【作者】 刘杰；

【导师】 司建国；

【作者基本信息】 山东大学 ， 基础数学， 2015， 博士

【摘要】 本文主要研究非线性Schrodinger方程.它在量子力学中有着广泛的应用.自无穷维KAM理论产生以来,作为一个Hamilton系统,人们渐渐开始利用KAM理论研究非线性Schrodinger方程有限维不变环面(对应于拟周期解)以及无穷维不变环面(对应于概周期解)的存在性.对于可积Hamilton系统而言,它的动力学行为是清楚明了的.然而,现实中可积系统少之又少,更多的是近可积系统.近可积Hamilton系统的动力学行为问题被Poincare称为“动力学基本问题”.上世纪五六十年代,三位国际著名数学家A.N. Kolmogorov[64], V.I. Arnold[1]和J.K. Moser[93]建立了经典KAM理论.因其重要的应用价值,KAM理论被视作20世纪最重要的数学成就之一.八十年代末九十年代初S.B. Kuksin[66][67][68][69], W. Craig和C.E. Wayne[29] [30]将有限维KAM理论推广应用至Hamilton型偏微分方程,发展出无穷维KAM理论.后来J. Poschel[105]重新整理加以描述成一个容易理解的无穷维KAM定理,并成功将其应用于Dirichlet边界条件下的非线性Schrodinger方程[70]和非线性波动方程[106].同时J. Bourgain[19][20]又将这一想法推广至一般的Hamilton偏微分方程.鉴于以上均在有界扰动情形下讨论,Poschel [59]通过引入广义正规形并利用Kuksin[71]求解变系数同调方程的引理,将有界扰动情形下的无穷维KAM定理推广至无界扰动的情形.然而仍有一大批重要的偏微分方程不能适用上面的定理,例如带导数的非线性波动方程,带导数的非线性Schrodinger方程等.最近,刘建军和袁小平[80]通过修改Kuksin的引理,得到了临界情形下的KAM定理(见[81]).进而刘建军和袁小平[81][82],张静,高美娜和袁小平[132]分别研究了几类带导数的非线性Schrodinger方程.这些方程中既有Hamilton系统也有反转系统.与Dirichlet边界条件不同,在周期边界条件下由于重特征值的出现,前面Poschel的KAM定理失效Chierchia和尤建功[27]最早用KAM方法解决了非线性波动方程在周期边界条件下拟周期解的存在性问题.事实上在他们的结果出现之前,Craig和Wayne[39]已经利用推广的Lyapunov-Schmidt分解和Frohlich, Spence技巧,证明了周期解的存在性.这种由Craig和Wayne29][30]发展并由Bourgain[15][18][19]改进的方法称为C-W-B方法.对于高维Hamilton偏微分方程,难度较大,进展也比较缓慢.最早Bourgain[18]研究了二维Schrodinger方程的小振幅拟周期解.这方面的主要进展参见Bour-gain耿建生和尤建功[43]Eliasson和Kuksin[33],耿建生,徐新东和尤建功[46],耿建生和尤建功[49],Eliasson, Grebert和Kuksin [34][35]等.在实际应用中,有三类著名的带导数的非线性Schrodinger方程：方程),方程),方程).2012年耿建生与吴健[48]在周期边界条件下研究了带导数的非线性Schrodinger方程并得到了具有2个频率的实解析拟周期解.同时刘建军和袁小平[82]在周期边界条件下研究了并得到了具有N个频率的光滑拟周期解.需要说明的是,因采取了两种完全不同的策略.以上两个结果才会有差别.耿建生与吴健通过引入“紧性形式”和“不变性质”.将变系数的同调方程变为常系数,然后利用KAM迭代得到了实解析的拟周期解.而刘建军和袁小平利用修改的求解变系数同调方程的引理并同时假设扰动满足特殊形式,通过KAM迭代得到了光滑的拟周期解.特别需要指出的是,由于“紧性形式”和“不变性质”的限制使得耿建生与吴健所得拟周期解的频率仅为两个,而刘建军和袁小平的结果允许频率的个数是任意整数.本文首先利用耿建生与吴健[48]的方法在周期边界条件下研究了第二类带导数的非线性Schrodinger方程Chen-Liu-Lee方程通过利用“紧性形式”和“不变性质”我们得到了具有两个频率的实解析拟周期解.当扰动项具有拟周期强迫时,相应的系统是非自治系统.对于具有周期强迫的完全共振波动方程的周期解首先由Rabinowitz利用整体变分方法和Lyapunov-Schmidt分解得到.后来Berti和Procesi[11]将Lyapunov-Schmidt分解和Nash-Moser迭代结合起来研究了带周期强迫的完全共振波动方程具有两个频率的拟周期解的存在性.再后来焦蕾和王奕倩[54]用Birkhoff标准形和KAM迭代方法证明了拟周期强迫下的非线性Schrodinger方程拟周期解的存在性.近些年,张敏和司建国[133],司建国[118]分别研究了具有拟周期强迫的非线性波动方程在Dirichlet边界条件下和周期边界条件下拟周期解的存在性.王怡和司建国[119],芮杰和司建国[109]分别研究了具有拟周期强迫的非线性梁方程和非线性Schrodinger方程拟周期解的存在性.需要注意的是以上讨论均是在有界扰动的情形下.当扰动无界时,特别是临界情形,即使对于自治系统也是比较困难的.当扰动是具有拟周期强迫的无界扰动时,这方面的结果非常少.弭鲁芳与张康康[89]利用KAM方法研究了具有拟周期扰动的Benjamin-Ono方程,Baldi, Berti和Montalto[7]将Nash-Moser迭代和KAM迭代结合起来研究了具有拟线性扰动的线性Airy方程.利用同样的方法,R. Feola和M. Procesi[39]研究了完全非线性反转Schrodinger方程.论文中作者研究了具有拟周期强迫的带导数的非线性Schrodinger方程在满足周期边界条件的不变环面的存在性.这里B为正常数,g是实解析函数,关于时间变量t拟周期,频率向量为β=(β1,β2,…,βm).我们的方法基于Birkhoff正规形理论和KAM迭代.需要说明的是,在周期边界条件下,由于特征值是二重的,因此不能使用Poschel[59]关于无界扰动的无穷维KAM定理.我们沿着刘建军和袁小平[81][82]的思路,考虑广义正规形,同时还假设扰动满足类似的特殊形式：扰动项P(φ,q,q)仅包含单项式这里然后利用求解变系数同调方程的引理,证明了不变环面的存在性.论文中我们分别在两种情形下讨论了该问题：(1)β为任意实数向量；(2)β与指定频率β“共线”(co-linear)即β=入β∈Rm,λ∈[1/2,3/2].对于第一种情形我们利用修改的满足特殊形式的刘建军和袁小平[82]的KAM定理得到了具有任意正整数个频率的光滑拟周期解.而对于第二种情形,由于强迫项的频率是“固定”的,我们没有足够多的参数,因此文献[81][82]中关于参数集测度估计的方法并不适用.通过采用不同的测度估计方法,我们得到了具有任意正整数个频率的光滑拟周期解.本论文共分为五章,主要内容如下：第一章我们给出Hamilton系统和KAM理论的相关知识.这一章又包括以下四节.第一节介绍有限维Hamilton系统理论,包含Hamilton向量场及变换理论,可积Hamilton系统和Birkhoff正规形.第二节主要介绍经典KAM理论.第三节简单介绍了Hamilton系统扰动理论的三个主要研究方向：经典稳定性(Classical stability),几何稳定性(Geometric stability)和不稳定性(Instability).最后一节详细叙述了无穷维KAM理论及其应用于Hamilton型偏微分方程的研究现状,特别是非线性波动方程和非线性Schrodinger方程.第二章我们罗列了一些在KAM理论应用中常用的定义与结论,如拟周期函数.实解析函数Cauchy估计等等.第三章我们利用耿建生与吴健[48]方法在周期边界条件下研究了Chen-Liu-Lee方程利用“紧性形式”和“不变性质”我们得到了具有两个频率的实解析拟周期解.第四章与第五章研究了具有拟周期强迫的带导数的非线性Schrodinger方程在周期边界条件下不变环面的存在性.第四章在β为任意实数向量的情形下讨论该问题.而第五章在β与指定频率β“共线”的情形下讨论该问题.更多还原

【Abstract】 In this dissertation, we consider nonlinear Schrodinger equations which have a wide range of applications in quantum mechanics. Since the infinite dimen-sional KAM theory has been set up, more and more authors began to study the existence of finite dimensional invariant tori(quasi-periodic solutions) and infinite dimensional invariant tori (almost periodic solutions) of the nonlinear Schrodinger equations as Hamiltonian systems.To integrable systems, the dynamics are clear. However, the integrable Sys-tems are few and more systems are nearly integrable in reality. Poincare saw the dynamics of nearly Hamiltonian systems as "Fundamental Problem of Dynamics" In 1950s to 1960s, the classical KAM theory was set up by three international cel-ebrated mathematician:A.N. Kolmogorov[64], V.I. Arnold[1] and J.K. Moser[93]. Because of its important value in application, KAM theory was seen as one of the high points of 20th century mathematics. In the late 1980s and early 1990s, the finite dimensional KAM theory was generalized and applied to Hamiltonian partial differential equations by S.B. Kuksin[66][67][68][69], W. Craig and C.E. Wayne[29] [30], and then they set up the infinite dimensional KAM theory. Later, a very explicit KAM-like theorem for PDEs was developed by J. Poschel[105] and was applied to nonlinear Schrodinger equations [70] and nonlinear wave equa-tions [106] under Dirichlet boundary conditions successfully, while J. Boungain [20] extended the concepts to more general PDEs. As all mentioned above are on the bounded perturbations, Poschel [59] generalized the KAM theorem for unbounded perturbation by using generalized normal forms and Kuksin’s lemma solving variant coefficient homological equations[71]. However, there still are a large class of important PDEs which can not be covered by above theorems, such as derivative nonlinear wave equations and derivative nonlinear Schrodinger equa-tions etc. Recently, Liu and Yuan[80] generalized Kuksin’s lemma and obtained the KAM theorem on limit case(see [81]). Then Liu and Yuan[81][82], Zhang, Gao and Yuan[132] considered several classes of derivative nonlinear Schrodinger equations, in which there are either Hamilton systems or reversible systems.Different form Dirichlet boundary conditions, above Poschel’s KAM the-orems are out of work under periodic boundary conditions as they give rise to double eigenvalues. For the existence of quasi-periodic solutions of nonlinear wave equations under periodic boundary conditions, the first result by KAM methods is due to Chierchia and You[27]. In fact, before them, Craig and Wayne[29] had already proved the existence of periodic solutions by Lyapunov-Schmidt reduc-tion method and techniques by Frohlich and Spencer. The method developed by Craig and Wayne[29][30] and improved by Bourgain[15][18][19] is called C-W-B method.For higher dimensional Hamiltonian PDEs, the difficulty is greater and the progress is slower. The earliest result is due to Bourgain[18] investigated the small amplitude periodic solutions for 2D Schrodinger equations. In this direction we refer to Bourgain[19], Geng and You [43], Eliasson and Kuksin[33]. Geng, Xu and You [46], Geng and You [49], Eliasson, Grebert and Kuksin[34][35] etc.There are three famous classes of derivative nonlinear Schrodinger equations in practical applications: In 2012, Geng and Wu [48] considered the derivative nonlinear Schrodinger Equa-tions under periodic boundary conditions: and obtained the real analytic quasi-periodic solutions with two frequencies. Meanwhile, Liu and Yuan [82] considered a similar equation under periodic bound-ary conditions: and got smooth quasi-periodic solutions with N frequencies. It should be pointed out that above two results are different as two different strategies they used. By introduced "Compact Form" and "Gauge Property", Geng and Wu reduced the variant coefficient homological equations to constant ones, then via KAM itera-tion obtained the real analytic quasi-periodic solutions. On the other hand, by using the improved lemma solving variant coefficient homological equations and assuming the perturbation satisfying special form, Liu and Yuan got the smooth quasi-periodic solutions via KAM iteration. Especially it should be noted that because of "Compact Form" and "Gauge Property", the number of frequencies of quasi-periodic solutions obtained by Geng and Wu is only two, while the num-ber is any positive integer in Liu and Yuan’s result. In this dissertation, we first consider the second class of derivative Schrodinger equations:Chen-Liu-Lee equation: under periodic boundary conditions by Geng and Wu’ method[48]. By "Com-pact Form" and "Gauge Property", we obtained the real analytic quasi-periodic solutions with two frequencies.For the perturbation with quasi-periodic forcing, the systems are nonau-tonomous ones. For completely resonant wave equations with periodic forcing, the periodic solutions are obtained by Rabinowitz using the variational method and the Lyapunov-Schmidt reduction. Later, combing with Lyapunov-Schmidt reduction and Nash-Moser iteration, Berti and Procesi[11] investigated the ex-istence of quasi-periodic solutions with two frequencies for completely resonant wave equations with periodic forcing. Then Jiao and Wang[54] proved the exis-tence of quasi-periodic solutions of nonlinear Schrodinger equations with quasi-periodic forcing via Birkhoff normal forms and KAM method. In recent years, Zhang and Si[133], Si[118] discussed the existence of quasi-periodic solutions of nonlinear wave equations with quasi-periodic forcing under Dirichlet bound-ary conditions and periodic boundary conditions respectively. Then Wang and Si[119], Rui and Si [109] considered the quasi-periodic solutions for the nonlinear beam equations and Schrodinger equations with quasi-periodic forcing respec-tively. It should be noted that all above perturbations are bounded. When the perturbations are unbounded, especially limit case, it is difficult even the sys-tems are autonomous. For the quasi-periodic forced unbounded perturbations, the results are few. First, Mi and Zhang[89] considered the Benjamin-Ono equa-tions with quasi-periodic forced perturbation via KAM method. Baldi, Berti and Montalto [7] considered linear Airy equations with quasi-linear perturbation by Nash-Moser iteration and KAM iteration. Using the same method, R. Feola and M. Procesi[39] discussed fully nonlinear forced reversible Schrodinger equations. In this paper, the author considers the derivative nonlinear Schrodinger equations with quasi-periodic forced perturbations: under periodic boundary conditions and proves the existence of invariant tori. Where B is a positive constant, g is real analytic and quasi-periodic on time t with frequency vector β=(β1,β2, ...,βm). The proof is based on Birkhoff normal forms and KAM iteration. It should be pointed out that under periodic boundary conditions, as the eigenval-ues are double, we can not use Poschel’s infinite dimensional KAM theorem for unbounded perturbations. Following Liu and Yuan[81][82], we consider general normal forms and suppose the perturbation satisfying the similar special form: perturbation P only consists of monomials where Then we proved the existence of invariant tori by solving variant coefficient ho-mological equations. We divided it into two cases:(1) β is any real vector; (2) β is co-linear with fixed β, that is β= λβ Rm, λ ∈ [1/2,3/2]. For the first case, we can use an improved Liu and Yuan’s KAM theorem satisfying special perturbations directly and get smooth quasi-periodic solutions which the number of frequencies is any positive integer. But for the second case, as the frequency of perturbation is "fixed", we do not have enough parameters and then the mea-sure estimate methods in [81] [82] can not be used in our problem. By different methods of measure estimate, we get smooth quasi-periodic solutions which the number of frequencies is any positive integer.This dissertation is divided into five chapters as following:In chapter one, we give the related theory on Hamiltonian systems and KAM theory. There are also four sections in this chapter. We introduce finite dimen-sional Hamilton systems theory in the first section, which consists of Hamilto-nian vector fields and transformation theory, integrable Hamiltonian systems and Birkhoff normal forms. In section 2, we mainly introduce classical KAM theory. Then we describe the three main parts in the study of Hamiltonian perturbation theory in section 3 briefly:Classical stability, Geometric stability and Instability. In the last section, we describe in detail the infinite dimensional KAM theory and applications in the study of Hamiltonian partial differential equations, especially nonlinear wave equations and nonlinear Schrodinger equations.In the second chapter, we list some basic definitions and conclusions in the application of KAM theory:such as quasi-periodic function, real analytic function and Cauchy estimate etc.In chapter three, we use Geng and Wu’s method [48] discuss the Chen-Liu-Lee equations under periodic boundary conditions: By "Compact Form" and "Gauge Property" we obtain the real analytic quasi-periodic solutions with two frequencies.In the forth and fifth chapter we study the derivative nonlinear Schrodinger equations with quasi-periodic forced perturbations and the existence of invariant tori under periodic boundary conditions. We con-sider this problem in the case where β is any real vector in chapter four and in the case where β is co-linear with fixed β in chapter five.更多还原