Mathematics PhD Defense: The interplay of dispersion and nonlinearity often leads to formation of nonlinear waves. The Ph.D. thesis focuses on existence, stability and dynamic evolution of several different types of these waveforms in spatially discrete nonlinear systems. In the first part of my thesis, I consider solitary traveling waves, localized nontopological excitations that propagate with amplitude-dependent constant velocity, in a lattice with nonlinear nearest-neighbor bonds and all-to-all harmonic long-range interactions whose strength decays exponentially with distance.

The second part of the thesis concerns another type of nonlinear lattice waves, moving discrete breathers. In addition to propagation velocity, these solutions are characterized by internal oscillations and possess nontrivial oscillatory wings. In the third part of the thesis, it turns to stationary discrete breathers, time-periodic spatially localized waveforms, in a flexible mechanical metamaterial.

Advisor: Anna Vainchtein

Comittee Members:

• Dehua Wang
• Ming Chen
• Panayotis Kevrekisis (U Mass Amherst)