systems

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Dr. Zhou, the principal investigator, proposes to develop computational theory and methods for finding multiple unstable solutions to PDE (eigen) systems. Multiple unstable solutions, lowly or highly, singly or multiply excited, to many nonlinear systems have been observed with very different instabilities, maneuverabilities and configurations. Hence they imply a variety of applications. When multi-bodies (particles, molecules, species, etc.) are involved in a system, they may interact with each other in many different ways, such as strongly coupled vs. weakly coupled, cooperative vs. non-cooperative, definite vs. strongly indefinite, variational vs. non-variational, etc. Thus mathematically systems can be classified into many different types. Based on the results obtained from the previous NSF funded project and several successful preliminary investigations, the PI proposes a new game theory methodology that treats each body in a system as a player. According to the interactions among all bodies, two-level local optimization methods will be developed to solve proposed problems and mathematical justification of those methods will be established. Several important model problems in Bose-Einstein condensates, nonlinear optics, biology and non-Darcian/Newtonian fluids/materials are proposed to solve for testing the new theory and methods. The instability index of unstable solutions is important information for application and thus will be analyzed. Unstable solutions appear only in a short time period, but may have much higher performance index. They used to be considered too hard to catch by traditional technologies or numerical algorithms and thus too difficult to apply. Thanks to new technologies (synchrotronic, laser), scientists are now able to induce or reach various unstable solutions and search for NEW applications, in particular, in system design and control of EMERGENCY machineries for MISSION CRITICAL SITUATIONS. So far, understanding of such solutions is still quite limited and analytic solutions are too difficult to obtain. Thus, development of efficient and reliable numerical methods to solve such problems for multiple solutions in an order becomes very important to both research and applications. However, there is no theory in literature to devise such a method. The outcome of this project will (1) greatly enhance understanding of unstable solutions by mathematical characterizations and lay a solid foundation for future study, (2) provide efficient and reliable algorithms to compute multiple solutions so that people can compare and then select the best one, and (3) promote new applications with new solution configurations.