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

文章全文 : translate this paragraph

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
[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

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

[1]	齐乐天, 王世元, 沈明琳, 黄刚毅. 基于Nyström柯西核共轭梯度算法的混沌时间序列预测. , 2022, 71(10): 108401. doi: 10.7498/aps.71.20212274
[2]	李晓静, 严静, 陈绚青, 曹毅. 具有一般非线性弹性力和广义阻尼力的相对转动非线性系统的周期解问题. , 2014, 63(20): 200202. doi: 10.7498/aps.63.200202
[3]	陈丽娟, 鲁世平. 零维气候系统非线性模式的周期解问题. , 2013, 62(20): 200201. doi: 10.7498/aps.62.200201
[4]	李晓静, 陈绚青, 严静. 具时变刚度的相对转动非线性动力系统的周期解问题. , 2013, 62(9): 090202. doi: 10.7498/aps.62.090202
[5]	吴钦宽. 输电线非线性振动问题的同伦映射近似解. , 2011, 60(6): 068802. doi: 10.7498/aps.60.068802
[6]	钱存, 王亮亮, 张解放. 变系数非线性Schrödinger方程的孤子解及其相互作用. , 2011, 60(6): 064214. doi: 10.7498/aps.60.064214
[7]	张民仓, 皇甫国庆. 环状非有心势场中Schr?dinger方程的精确解. , 2010, 59(10): 6819-6823. doi: 10.7498/aps.59.6819
[8]	罗香怡, 刘学深, 丁培柱. 立方非线性Schr?dinger方程的动力学性质研究及其解模式的漂移. , 2007, 56(2): 604-610. doi: 10.7498/aps.56.604
[9]	宗丰德, 戴朝卿, 杨琴, 张解放. 光纤中变系数非线性Schr?dinger方程的孤子解及其应用. , 2006, 55(8): 3805-3812. doi: 10.7498/aps.55.3805
[10]	蔡长英, 任中洲, 鞠国兴. 指数型变化有效质量的三维Schr?dinger方程的解析解. , 2005, 54(6): 2528-2533. doi: 10.7498/aps.54.2528
[11]	蒲利春, 林宗兵, 张雪峰, 王本菊, 姜毅, 严天艳. 非线性Schr?dinger方程组的精确解组. , 2005, 54(10): 4472-4477. doi: 10.7498/aps.54.4472
[12]	刘成仕, 杜兴华. 耦合Klein-Gordon-Schr?dinger方程新的精确解. , 2005, 54(3): 1039-1043. doi: 10.7498/aps.54.1039
[13]	沈守枫, 潘祖梁, 张隽, 叶彩儿. Manakov型非线性Schr?dinger方程的Jacobi椭圆函数包络解. , 2004, 53(7): 2056-2059. doi: 10.7498/aps.53.2056
[14]	李向正, 张金良, 王跃明, 王明亮. 非线性Schr?dinger方程的包络形式解. , 2004, 53(12): 4045-4051. doi: 10.7498/aps.53.4045
[15]	张善卿, 李志斌. 非线性耦合SchrOdinger-KdV方程组新的精确解析解. , 2002, 51(10): 2197-2201. doi: 10.7498/aps.51.2197
[16]	段一士, 俞重远, 吴振森. 双折射光纤中非线性耦合Schr?dinger方程的小振幅孤波解. , 1997, 46(12): 2359-2362. doi: 10.7498/aps.46.2359
[17]	曹庆杰, 张天德, 李久平, G.W.PRICE, K.DJIDJELI, E.H.TWIZELL. 一类广义N维非线性Schr?dinger方程的孤子解及其性质研究. , 1997, 46(11): 2166-2173. doi: 10.7498/aps.46.2166
[18]	周光辉. 无反射势阱的Schr?dinger方程的精确解. , 1993, 42(2): 173-179. doi: 10.7498/aps.42.173
[19]	刘中柱, 黄念宁. 用广田直接法求带高阶修正的扩充的非线性Schr?dinger方程的孤子解. , 1991, 40(1): 1-7. doi: 10.7498/aps.40.1
[20]	刘韵吉, 郭常霖. 一种用加权高斯-柯西组合函数拟合对称衍射线形测定晶粒大小的方法. , 1985, 34(9): 1156-1165. doi: 10.7498/aps.34.1156

计量

文章访问数: 3197
PDF下载量: 134
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板