### Abstract or Additional Information

We study the 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 non-convex 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. Joint work with Y. Wang and A. Vainchtein.