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D. Pelinovsky and A. Sakovich

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

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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.