In this paper, coupled systems of Korteweg–de Vries type are considered, where , are real-valued functions and where . Here, subscripts connote partial differentiation and are quadratic polynomials in the variables and . Attention is given to the pure initial-value problem in which and are both specified at , namely, for . Under suitable conditions on and , global well-posedness of this problem is established for initial data in the -based Sobolev spaces for any .
"Global well-posedness for a system of KdV-type equations with coupled quadratic nonlinearities." Nagoya Math. J. 215 67 - 149, September 2014. https://doi.org/10.1215/00277630-2691901