Geometry and the nonlinear elasticity of defects in smectic liquid crystals

Rotation invariance imposes nonlinearities in the elastic strain of smectic liquid crystals. Though often neglected, in smectic liquid crystals these nonlinearities can generate new qualitative behavior, especially in the presence of defects such as dislocations. By exploiting geometry, I describe exact results on edge dislocations, and demonstrate a nonlinear superposition principle for certain multiple defect configurations. Though there are few exact results analogous to those of edge dislocations, results on twist-grain boundaries hint at an approximate superposition principle for multiple screw dislocations also. These superpositions, which appear to be related to the theory of minimal surfaces, exhibit unexpected symmetries that are still poorly understood.