On the Cauchy problem for a coupled system of KdV equations

Felipe Linares; M. Panthee

doi:10.3934/cpaa.2004.3.417

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2004, Volume 3, Issue 3: 417-431. Doi: 10.3934/cpaa.2004.3.417

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On the Cauchy problem for a coupled system of KdV equations

Felipe Linares^1, and
M. Panthee^2,

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

Received: July 31, 2003

Revised: September 30, 2003

Published: August 2004

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Abstract

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.

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Mathematics Subject Classification: 35Q35, 35Q53.

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