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

Abstract

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 $3/4<s\le 1$. We introduce some Bourgain-type spaces $X_{s,b}^a$ for $a\ne 0$, $s,b\in \mathbb{R}$ to obtain local well-posedness for the Gear-Grimshaw system in $H^s(\mathbb{R})\times H^s(\mathbb{R})$ for $s>-3/4$, 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 $|{\alpha }_{+}|=|{\alpha }_{-}|\ne 0$.

26 pages

References in corpus (1)

• Coupled KdV equations of Hirota-Satsuma type

Cited by in corpus (2)

• Local Well-posedness of the Coupled KdV-KdV Systems on $\mathbb{R}$
• Well-posedness and Critical Index Set of the Cauchy Problem for the Coupled KdV-KdV Systems on $\mathbb{T}$