本文主要研究内容
作者薛帅帅（2019）在《非线性薛定谔方程的KAM理论》一文中研究指出：我们关心的问题是加了哈密顿扰动后的线性方程或可积方程的拟周期解的存在性。在哈密顿偏微分方程的KAM(Kolmogorov-Arnold-Moser)理论中已经有许多显著的结果。哈密顿偏微分方程的KAM理论,主要有两种方法。一种是由经典KAM理论发展来的[1,9,11,12,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,30,31,32,35],另一种是由Craig,Wayne,Bourgain通过牛顿迭代技巧,发展完善而来的CWB方法[2,3,4,5,6,7,8,10,29]。前者的方法优点是在拟周期解附近,一局部的Birkhoff标准形获得从而得到拟周期解的线性稳定性和零Lyapunov指数,这对于理解拟周期解附近的动力学性态是非常有用的。而CWB方法长处在于,它通过解和角变量有关的同调方程避免了繁琐的第二Melnikov条件,使得它相比于KAM理论更适于解重法频共振的情况,从而对周期边界条件的哈密顿偏微分方程及高维偏微分方程很有效,而缺点是它不能给出拟周期解附近的动力学性态,使得我们无从获知以拟周期解附近点为初值的解的长时间行为。这些方法对于一维哈密顿偏微分方程都有很好的应用。尽管如此,这些方法在处理高维哈密顿偏微分方程中却遇到困难。Bourgain[2]证明了二维非线性薛定谔方程有小振幅拟周期解。后来,他在[5]中,改进了他的方法,证明了高维非线性薛定谔方程和波方程有小振幅拟周期解。通过有限维KAM理论构造高维哈密顿偏微分方程的拟周期解的方法后来才出现。Geng-You[16,17]证明了高维非线性梁方程和非局部薛定谔方程有小振幅线性稳定拟周期解。Eliasson-Kuksin[12]通过修改的KAM方法构造了更有趣的高维非线性薛定愕方程的小振幅线性稳定的拟周期解。对于在周期边界条件下的二维的三次薛定谔方程iut—△u+｜u｜2u=0,x ∈T2,t ∈R,Gcng-Xu-You[14]给了拟周期解的证明。他们通过小心选择切点集合｛i1,…,ib｝∈Z2,他们证明了上述非线性薛定谔方程有一族小振幅拟周期解(也看[28])。在本论文中,通过一个改进的KAM机制和非线性项的衰减性,我们致力于研究非线性薛定谔方程(NLS)的拟周期解的存在性。我们关注非线性项是否与外频或者空间变量相关。更详细的说,本文给出了下面的结果:1.有外力驱动的高维非线性薛定谔方程iut—△u+Mu+f{ωt)｜u｜2u=0,t ∈ R,x∈Td在周期边界条件下,这里Mε是傅里叶乘子,f(θ(θ=ωt)是实解析的,驱动频率ω是固定的丢番图向量。我们证明方程存在一族实解析小振幅线性稳定拟周期解。2.有非扰的外力驱动的二维非线性薛定谔方程iut—△u+φ(ωt)u+φ(ωt)｜u｜2u=0,t ∈ R,x∈T2在周期边界条件下,这里φ以(ωt)对于θ=ωt以是实解析的,驱动频率ω是固定的丢番图向量,并且满足我们这里强调φ(ωt)不是小的扰动。通过一个无限维的KAM定理,我们获得这个方程一族Whitney光滑的部分双曲的小振幅拟周期解。3.二维非线性五次薛定谔方程iut—△u+｜u｜4u=0,t ∈R,x ∈T2在周期边界条件下,我们证明一个无限维的KAM定理。作为应用,我们获得这个方程一族Whitney光滑的部分双曲的小振幅拟周期解。4.二维非线性薛定谔方程在周期边界条件下,这里非线性项f(x,u,u)=∑j,lj+l≥6αjl(x)uju-l,ajl=alj在原点的一个邻域内是实解析的。我们证明这个方程一族Whitney光滑的小振幅拟周期解。
Abstract
wo men guan xin de wen ti shi jia le ha mi du rao dong hou de xian xing fang cheng huo ke ji fang cheng de ni zhou ji jie de cun zai xing 。zai ha mi du pian wei fen fang cheng de KAM(Kolmogorov-Arnold-Moser)li lun zhong yi jing you hu duo xian zhe de jie guo 。ha mi du pian wei fen fang cheng de KAMli lun ,zhu yao you liang chong fang fa 。yi chong shi you jing dian KAMli lun fa zhan lai de [1,9,11,12,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,30,31,32,35],ling yi chong shi you Craig,Wayne,Bourgaintong guo niu du die dai ji qiao ,fa zhan wan shan er lai de CWBfang fa [2,3,4,5,6,7,8,10,29]。qian zhe de fang fa you dian shi zai ni zhou ji jie fu jin ,yi ju bu de Birkhoffbiao zhun xing huo de cong er de dao ni zhou ji jie de xian xing wen ding xing he ling Lyapunovzhi shu ,zhe dui yu li jie ni zhou ji jie fu jin de dong li xue xing tai shi fei chang you yong de 。er CWBfang fa chang chu zai yu ,ta tong guo jie he jiao bian liang you guan de tong diao fang cheng bi mian le fan suo de di er Melnikovtiao jian ,shi de ta xiang bi yu KAMli lun geng kuo yu jie chong fa pin gong zhen de qing kuang ,cong er dui zhou ji bian jie tiao jian de ha mi du pian wei fen fang cheng ji gao wei pian wei fen fang cheng hen you xiao ,er que dian shi ta bu neng gei chu ni zhou ji jie fu jin de dong li xue xing tai ,shi de wo men mo cong huo zhi yi ni zhou ji jie fu jin dian wei chu zhi de jie de chang shi jian hang wei 。zhe xie fang fa dui yu yi wei ha mi du pian wei fen fang cheng dou you hen hao de ying yong 。jin guan ru ci ,zhe xie fang fa zai chu li gao wei ha mi du pian wei fen fang cheng zhong que yu dao kun nan 。Bourgain[2]zheng ming le er wei fei xian xing xue ding e fang cheng you xiao zhen fu ni zhou ji jie 。hou lai ,ta zai [5]zhong ,gai jin le ta de fang fa ,zheng ming le gao wei fei xian xing xue ding e fang cheng he bo fang cheng you xiao zhen fu ni zhou ji jie 。tong guo you xian wei KAMli lun gou zao gao wei ha mi du pian wei fen fang cheng de ni zhou ji jie de fang fa hou lai cai chu xian 。Geng-You[16,17]zheng ming le gao wei fei xian xing liang fang cheng he fei ju bu xue ding e fang cheng you xiao zhen fu xian xing wen ding ni zhou ji jie 。Eliasson-Kuksin[12]tong guo xiu gai de KAMfang fa gou zao le geng you qu de gao wei fei xian xing xue ding e fang cheng de xiao zhen fu xian xing wen ding de ni zhou ji jie 。dui yu zai zhou ji bian jie tiao jian xia de er wei de san ci xue ding e fang cheng iut—△u+｜u｜2u=0,x ∈T2,t ∈R,Gcng-Xu-You[14]gei le ni zhou ji jie de zheng ming 。ta men tong guo xiao xin shua ze qie dian ji ge ｛i1,…,ib｝∈Z2,ta men zheng ming le shang shu fei xian xing xue ding e fang cheng you yi zu xiao zhen fu ni zhou ji jie (ye kan [28])。zai ben lun wen zhong ,tong guo yi ge gai jin de KAMji zhi he fei xian xing xiang de cui jian xing ,wo men zhi li yu yan jiu fei xian xing xue ding e fang cheng (NLS)de ni zhou ji jie de cun zai xing 。wo men guan zhu fei xian xing xiang shi fou yu wai pin huo zhe kong jian bian liang xiang guan 。geng xiang xi de shui ,ben wen gei chu le xia mian de jie guo :1.you wai li qu dong de gao wei fei xian xing xue ding e fang cheng iut—△u+Mu+f{ωt)｜u｜2u=0,t ∈ R,x∈Tdzai zhou ji bian jie tiao jian xia ,zhe li Mεshi fu li xie cheng zi ,f(θ(θ=ωt)shi shi jie xi de ,qu dong pin lv ωshi gu ding de diu fan tu xiang liang 。wo men zheng ming fang cheng cun zai yi zu shi jie xi xiao zhen fu xian xing wen ding ni zhou ji jie 。2.you fei rao de wai li qu dong de er wei fei xian xing xue ding e fang cheng iut—△u+φ(ωt)u+φ(ωt)｜u｜2u=0,t ∈ R,x∈T2zai zhou ji bian jie tiao jian xia ,zhe li φyi (ωt)dui yu θ=ωtyi shi shi jie xi de ,qu dong pin lv ωshi gu ding de diu fan tu xiang liang ,bing ju man zu wo men zhe li jiang diao φ(ωt)bu shi xiao de rao dong 。tong guo yi ge mo xian wei de KAMding li ,wo men huo de zhe ge fang cheng yi zu Whitneyguang hua de bu fen shuang qu de xiao zhen fu ni zhou ji jie 。3.er wei fei xian xing wu ci xue ding e fang cheng iut—△u+｜u｜4u=0,t ∈R,x ∈T2zai zhou ji bian jie tiao jian xia ,wo men zheng ming yi ge mo xian wei de KAMding li 。zuo wei ying yong ,wo men huo de zhe ge fang cheng yi zu Whitneyguang hua de bu fen shuang qu de xiao zhen fu ni zhou ji jie 。4.er wei fei xian xing xue ding e fang cheng zai zhou ji bian jie tiao jian xia ,zhe li fei xian xing xiang f(x,u,u)=∑j,lj+l≥6αjl(x)uju-l,ajl=aljzai yuan dian de yi ge lin yu nei shi shi jie xi de 。wo men zheng ming zhe ge fang cheng yi zu Whitneyguang hua de xiao zhen fu ni zhou ji jie 。
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论文作者分别是来自南京大学的薛帅帅，发表于刊物南京大学2019-11-14论文，是一篇关于薛定谔方程论文,环面论文,拟周期解论文,南京大学2019-11-14论文的文章。本文可供学术参考使用，各位学者可以免费参考阅读下载，文章观点不代表本站观点，资料来自南京大学2019-11-14论文网站，若本站收录的文献无意侵犯了您的著作版权，请联系我们删除。
标签：薛定谔方程论文; 环面论文; 拟周期解论文; 南京大学2019-11-14论文;