原子链中的非线性晶格动力学
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摘要
近年来，完整晶格的非线性局域问题是晶格动力学的新的
热门研究领域。就理论分析而言，单原子链常常因为模型简单、
物理图象清晰，并且存在各种线性、非线性元激发而被普遍重
视和研究。考虑到最近邻格点间谐振和三次方、四次方非谐相
互作用，单原子链中既存在包络孤子，又存在反对称的内部局
域模式——扭状、反扭状包络孤子。当计及最近邻相互作用的
理论基本成熟后，同时包含高次非谐相互作用和各种长程关联
谐振特别是次近邻关联谐振的晶格动力学问题就显得日益重要
了。
本文，我们主要通过引进多重尺度，并利用准离散近似方
法对单原子链中的晶格动力学作了一些研究。首次得到了一维
晶格的CMKdV方程和PDNLS方程，取得了一系列关于一维
晶格中的孤立子的动力学性质的一些有意义的结果。全文分为
五章：第一章为基本理论，介绍了孤立子的发展历史、多重尺
度结合准离散法及其关联近似方法。第二章研究了涉及次近邻
谐振和四次非谐相互作用下的单原子链中的波动问题，得到了
新的色散关系。且首次推出单原子链中的CMKdV方程，在讨
论考虑次近邻谐振和三次方、四次方非谐相互作用下单原子链
的孤立子时，解释了自局域结构的幅度只取决于点阵中的固有
参数的实验现象。第三章研究了长程关联作用下的单原子链中，
不仅仅在Brillouin边界处存在非传播的内在局域模：而且
Brillouin边界之外也存在非传播的内在局域模。当计及l重近
邻相互作用时，在Brillouin内就有7处存在非传播的孤立子。
第四章，我们研究弱阻尼作用下非线性单原子链中的波动行为，
首次在晶格动力学中得到了参数耗散型非线性Schrodinger方程
oDNLS方程人 以及阻尼对单原子链中孤立子的影响。在最
后一章，总结了我们的研究成果以及对学术进行展望。
Abstract
The nonlinear localized mode of perfect lattice has become a
new popular investigating field of lattice dynamics last several
years. As to theory analysis, much atteniion has usually been
largely paid to a monatomic chain, because of itS simPle model, its
clear physics image and the exiting various linear or nonlinear
elementary excitations. Research showed that it exhibits intrinsic
localized mode in a monatomic chain with the nearest-neighbor
harmonic and cubic, quartic anharmonic interaction. That is, there
exits envelope, envelope-kink, envelope-antikink solitons in the
chain. Since the theory what the lattice dynamics of a monatomic
chain with nearest-neighbor interaction has been matllred, the issue
of lattice dynamics of a monatomic chain with higher anhimonic
and long-range interaction, especial with second nearest-neighbor
interaction, Iooks like more important.
In this paPer, lattice dynamics of a monatomic chain are
studied by employing the method of multiple scales combined with
quasi-discreteness aPproximation. We get CMKdV and PDNLS
equations of the one-dimensiona1 lattices and obtain a series of
dynamic properties of soliton in the monatomic chain. The thesis
consists of five chapters. ln chapter l, we mainly introduce
fundamental theories about the developed process of soliton, the
method of multiple scales combined with quasi-discreteness
approximation and other relative methods. In chaPter 2, these
methods are emPloyed to inveStigate the wave motive equation of a
monatomic chain Wth second nearest-neighbor harmonic and
quartic anharmonic interactions get a new linear dispersion relation.
Further more, we first obtain the CMKdV equation of the
monatomic chain. Then, soliton of the chain with second nearest-neighbor harmonic, cubic and quartic anharmonic interactions are discussed explain our result show that the kink amplitude of the self-localized structures are determined only by the intrinsic properties of its lattices, which is agreement with the experimental phenomena. In chapter 3, we study monatomic chain with power-law long-range interactions and get non-propagating intrinsic localized mode in other wave-vector as well as that in the boundary of the phonon Brillouin zone. There are 1 point non-propagating soliton in the phonon Brillouin zone when considering the lth nearest-neighbor inter-actions. In chapter 4, we investigated the behavior of wave in a nonlinear monatomic chain with damping. Parameter dissipation nonlinear Schrodinger (PDNLS) equation of lattice dynamics is first obtained. Then we discuss effect of damping on soliton of monatomic chain. In last chapter, our results are summarized and studied prospects are made out.
热门研究领域。就理论分析而言，单原子链常常因为模型简单、
物理图象清晰，并且存在各种线性、非线性元激发而被普遍重
视和研究。考虑到最近邻格点间谐振和三次方、四次方非谐相
互作用，单原子链中既存在包络孤子，又存在反对称的内部局
域模式——扭状、反扭状包络孤子。当计及最近邻相互作用的
理论基本成熟后，同时包含高次非谐相互作用和各种长程关联
谐振特别是次近邻关联谐振的晶格动力学问题就显得日益重要
了。
本文，我们主要通过引进多重尺度，并利用准离散近似方
法对单原子链中的晶格动力学作了一些研究。首次得到了一维
晶格的CMKdV方程和PDNLS方程，取得了一系列关于一维
晶格中的孤立子的动力学性质的一些有意义的结果。全文分为
五章：第一章为基本理论，介绍了孤立子的发展历史、多重尺
度结合准离散法及其关联近似方法。第二章研究了涉及次近邻
谐振和四次非谐相互作用下的单原子链中的波动问题，得到了
新的色散关系。且首次推出单原子链中的CMKdV方程，在讨
论考虑次近邻谐振和三次方、四次方非谐相互作用下单原子链
的孤立子时，解释了自局域结构的幅度只取决于点阵中的固有
参数的实验现象。第三章研究了长程关联作用下的单原子链中，
不仅仅在Brillouin边界处存在非传播的内在局域模：而且
Brillouin边界之外也存在非传播的内在局域模。当计及l重近
邻相互作用时，在Brillouin内就有7处存在非传播的孤立子。
第四章，我们研究弱阻尼作用下非线性单原子链中的波动行为，
首次在晶格动力学中得到了参数耗散型非线性Schrodinger方程
oDNLS方程人 以及阻尼对单原子链中孤立子的影响。在最
后一章，总结了我们的研究成果以及对学术进行展望。
Abstract
The nonlinear localized mode of perfect lattice has become a
new popular investigating field of lattice dynamics last several
years. As to theory analysis, much atteniion has usually been
largely paid to a monatomic chain, because of itS simPle model, its
clear physics image and the exiting various linear or nonlinear
elementary excitations. Research showed that it exhibits intrinsic
localized mode in a monatomic chain with the nearest-neighbor
harmonic and cubic, quartic anharmonic interaction. That is, there
exits envelope, envelope-kink, envelope-antikink solitons in the
chain. Since the theory what the lattice dynamics of a monatomic
chain with nearest-neighbor interaction has been matllred, the issue
of lattice dynamics of a monatomic chain with higher anhimonic
and long-range interaction, especial with second nearest-neighbor
interaction, Iooks like more important.
In this paPer, lattice dynamics of a monatomic chain are
studied by employing the method of multiple scales combined with
quasi-discreteness aPproximation. We get CMKdV and PDNLS
equations of the one-dimensiona1 lattices and obtain a series of
dynamic properties of soliton in the monatomic chain. The thesis
consists of five chapters. ln chapter l, we mainly introduce
fundamental theories about the developed process of soliton, the
method of multiple scales combined with quasi-discreteness
approximation and other relative methods. In chaPter 2, these
methods are emPloyed to inveStigate the wave motive equation of a
monatomic chain Wth second nearest-neighbor harmonic and
quartic anharmonic interactions get a new linear dispersion relation.
Further more, we first obtain the CMKdV equation of the
monatomic chain. Then, soliton of the chain with second nearest-neighbor harmonic, cubic and quartic anharmonic interactions are discussed explain our result show that the kink amplitude of the self-localized structures are determined only by the intrinsic properties of its lattices, which is agreement with the experimental phenomena. In chapter 3, we study monatomic chain with power-law long-range interactions and get non-propagating intrinsic localized mode in other wave-vector as well as that in the boundary of the phonon Brillouin zone. There are 1 point non-propagating soliton in the phonon Brillouin zone when considering the lth nearest-neighbor inter-actions. In chapter 4, we investigated the behavior of wave in a nonlinear monatomic chain with damping. Parameter dissipation nonlinear Schrodinger (PDNLS) equation of lattice dynamics is first obtained. Then we discuss effect of damping on soliton of monatomic chain. In last chapter, our results are summarized and studied prospects are made out.
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