On the Cauchy problem for a coupled system of KdV equations

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September 2004, 3(3): 417-431. doi: 10.3934/cpaa.2004.3.417

On the Cauchy problem for a coupled system of KdV equations

1.	IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil
2.	Instituto de Matemática Pura e Aplicada, 22460-320, Rio de Janeiro, Brazil

Received August 2003 Revised October 2003 Published June 2004

Abstract
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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.

Keywords: Well-posedness, dispersive equations, Korteweg-de Vries equation.

Mathematics Subject Classification: 35Q35, 35Q53.

Citation: Felipe Linares, M. Panthee. On the Cauchy problem for a coupled system of KdV equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 417-431. doi: 10.3934/cpaa.2004.3.417

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