# Local Well-posedness of the Coupled KdV-KdV Systems on $\mathbb{R}$

## TL;DR

This paper classifies the well-posedness of coupled KdV systems on the real line in Sobolev spaces, identifying critical indices and establishing sharp bilinear estimates for different coefficient configurations.

## Contribution

It provides a complete classification of coupled KdV systems' well-posedness based on system coefficients and Sobolev space regularity, extending known results for single KdV.

## Key findings

- Identified four critical regularity indices for coupled KdV systems.
- Established sharp bilinear estimates for all relevant cases.
- Classified systems into four well-posedness categories based on coefficients.

## Abstract

Inspired by the recent successful completion of the study of the well-posedness theory for the Cauchy problem of the Korteweg-de Vries (KdV) equation \[ u_t +uu_x +u_{xxx}=0, \quad \left. u \right |_{t=0}=u_{0} \] in the space $H^{s} (\mathbb{R})$ (or $H^{s} (\mathbb{T})$), we study the well-posedness of the Cauchy problem for a class of coupled KdV-KdV (cKdV) systems \[\left\{\begin{array}{rcl} u_t+a_{1}u_{xxx} &=& c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x},\\ v_t+a_{2}v_{xxx}&=& c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x},\\ \left. (u,v)\right |_{t=0} &=& (u_{0},v_{0}) \end{array}\right.\] in the space $\mathcal{H}^s (\mathbb{R}) := H^s (\mathbb{R})\times H^s (\mathbb{R})$. Typical examples include the Gear-Grimshaw system, the Hirota-Satsuma system and the Majda-Biello system, to name a few. In this paper we look for those values of $s\in \mathbb{R}$ for which the cKdV systems are well-posed in $\mathcal{H}^s (\mathbb{R})$. Our findings enable us to provide a complete classification for the cKdV systems in terms of the analytical well-posedness in $\mathcal{H}^s (\mathbb{R})$ based on its coefficients $a_i$, $c_{ij}$ and $d_{ij}$ for $i,j=1,2$. The key ingredients in the proofs are the bilinear estimates under the Fourier restriction space norms. There are four types of the bilinear estimates that need to be investigated. Sharp results are established for all of them. In contrast to the lone critical index $-\frac{3}{4}$ for the single KdV equation, the critical indexes for the cKdV systems are $-\frac{13}{12}$, $-\frac{3}{4}$, $0$ and $\frac{3}{4}$. As a result, the cKdV systems are classified into four classes, each of which corresponds to a unique index $s^{*}\in\{-\frac{13}{12},\,-\frac{3}{4},\,0,\,\frac{3}{4}\}$ such that any system in this class is locally analytically well-posed if $s>s^{*}$ while the bilinear estimate fails if $s<s^{*}$.