*** Special LCDS & PDE Seminar ***
Lefschetz Center for Dynamical Systems Seminar
Abstract: The following scenario has been seen in many non-integrable, dispersive, nonlinear PDE over the last 25 years: two solitary waves are propagated on a collision course. Above some critical velocity vc, they simply bounce off each other. Below vc they may be captured and merge into a single localized mass, or they may interact a finite number of times before escaping each other's embrace. Whether they are captured, and how many times the solitary waves interact before escape, depends on the initial velocity in a complicated manner, often remarked, though never previously shown, to be a fractal (a chaotic scattering process). This has been observed in coupled NLS, sine-Gordon, phi-4, and others. These PDE systems are commonly studied by (nonrigorously) deriving a reduced set of ODE that numerically reproduce this behavior. Using matched asymptotics and Melnikov integrals, we give asymptotic formulas for the critical and for certain salient features of the fractal structure. We derive a discrete-time iterated map through which the entire structure can be unravelled. Surprising connections are made to other well-known dynamical systems.
Joint Solid Mechanics / Applied Math Seminar Series
Note: There will be a dinner for Prof. Sung after the seminar. Please contact Ms. Pat Capece at X1501 if you wish to attend. |
Abstract: Due to structural connectivity and flexibility, certain biological systems at the mesoscale manifest interesting cooperative dynamics under certain barriers caused by external fields and confining and constraining environments. Cooperative dynamics is important, not only in understanding how a biological system self-organizes by manipulating its flexible degrees of freedom, but also in a multitude of biotechnological applications. Nature utilizes the ambient fluctuations in biological soft-condensed matter to facilitate the crossing of seemingly insurmountable barriers, using shape changes and coupling of the collective modes of the fluctuations. As examples I will talk about polymer dynamics through membranes and potential barriers, bubble formation in double-stranded DNA, membrane fusion, and, if time permits, blood flow within narrow vessels, which we have studied.
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