Operator splitting for generalized KdV equations

We apply the method of operator splitting on the generalized Korteweg--de Vries (KdV) equation $u_{t}+f(u)_{x} +\eps u_{xxx} = 0$, by solving the nonlinear conservation law $u_{t}+f(u)_{x}=0$ and the linear dispersive equation $u_{t}+\eps u_{xxx} = 0$ sequentially. We prove that {\it if} the approximation obtained by operator splitting converges, then the limit function is a weak solution of the generalized KdV equation. Convergence properties are analyzed numerically by studying the effect of combining different numerical methods for each of the simplified problems.