D. Pelinovsky and A. Sakovich

Global well-posedness of the short-pulse and sine-Gordon equations in energy space

Communications in PDE 35, 613-629 (2010)

Abstract:
We prove global well-posedness of the short-pulse equation with small initial data in Sobolev space H^s for an integer s >= 2. Our analysis relies on local well-posedness results of Schafer & Wayne (2004), 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 vector space.

Keywords:
Short-pulse equation, sine-Gordon equation in characterstic coordinates, well-posedness, conserved quantities, wave breaking.