Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities.

In this paper, coupled systems ut + uxxx + P(u, v)x = 0, vt + vxxx + Q(u, v)x = 0, of KdV-type are considered, where u = u(x, t), v = v(x, t) are real-valued functions and x, t ∈ R. Here, subscripts connote partial differentiation and P(u, v) = Au2 + Buv + Cv2 and Q(u, v) = Du2 + Euv + F v2 are quadratic polynomials in the variables u and v. Attention is given to the pure initial-value problem in which u(x, t) and v(x, t) are both specified at t = 0, viz. u(x, 0) = u0(x) and v(x, 0) = v0(x) for x ∈ R. Under suitable conditions on P and Q, global well-posedness of this problem is established for initial data in the L2 -based Sobolev spaces Hs (R) × Hs (R) for any s > − 3 4 .