The Cauchy Problem for Two Nonlinear Evolution Equations

In this work, we study the periodic Cauchy problem for two nonlinear evolution equations: The modified Hunter-Saxton equation and the Euler-Poisson equation. Modifying the techniques developed for Euler equations of hydrodynamics, we prove local well-posedness results in Sobolev spaces.

We also investigate the analytic regularity of solutions to these equations and prove Cauchy-Kowalevski type results.

Finally we describe the Hamiltonian structure of the Euler-Poisson equation on a semidirect product space.