Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces [formula omitted] and applications

Abstract

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 ∂ t u − i P ( 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 | σ u for some constant σ > 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 Schrödinger equation i ∂ t u + a △ u + b △ 2 u = c | u | σ u .