˖ڍመڙጲ
http://www.paper.edu.cn
四阶非线性薛定谔方程的拟周期解
高美娜
1
，刘建军
2
1
上海第二工业大学文理学部，上海 201209
2
四川大学数学学院，成都 610065
摘要：本文研究周期边条件的立方非线性项的四阶薛定谔方程：
iu
t
= ∂
4
x
u ± |u|
2
u, x ∈ T := R/2πZ.
得到小振幅拟周期解构成的不变康托流形。
关键词：基础数学，薛定谔方程，拟周期解
中图分类号： O175.14, O175.29
Quasi-periodic solutions for the fourth order
nonlinear Schr
¨
odinger equation
GAO Meina
1
, LIU Jianjun
2
1
College of Arts and Sciences, Shanghai Polytechnic University, Shanghai 201209
2
School of Mathematics, Sichuan University, Chengdu 610065
Abstract: This paper is concerned with the cubic fourth order nonlinear Schr
¨
odinger
equation with periodic boundary conditions
iu
t
= ∂
4
x
u ± |u|
2
u, x ∈ T := R/2πZ.
We show the above equation possesses Cantor families of quasi-periodic solutions of small
amplitude.
Key words: Fundamental Mathematics, Schr
¨
odinger Equation, Quasi-periodic Solution
0 Introduction
This paper is concerned with the cubic fourth order nonlinear Schr
¨
odinger equation with
periodic boundary conditions
iu
t
= ∂
4
x
u ± |u|
2
u, (t, x) ∈ T × R, (1)
where u is a complex-valued function on T × R. The equation is also called the biharmonic
NLS and it was studied in [1, 2] in the context of stability of solitons in magnetic materials.
Foundations: National Research Foundation for the Doctoral Program of Higher Education of China (20130181120104)
Author Introduction: GAO Meina（1983-），female，associate professor，major research direction：dynamical system，
email：mngao@sspu.edu.cn. Correspondence author：LIU Jianjun （1983-），male，professor，major research direction：
dynamical system，email：071018018@fudan.edu.cn.
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