  We provide a new analytical approach to operator splitting for equations of
the type $u_t=Au+u u_x$ where $A$ is a linear differential operator such that
the equation is well-posed. Particular examples include the viscous Burgers'
equation, the Korteweg-de Vries (KdV) equation, the Benney-Lin equation, and
the Kawahara equation. We show that the Strang splitting method converges with
the expected rate if the initial data are sufficiently regular. In particular,
for the KdV equation we obtain second-order convergence in $H^r$ for initial
data in $H^{r+5}$ with arbitrary $r\ge 1$.
