Venue: 1084 East Hall
Bio: Anna Vainchtein is a professor in the Department of Mathematics at the University of Pittsburgh. She is generally interested in mathematical modeling and analysis of nonlinear phenomena in materials science, physics and biology. Examples include dynamics of phase boundaries, cracks and dislocations in crystals, hysteresis in phase-transforming materials, solitary and heteroclinic traveling waves in nonlinear lattices and DNA overstretching. The resulting mathematical problems typically involve minimization of nonconvex functionals, nonlinear PDEs that change type, dynamical systems with many degrees of freedom and functional differential equations. Thus nonstandard analytical and numerical techniques are required.
We consider a nonlinear mass-spring chain with first and second-neighbor interactions and show that there is a parameter range where solitary waves in this system are strictly supersonic. In these regimes standard quasicontinuum theories, targeting long-wave limits of lattice models, are not adequate since even weak strictly supersonic solitary waves are of envelope type and crucially involve a microscopic scale in addition to the mesoscopic scale of the envelope. To capture this effect in a continuum setting it is necessary to employ unconventional, higher-order quasicontinuum approximations carrying more than one length scale. This talk is based on recent joint work with Lev Truskinovsky (ESPCI).
This seminar is co-sponsored by the Applied & Interdisciplinary Mathematics program. Prof. Vainchtein is being hosted by Prof. Garikipati (ME). If you would like to meet with her during her visit, please send an email to firstname.lastname@example.org.