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

TL;DR

This paper establishes local well-posedness results for certain coupled KdV systems in Sobolev spaces, using Bourgain-type spaces and bilinear estimates, extending the understanding of these nonlinear dispersive equations.

Contribution

It introduces new Bourgain-type spaces and bilinear estimates to prove local well-posedness for the Hirota-Satsuma and Gear-Grimshaw systems in Sobolev spaces.

Findings

Proves local well-posedness for Hirota-Satsuma system in H^s for 3/4<s≤1.

Establishes local well-posedness for Gear-Grimshaw system in H^s for s>-3/4.

Develops new mixed-bilinear estimates involving adapted Bourgain-type spaces.

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 $3/4<s\le1$. We introduce some Bourgain-type spaces $X_{s,b}^a$ for $a\not =0$, $s,b \in \mathbb{R}$ 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 $|\alpha_+|=|\alpha_-|\not = 0$.