We study nonlinear dispersive equations of the form \[ ∂ t u + ∂ x 2 j + 1 u + P ( u , ∂ x u , … , ∂ x 2 j u ) = 0 , x , t ∈ R , j ∈ Z + , {\partial _t}u + \partial _x^{2j + 1}u + P(u,{\partial _x}u, \ldots ,\partial _x^{2j}u) = 0,\qquad x,t \in \mathbb {R},\quad j \in {\mathbb {Z}^ + }, \] where P ( ⋅ ) P( \cdot ) is a polynomial having no constant or linear terms. It is shown that the associated initial value problem is locally well posed in weighted Sobolev spaces. The method of proof combines several sharp estimates for solutions of the associated linear problem and a change of dependent variable which allows us to consider data of arbitrary size.