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

In this paper we establish local well-posedness in the Sobolev space Hs(Rn)Hs(Rn) with s>s0s>s0 for a general class of nonlinear dispersive equations of the type ∂tu−iP(Dx)u=F(u), where P(Dx)P(Dx) is an elliptic differential operator on RnRn with a real symbol, F(u)F(u) is a nonlinear function which behaves like |u|σu|u|σu for some constant σ>0σ>0, and s0s0 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 Schrödinger equation i∂tu+a△u+b△2u=c|u|σu.