Research
Featured Research
Anna Vainchtein - Nonlinear waves in nonintegrable lattices
Many physical and biological systems are both spatially discrete and nonlinear. Examples include crystal lattices, biopolymers, and nanoelectromechanical devices. The interplay of the discrete structure and nonlinearity often leads to the formation of spatially localized nonlinear waves that carry or trap energy. My research work, funded by the National Science Foundation, aims to advance fundamental understanding of properties and stability of nonlinear waves in discrete structures.
As part of this research program, I am developing a new analytical approach to illuminate the essential role of discreteness in the dynamics of solitary waves in nonintegrable Hamiltonian lattices. The goal is to clarify how this dynamics is affected by lattice dimension, nonlocality, nonconvexity, anisotropy, and heterogeneity, which often play an important role in real physical and biological discrete systems. The work is accomplished in part by constructing semi-analytical solutions for one- and two-dimensional lattices with piecewise linear interactions. This provides insight into fully nonlinear lattice problems, which are studied through a combination of asymptotic and numerical methods. I am also studying a particularly interesting class of nonintegrable lattices corresponding to locally resonant granular crystals, a new type of granular material that has additional internal degrees of freedom. Through a combination of analytical, numerical, and experimental work, my collaborators and I investigate the dynamics of solitary waves, breathers, and other nonlinear waves in these systems, focusing on the phenomena of energy trapping, wave redirection, and attenuation, which are of both fundamental and practical interest.
New Research Grants
Steady stratified water waves and asymptotic models
Principle Investigator: Ming Chen
Sponsor: NSF
Convex geometry, algebraic geometry and symplectic geometry
Principle Investigator: Kiumars Kaveh
Sponsor NSF
Convex geometry, algebraic geometry and symplectic geometry
Principle Investigator: Kiumars Kaveh
Sponsor: Simons
CDS&E: Appraisal of Subgrid Scale Closures in Reacting
Principle Investigator: William Layton
Sponsor: NSF
Singular limits with geometric effects
Principle Investigator: Marta Lewicka
Sponsor: NSF
Multiple time scales, coupling properties, and network interactions in respiratory rythmicity
Principle Investigator: Jonathan Rubin
Sponsor: NSF
Hyperbolic Conservation Laws and Applications
Principle Investigator: Dehua Wang
Sponsor: NSF
Some Mathematical Finance Problems Under Model Uncertainty
Principle Investigator: Song Yao
Sponsor: NSF