一类非线性Schrödinger-KdV微扰系统的初值问题

裴一潼; 王锦坤; 郭柏灵; 刘伍明

doi:10.7498/aps.72.20230241

摘要

Korteweg-de Vries (KdV)方程是一种数学模型, 用于描述色散介质中长波的传播. 而非线性薛定谔(NLS)方程模拟了由短色散波组成的窄带宽波包的动态, 它是描述许多物理系统的有用模型, 包括玻色-爱因斯坦凝聚、光纤和水波等. 将KdV和NLS方程耦合起来的系统可以模拟长波和短波的相互作用. 这个系统在物理和数学上很有吸引力, 它结合了两个模型的优点. KdV方程描述的长波可以影响NLS方程描述的短波的行为, 而短波反过来也可以影响长波的行为. 这样一个耦合系统在过去的几十年中得到了广泛的研究, 并为许多物理系统带来了重要的影响. 本文在Bernard等工作(Bernard D, Nghiem V N, Benjamin L S 2016 J. Phys. A: Math. Theor. 49 415501)的基础上考虑了KdV非线性Schrödinger微扰系统柯西问题局部解的存在性, 并给出了解的存在空间.

关键词:

Abstract

The Korteweg-de Vries (KdV) equation is a mathematical model that describes the propagation of long waves in dispersive media. It takes into account both nonlinearity and dispersion, and is particularly useful for modeling phenomena like solitons. The nonlinear Schrödinger (NLS) equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. It is a useful model for describing many physical systems, including Bose-Einstein condensates, optical fibers, and water waves. A system that couples the KdV and NLS equations can model the interaction of long and short waves. This system combines the strengths of both models. The long waves described by the KdV equation can affect the behavior of the short waves described by the NLS equation, while the short waves can in turn affect the behavior of the long waves. Such a coupled system has been studied extensively over the last few decades, and has led to important insights into many physical systems. This paper considers the existence of local solutions to the Cauchy problem of KdV-Schrödinger nonlinear system on the basis of literature (Bernard D, Nghiem V N, Benjamin L S 2016 J. Phys. A: Math. Theor. 49 415501), and also gives the existence space of the local solutions.

Keywords:

作者及机构信息

1.
南京航空航天大学数学学院, 南京　211106

2.
中国科学院物理研究所, 北京凝聚态物理国家研究中心, 北京　100190

3.
北京应用物理与计算数学研究所, 北京　100088

通信作者: 刘伍明, wmliu@iphy.ac.cn

基金项目: 国家重点研发计划(批准号: 2021YFA1400900, 2021YFA0718300, 2021YFA1402100)、国家自然科学基金(批准号: 61835013, 12234012)和中国载人航天计划空间应用系统资助的课题

Authors and contacts

1.
School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

2.
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

3.
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Corresponding author: Liu Wu-Ming, wmliu@iphy.ac.cn

Funds: Project supported by the National Key R&D Program of China (Grant Nos. 2021YFA1400900, 2021YFA0718300, 2021YFA1402100), the National Natural Science Foundation of China (Grant Nos. 61835013, 12234012), and the Space Application System of China Manned Space Program

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参考文献

[1]	Gardner C S, Greene J M, Kruskal M D, Miura R M 1967 Phys. Rev. Lett. 19 1095 Google Scholar
[2]	Shabat A, Zakharov V 1972 Sov. Phys. JETP 34 62
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[16]	Pei Y, Guo B, Liu W 2022 Math. Methods Appl Sci. 45 4853 Google Scholar
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施引文献

[1]	Gardner C S, Greene J M, Kruskal M D, Miura R M 1967 Phys. Rev. Lett. 19 1095 Google Scholar
[2]	Shabat A, Zakharov V 1972 Sov. Phys. JETP 34 62
[3]	Bernard D, Nghiem V N, Benjamin L S 2016 J. Phys. A: Math. Theor. 49 415501 Google Scholar
[4]	Albert J, Bhattarai S 2013 Adv. Differ. Equ. 18 112964 Google Scholar
[5]	Angulo Pava J 2006 Electron. J. Differ. Equ. 72 1 Google Scholar
[6]	Angulo Pava J, Matheus C, Pilod D 2009 Commun. Pure Appl. Anal. 8 81544 Google Scholar
[7]	Chen L 1999 J. Partial Differ. Equ. 12 11
[8]	Corcho A J, Linares F 2007 Trans. Am. Math. Soc. 359 4089106 Google Scholar
[9]	Dias J P, Figueira M, Oliveira F 2010 Nonlinear Anal. 73 268698 Google Scholar
[10]	Appert K, Vaclavik J 1977 Phys. Fluids 20 18459 Google Scholar
[11]	Ikezi H, Nishikawa K, Hojo H, Mima K 1974 Proc. 5th Int. Conf. on Plasma Phys. and Contr. Nucl. Fusion Res. (Tokyo) 23948
[12]	Nishikawa K, Hojo H, Mima K, Ikezi H 1974 Phys. Rev. Lett. 33 14851 Google Scholar
[13]	Bubnov B A 1979 Differ. Equ. 15 17
[14]	Bubnov B A 1980 Differ. Equ. 16 24
[15]	Zhang B Y 1994 Int. Ser. Numer. Math. 118 371
[16]	Pei Y, Guo B, Liu W 2022 Math. Methods Appl Sci. 45 4853 Google Scholar
[17]	Capistrano-Filho R A, Sun S M, Zhang B Y 2018 Math. Control Relat. Fields 8 583 Google Scholar
[18]	Bona J L, Dougalis V A 1980 J. Math. Anal. Appl. 75 503 Google Scholar
[19]	Kumar V, Alqahtani A 2017 Nonlinear Dyn. 90 2903 Google Scholar
[20]	Kumar V, Alqahtani A 2019 Adv. Differ. Equ. 1 1 Google Scholar

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