Global existence of small solutions for the fourth-order nonlinear Schrödinger equation

Abstract

We consider the Cauchy problem for the fourth-order nonlinear Schrodinger equation Open image in new window where \(n=1,2\). We prove global existence of small solutions under the growth condition of \(f\left( u\right) \) satisfying \(\left| \partial _{u}^{j}f\left( u\right) \right| \le C\left| u\right| ^{p-j},\) where \(p>1+\frac{4}{n},0\le j\le 3\).