Local well-posedness for higher order nonlinear dispersive systems

We study nonlinear dispersive systems of the form where k=1, …, n, j ∈ ℤ⁺, and P_k(·) are polynomials having no constant or linear terms. We show that the associated initial value problem is locally well-posed in weighted Sobolev spaces. The method we use is a combination of the smoothing effect of the operator ∂_t + ∂ _x ^(2j+1) and a gauge transformation performed on a linear system, which allows us to consider initial data with arbitrary size. Staffilani was partially supported by NSF grant DMS9304580.