In this paper we study the smoothness properties of solutions of some nonlinear equations of Korteweg–de Vries (KdV) type, which are of the form∂tu=a(x, t)u3+f(u2, u1, u, x, t),(1)wherex∈R,uj=∂jxu, andkandjare nonnegative integers. Our principal condition is thata(x, t) be positive and bounded, so that the dispersion is dominant. It is shown under certain additional conditions onaandfthatC∞solutionsu(x, t) are obtained fort>0 if the initial datau(x, 0) decays faster than it does polynomially onR−and has certain initial Sobolev regularity. A quantitative relationship between the rate of decay and the amount of gain of smoothness is given. Lets0be the Sobolev index. If∫Ru2(x, 0)(1+|x−|m)dx<∞(2)for an integerm⩾0 and the solution obeys ‖u‖Hs0<∞ for an existence time 0<t<T, thenu(x, t)∈Hmloc(R) for all 0<t⩽T, andu(x, t)∈L1([0, T];H(m+1)loc(R)). Our method can also be extended to address the fully nonlinear dispersive equations related to (1).