Using the theory developed by Kenig, Ponce, and Vega, we prove that the Hirota-Satsuma system is locally well-posed in Sobolev spaces H(R)× H(R) for 3/4 < s ≤ 1. We introduce some Bourgain-type… (More)

Using the theory of almost conserved energies and the “I-method” developed by Colliander, Keel, Staffilani, Takaoka and Tao, we prove that the initial value problem for a higher order Schrödinger… (More)

We address the question of the uniqueness of solution to the initial value problem associated to the equation ∂tu+ iα∂ 2 xu+ β∂ 3 xu+ iγ|u| 2 u+ δ|u|∂xu+ ǫu 2 ∂xu = 0, x, t ∈ R, and prove that a… (More)

In this work we obtain some a priori estimates for a higher order Schrödinger equation and in particular we obtain some a priori estimates for the modified Korteweg-de Vries equation.