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

Abstract

In this paper, coupled systems $$u_t+u_{xxx}+P(u,v{)}_x=0,$$ $$v_t+v_{xxx}+Q(u,v{)}_x=0$$ of Korteweg–de Vries type are considered, where $u=u(x,t)$, $v=v(x,t)$ are real-valued functions and where $x,t\in \mathbb{R}$. Here, subscripts connote partial differentiation and $$P(u,v)=A{u}^2+Buv+C{v}^2\text{and}Q(u,v)=D{u}^2+Euv+F{v}^2$$ 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 $t=0$, namely, $$u(x,0)=u_0\left(x\right)\text{and}v(x,0)=v_0\left(x\right),$$ for $x\in \mathbb{R}$. Under suitable conditions on $P$ and $Q$, global well-posedness of this problem is established for initial data in the $L^2$-based Sobolev spaces $H^s\left(\mathbb{R}\right)\times H^s\left(\mathbb{R}\right)$ for any $s>-3/4$.