Modified scattering for the higher-order anisotropic nonlinear Schrödinger equation in two space dimensions

Abstract

We study the asymptotic behavior of solutions to the Cauchy problem for the higher-order anisotropic nonlinear Schrödinger equation in two space dimensions $i{\partial }_{t}u+\frac{1}{2}\mathrm{\Delta }u-\frac{1}{4}{\partial }_{x_1}^{4}u=\lambda \left|u\right|u$ , t > 0, $x\in {\mathbb{R}}^2,$ with initial data $u\left(0,x\right)=u_0\left(x\right)$ , $x\in {\mathbb{R}}^2,$ where $\lambda \in \mathbb{R}$ . We will show the modified scattering for solutions. We continue to develop the factorization techniques, which were started in the papers of N. Hayashi and P. I. Naumkin [Z. Angew. Math. Phys. 59(6), 1002–1028 (2008); J. Math. Phys. 56(9), 093502 (2015)], N. Hayashi and T. Ozawa [Ann. I.H.P.: Phys. Theor. 48, 17–37 (1988)], and T. Ozawa [Commun. Math. Phys. 139(3), 479–493 (1991)]. The crucial point of our approach presented here is the L²-boundedness of the pseudodifferential operators.