Well-posedness of higher-order nonlinear Schrodinger equations in Sobolev spaces Hs (Rn) and applications

In this paper we establish local well-posedness in the Sobolev space H-s (R-n) with s > s(0) for a general class of nonlinear dispersive equations of the type partial derivative(t)u - iP(D-x)u = F(u), where P(D-x) is an elliptic differential operator on R-n with a real symbol, F(u) is a nonlinear function which behaves like \u\(sigma) u for some constant sigma > 0, and s(0) is a critical index suggested by a standard scaling argument. By using such local result and conservation laws, we improve the known and obtain some new global well-posedness results for the fourth-order nonlinear Schrodinger equation i partial derivative(t)u + a Delta u + b Delta(2) u = c\u\(sigma) u. (C) 2006 Elsevier Ltd. All rights reserved.