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# On the Cauchy problem for a coupled system of KdV equations

• We study some questions related to the well-posedness for the initial value problem associated to the system

$u_{t}+u_{x x x}+a_3 v_{x x x}+u u_{x}+a_1 v v_{x}+a_2(uv)_x =0,$

$b_1 v_{t}+v_{x x x}+b_2 a_3 u_{x x x}+v v_{x}+b_2 a_2 u u_{x}+b_2 a_1(uv)_x=0.$

Using recent methods, we prove a sharp local result in Sobolev spaces. We also prove global result under some conditions on the coefficients.

Mathematics Subject Classification: 35Q35, 35Q53.

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