Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems \[ u_t+a_1u_{ xxx} = c_{ 11}uu_x+c_{ 12}vv_x+d_{ 11}u_{ x}v+d_{ 12}uv_{ x}, \quad u(x,0)= u_0(x) \] \[ v_t+a_2v_{ xxx}= c_{ 21}uu_x+c_{ 22}vv_x +d_{ 21}u_{ x}v+d_{ 22}uv_{ x}, \quad v(x,0)=v_0(x)\] posed on the torus $\mathbb{ T}$ in the spaces \[ { \cal H}^s_1:=H^s_0 (\mathbb{ T})\times H^s_0 (\mathbb{ T}), \quad { \cal H}^s_2:=H^s_0 (\mathbb{ T})\times H^s(\mathbb{ T}), \quad { \cal H}^s_3:=H^s (\mathbb{ T})\times H^s_0 (\mathbb{ T}), \quad { \cal H}^s_4:=H^s (\mathbb{ T})\times H^s (\mathbb{ T}).\] For $k=1,2,3,4$, it is shown that for given $a_1$, $a_2$, $(c_{ ij})$ and $(d_{ ij})$, there exists a unique $s^*_k \in (-\infty, +\infty]$, called the critical index, such that the system is analytically well-posed in $\cal{ H}^s_k$ for $s>s^*_k$ while the bilinear estimate, the key for the proof of the analytical well-posedness, fails if $s<s^{ *}_k$. Viewing the critical index $s^*_k$ as a function of the coefficients $a_1$, $a_2$, $(c_{ ij})$ and $(d_{ ij})$, its range $\cal{ C}_k$ is called the critical index set for the analytical well-posedness of the system in the space $\cal{ H}^s_k$. Invoking some classical results of Diophantine approximation in number theory, we are able to identify that \[ \mbox{ $ { \cal C}_1= \left \{ -\frac12, \infty \right\} \bigcup \left \{ \alpha: \frac12\leq \alpha\leq 1 \right \}$ } \quad\text{ and}\quad \mbox{ ${ \cal C}_q= \left \{ -\frac12, -\frac14, \infty \right\} \bigcup \left \{ \alpha: \frac12\leq \alpha\leq 1 \right \}$ $\quad$ for $\quad$ $q=2,3,4$.}\] This is in sharp contrast to the $R$ case in which the critical index set ${ \cal C}$ for the analytical well-posedness of in the space $H^s (R)\times H^s (R)$ consists of exactly four numbers: $ { \cal C}=\left \{ -\frac{ 13}{ 12}, -\frac34, 0, \frac34 \right \}.$
Categories: math.AP