Global Well-Posedness of the Short-Pulse and Sine–Gordon Equations in Energy Space

We prove global well-posedness of the short-pulse equation with small initial data in Sobolev space H 2. Our analysis relies on local well-posedness results of Schäfer and Wayne [15], the correspondence of the short-pulse equation to the sine–Gordon equation in characteristic coordinates, and a number of conserved quantities of the short-pulse equation. We also prove local and global well-posedness of the sine–Gordon equation in an appropriate function space.