Well-Posedness of a Nonlinear Wave Equations in Characteristic Coordinates

Abstract

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 Schafer and Wayne, the correspondence of the short-pulse equation to the sine-Gordon equation in characteristic coordinates, and conserved quantities of the short-pulse equation. We perform numerical computations to illustrate this result. We also prove local and global well-posedness of the sine-Gordon equation in an appropriate vector space.