Abstract: Is this Paper, the global existence of smooth solutions to the initial value problem for the fourth order nonlinear Schrödinger equation in the Lax hierarchy of the nonlinear Schrödinger equation(NLS equation) is established by using the so-called continuation method and delicate a priori estimate. In addition, the asymptotic properties of the solutions as |x|→+∞ are discussed.

Key words: Forth order nonlinear schrodinger equation, Initial value problem, Global existence, Asymptotic behavior