Local Well-posedness of the Derivative Schr¨odinger Equation in Higher Dimension for Any Large Data

In this paper, the authors consider the local well-posedness for the derivative Schr¨odinger equation in higher dimension u_t ? i△u + |u|²(?→γ · ?u) + u²(?→λ · ?u) = 0, (x, t) ∈ Rⁿ × R,?→γ ,?→λ ∈ Rⁿ; n ≥ 2.It is shown that the Cauchy problem of the derivative Schr¨odinger equation in higher dimension is locally well-posed in H^s(Rⁿ) (s > n/2) for any large initial data. Thus this result can compare with that in one dimension except for the endpoint space H^n/2.