THE SMOOTH SOLUTIONS OF A CLASS OF COUPLED KDV EQUATIONS

This paper is devoted to the study of the periodic initial boundary value problem and Cauchy problem for the coupled KdV equations. By the Galerkin method and sequential approximation, we get a series of a priori estimates and establish the existence of classical local solution to the periodic problem for the system. Then we obtain the existence and uniqueness of global smooth solution when the coefficients of the system satisfy certain conditions by energy method, conserved quantities and nonconservative quantity $ I(u, v) $.