The interplay of spatial discreteness and nonlinearity plays an important role in the dynamics of nonlinear lattice systems, as illustrated by two problems considered in this thesis.
The first problem concerns kinetics of a step propagating along a twin boundary in a cubic lattice undergoing an antiplane shear deformation. To model twinning, we consider a piecewise quadratic double-well interaction potential with respect to one component of the shear strain and harmonic interaction with respect to another. We construct semi-analytical traveling wave solutions that correspond to a steady step propagation and obtain the kinetic relation between the applied stress and the velocity of the step. We show that this relation strongly depends on the width of the spinodal region where the double-well potential is nonconvex and on the material anisotropy parameter. In the limiting case when the spinodal region degenerates to a point, we construct new solutions that extend the kinetic relation obtained in the earlier work of Celli, Flytzanis and Ishioka into the low-velocity regime. Numerical simulations suggest stability of some of the obtained solutions, including low-velocity step motion when the spinodal region is sufficiently wide. When the applied stress is above a certain threshold, nucleation and steady propagation of multiple steps are observed.
In the second part of the thesis we explore a novel locally resonant granular material bearing harmonic internal resonators in a chain of beads interacting via Hertzian elastic contacts. Dynamics of the system can range from strongly to weakly nonlinear, depending on the solution amplitude and the amount of static precompression. We provide numerical and analytical evidence of the existence of discrete dark breathers solutions, exponentially localized, time-periodic states mounted on top of a non-vanishing background. Our results are obtained by means of an asymptotic reduction to suitably modified versions of the discrete p-Schr\"{o}dinger and nonlinear Schr\"{o}dinger modulation equations in the strongly and weakly nonlinear regime, respectively. Stability and bifurcation structure of numerically computed exact dark breathers are examined. In some parameter regimes we also find small-amplitude periodic traveling waves, long-lived standing and traveling bright breathers, i.e., localized time-periodic solutions with exponentially decaying tails
