We consider the large time asymptotics of solutions to the Cauchy problem for the higher-order nonlinear Schrödinger equation with critical cubic nonlinearity

where \(\lambda \in \mathbb {R}\) and \(\alpha =4\) or \(\alpha \ge 5,\) since while estimating pseudodifferential operators below we need that the condition such that the symbol \(\Lambda \left( \xi \right) =\frac{1}{2}\xi ^{2}+\frac{1}{\alpha }\left| \xi \right| ^{\alpha }\in \mathbf {C}^{5}\left( \mathbb {R}\right) .\) We show that the modified scattering occurs in the uniform norm. We continue to develop the factorization techniques which was started in papers (Ozawa in Commun Math Phys 139(3):479–493, 1991; Hayashi and Ozawa in Ann IHP (Phys Théor) 48:17–37, 1988; Hayashi and Naumkin in Z Angew Math Phys 59(6):1002–1028, 2008; Hayashi and Kaikina in Math Methods Appl Sci 40(5):1573–1597, 2017; Hayashi and Naumkin in J Math Phys 56(9):093502, 2015). The crucial points of our approach presented here are the \(\mathbf {L}^{2}\)-boundedness of the pseudodifferential operators which are used to obtain estimates of nonlinear terms in weighted Sobolev space.

我们考虑具有临界三次非线性的高阶非线性薛定谔方程的柯西问题解的大时间渐近性

其中\(\lambda \in \mathbb {R}\)和\(\alpha =4\)或\(\alpha \ge 5,\)因为在估计下面的伪微分运算符时，我们需要条件使得符号\ (\Lambda \left( \xi \right) =\frac{1}{2}\xi ^{2}+\frac{1}{\alpha }\left| \xi \right| ^{\alpha }\在 \mathbf {C}^{5}\left( \mathbb {R}\right) .\)我们表明修改后的散射发生在均匀范数中。我们继续开发从论文中开始的分解技术（Ozawa in Commun Math Phys 139(3):479–493, 1991; Hayashi and Ozawa in Ann IHP (Phys Théor) 48:17–37, 1988; Hayashi and Naumkin在 Z Angew Math Phys 59(6):1002–1028, 2008; Hayashi and Kaikina in Math Methods Appl Sci 40(5):1573–1597, 2017; Hayashi and Naumkin in J Math Phys 56(9):093502, 20 ）。我们这里提出的方法的关键点是伪微分算子的\(\mathbf {L}^{2}\) -有界性，用于获得加权 Sobolev 空间中非线性项的估计。