Nonlinear Shape Oscillations in Compact Systems

Northwestern State University Louisiana, Natchitoches LA

Investigators

Abstract

PHY-0140274 Ludu This 2-year research plan is focused on modeling and understanding some classes of nonlinear phenomena at the surface of fluid objects, namely large amplitude shape oscillations like solitary waves on compact surfaces. Solitary waves are the only singular traveling stable excitations known in quantum field theory. However, the existence of such special solution on compact (sphere-like) objects is still an open question. These topics are related with PI's long experience in nonlinear physics, especially with the successful introduction of the "roton" model in order to explain spontaneous alpha- and cluster-decay of heavy ions (Phys. Rev. Lett. 80 (1998) 2125). The aim here is to extend this theoretical model and to apply it to any compact (almost spherical) system, in order to explain and predict its shape, modes, patterns, and its strange nonlinear behavior. In nonlinear science, finding an exact solution, or even formal classification of possible evolution paths of a system, is rather the exception than the rule. This is the basic reason to investigate such new exact solitary wave solutions for spherical objects, of different space scales from heavy nuclei, atomic clusters, biological cells, macroscopic liquid drops, shells, bubbles to neutron stars. Last but not the least, these results will be applied to the recently discovered new form of matter, Bose Einstein-Condensates (BEC drops). All in all we'll try to predict the evolution of such systems, their modes of motion, possible catastrophes, break-up processes, fission, etc., beyond the limits of linear approximation. This will be described theoretically in terms of causal fluid equations (also called Hamiltonian systems). Exact traveling singular stable solutions will be investigated (soliton theory) and later on the equations will be solved numerically using large computer codes. The behavior of the free surface of such compact fluid systems, that is drop-like objects or fluid layers is of special interest in many natural and industrial processes, from cell division in biology or spraying and atomization to exotic nuclear shapes, exotic radioactivity, or impact between stellar objects. The basic aim in the study of such nonlinear collective modes is the understanding of the interaction of the bulk with the boundaries. Another aim is the understanding of the transitions between different modes and patterns. A generalized KdV equation, valid for different geometries, including viscoelastic properties and damping is introduced and solved. The theoretical tools used in this research will have a certain degree of novelty, since there will be used self-similarity techniques in order to solve such NPDE. This research is expected to impact the teaching and learning enterprise through the student research, and through building international collaborations.