Maximisers for Strichartz inequalities on the torus

We study the existence of maximisers for a one-parameter family of Strichartz inequalities on the torus. In general, maximising sequences can fail to be precompact in ${L}^{2}(\mathbb{T})$, and maximisers can fail to exist. We provide a sufficient condition for precompactness of maximising sequences (after translation in Fourier space), and verify the existence of maximisers for a range of values of the parameter. Maximisers for the Strichartz inequalities correspond to stable, periodic (in space and time) solutions of a model equation for optical pulses in a dispersion-managed fiber.