On The Local Well-Posedness for Some Systems of Coupled KdV Equations

Details On The Local Well-Posedness for Some Systems of Coupled KdV Equations On The Local Well-Posedness for Some Systems of Coupled KdV Equations

2007-05-03

arxiv.org/abs/0705.0482v1

Mathematics

Analysis of PDEs

26 pages

Scientific paper

Using the theory developed by Kenig, Ponce, and Vega, we prove that the Hirota-Satsuma system is locally well-posed in Sobolev spaces $H^s(\mathbb{R}) \times H^{s}(\mathbb{R})$ for $3/4-3/4$, by establishing new mixed-bilinear estimates involving the two Bourgain-type spaces $X_{s,b}^{-\alpha_-}$ and $X_{s,b}^{-\alpha_+}$ adapted to $\partial_t+\alpha_-\partial_x^3$ and $\partial_t+\alpha_+\partial_x^3$ respectively, where $|\alpha_+|=|\alpha_-|\not = 0$.

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