复杂非线性系统中的孤子
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摘要
人们普遍认为,非线性科学是继量子力学、相对论之后的一次科学革命。现在,人们逐渐认识到,错综复杂的客观世界,线性系统只是客观存在的一种近似,非线性系统才能更好地接近自然的本质。非线性科学的研究,目前主要有三大领域：混沌、孤子和分形。本文的研究,是集中在孤子领域。
在本研究中,我们将主要集中于若干复杂非线性系统中的孤子传播问题,其中包括理想化的均匀系统和带有离散缺陷的非均匀系统。对于理想的均匀系统,我们将关注非线性发展方程精确解求解方法的研究,且具体研究其中的辅助方程法；而在带缺陷的非均匀系统中,我们将研究缺陷与孤子的相互作用,且将涉及两种类型的物理系统：连续的流体系统和离散的非线性传输线电路。
在第一章中,我们将对孤子进行简要地介绍,其中包括孤子发现的历史、研究孤子的常用方法,以及缺陷与孤子相互作用的研究现状等,并进一步提出本文的研究内容。
相比于非传播孤子,人们对流体中的传播孤子在离散缺陷系统中的传播问题还缺乏认识。在第二章中,我们将研究流体系统中离散缺陷与传播孤子的相互作用。具体我们将对水槽底部缺陷与水表面传播孤子(Korteweg-de Vries (KdV)孤子)之间的相互作用进行研究。首先,我们利用摄动方法从流体动力学方程中约化出近似的理论模型,然后对该模型进行数值计算。结果表明,根据缺陷极性的不同,其对传播水波孤子将产生减速或加速的影响。这与非传播水波孤子的情形一致。但是,在单底部缺陷对传播水波孤子的作用中,包含有一种偶极子的影响,这在目前人们研究单缺陷情形的非传播孤子中没有发现到。另外,我们还考察了不同强度的缺陷情形。结果表明,缺陷对非传播水波孤子的影响还与缺陷强度有关。
在第三章中,我们将研究非线性传输线电路中缺陷与孤子的相互作用。目前,关于非线性系统中缺陷与孤子的相互作用,在非传播孤子方面,在Frenkel-Kontorova (FK)模型和流体系统中取得丰富而且一致的结果,即根据缺陷极性的不同,缺陷可以吸引或排斥孤子。在传播孤子方面,我们通过第二章流体力学系统中的研究发现,缺陷可以改变孤子的传播速度。具体为,对于不同极性的缺陷,其将加速或减速孤子。在非线性传输线电路中,缺陷与孤子的相互作用有一定的研究成果,但缺少系统的了解。我们通过对一种新类型的非线性传输线电路中电容缺陷与孤子相互作用的研究,达到系统地理解非线性传输线电路系统中缺陷与孤子相互作用的原理。我们通过改变非线性电容器的线性电容来引入单缺陷,并从基于Kirchhoff定律的直接数值模拟和基于它的约化理论模型的数值计算两方面对系统的缺陷与孤子的相互作用进行研究。将我们的结果和已有的研究结果作比较和分析,表明对于非线性传输线电路系统中不同类型的参量缺陷,其与孤子的相互作用具有共同的机理,进而给出了非线性传输线电路中缺陷与孤子相互作用统一的物理机制。同时,我们将非线性传输线电路中的缺陷与孤子相互作用与FK模型和流体系统中的情形进行对比。结果表明,在这些不同的非线性系统中,缺陷与孤子的相互作用具有共同的机理,进而可以用一种统一的方式来进行理解。
另外,我们还研究了非线性传输线电路中线性电容单缺陷的不同强度情形。结果表明,缺陷对非线性传输线电路孤子的影响还与缺陷强度有关。此外,鉴于包含单缺陷的系统只是实际非均匀系统的一种初级近似,我们还尝试在均匀非线性传输线电路中考虑由若干线性电容单缺陷组合而成的对缺陷、偶极子缺陷和四极子缺陷情形,并研究了它们对孤子传播的影响。
在第四章中,我们采用解析的方法,探讨理想非线性系统精确解的求解方法。求解非线性方程的精确解在非线性科学中扮演着重要的作用。目前,人们已经提出许多求解方法,比如,反散射方法(inverse scattering method), Hirota双线性方法,Backlund变换法等。但是,由于线性叠加原理的失效,目前人们还没有办法给出各种非线性方程精确解的一般形式,通常只能针对各种不同的非线性问题而寻求不同的方法。在本章中,我们将集中研究求解非线性发展方程精确解中的辅助方程法。具体地,我们进一步推广Huang方法到一种至多含有8次非线性项的一阶常微分方程的形式,并在函数展开上将这种一阶常微分方程展开到范围更为广泛的展开式形式,从而发展了一种求解非线性发展方程精确解的方法。利用该方法,我们对Sharma-Tasso-Olver(STO)方程和广义的Camassa-Holm(GCH)方程进行求解,并获得了这两个非线性发展方程的一些精确解。
论文的最后,我们在第五章对全文进行了总结,并对后续的工作进行了展望。
It is now commonly acknowledged that nonlinear science is another scientific revolution after the Quantum mechanics and the theory of relativity. One has now gradually recognized that linear system is only approximate to the complex real physical world, but the nonlinear one can be better close to the nature of the world. At present, there are three main nonlinear science research fields:chaos, soliton and fractal. We will focus on the soliton in this study.
Here, we will mainly study the propagation of solitons in some complex nonlinear systems, including the ideal homogeneous systems and the inhomogeneous ones with discrete impurities. As for the homogeneous case, we will focus our interest on the research on developing method for solving exact solutions to nonlinear evolution equations (NEEs), and specifically, we will study the auxiliary equation method; and for the inhomogeneous case, we will study the impurity-soliton interactions (ISIs), and will investigate two types of nonlinear systems:the continuous hydrodynamic system and the discrete nonlinear electrical transmission line (NETL).
In Chapter 1, we will briefly introduce the soliton, including its history and research methods, the developments of ISIs, and so on. Then, we will further present the main research in this study.
At present, the propagation of hydrodynamic solitons influenced by discrete impurities, compared with that of non-propagating ones, still lacks the understanding. In Chapter 2, we will study the interactions between discrete impurity and propagating soliton in the hydrodynamics. Specifically, we will pay our attention to the interactions between depth defects of a trough and surface hydrodynamic propagating solitons (Korteweg-de Vries (KdV) solitons). First, we will use the perturbation method to derive the corresponding theoretical approximate ISI model from the hydrodynamic dynamic equations, and then do the numerical study based on this model. The results show that, the defect can decelerate or accelerate the propagating hydrodynamic solitons depending on its polarity, which is consistent with the case for the non-propagating ones. However, from the present study, we find that a dipole defect-induced effect is involved on propagating hydrodynamic solitons, which is not found in the present non-propagating cases influenced by single impurity. In addition, we also investigate different impurity intensity cases, and find that the ISIs in the hydrodynamics for propagating solitons are related to the impurity intensity.
In Chapter 3, we will study the ISIs in the NETL. As we know, the present ISIs both in the Frenkel-Kontorova (FK) model and hydrodynamics for the non-propagating solitons have got fruitful results and reached a good agreement each other, namely, an impurity or defect can attract or repel non-propagating solitons depending on its polarity. And from Chapter 2, we can get that, a defect can change the propagating speeds of the propagating hydrodynamic solitons, accelerating or decelerating also depending on its polarity. The present ISIs in the NETL lack the systemic understanding though they have also got some results. This Chapter is aimed to systemically understand the ISIs in the NETL by simulating the influence of a new capacitor impurity on the NETL solitons. Here, a capacitor with a slight difference in linear capacitance will serve as the single impurity. We will study the influence of the capacitor impurity on the NETL solitons both by the simulation which is based on the Kirchhoffs laws and by the calculation of the corresponding theoretical ISI model. The results show that, compared with the present ISI results of the NETL, the impurity-induced influence in the NETL for different types of parameter impurities has the same nature and essentially shares the same physical mechanism. At the same time, they also show that, compared with the present ISI results of the FK model and hydrodynamics, the impurity-induced influence in the NETL, FK model, and hydrodynamics essentially shares the same physical mechanism and thus can be understood in a unified way.
In addition, we also study the different intensity cases of the single linear capacitance impurity, and find that, the ISIs in the NETL are related to the impurity intensity. Besides, in view of the fact that single impurity is only a primary approximation to the real inhomogeneous systems, we tentatively consider the pair, dipole and quadrupole impurities in the homogeneous NETL, which are the different combination of the single linear capacitance impurity, to study the impurity-induced influence on the propagation of solitons.
In Chapter 4, we will analytically study the method for solving exact solutions to ideal nonlinear systems. Solving exact solutions to nonlinear equations play an important role in nonlinear science. Up to now, a wealth of powerful methods have been proposed, such as the inverse scattering method, Backlund transformation, the Hirota's bilinear method, and so on. However, because the superposition principle is no longer valid in nonlinear cases, one can not give the general solutions to the different nonlinear equations, and usually, he only can seek the different solving method depending on the different nonlinear problems. In this Chapter, we will focus our interest on studying the auxiliary equation method for solving nonlinear evolution equations (NEEs). Specifically, we will further extend the Huang's method to the form which possesses an auxiliary equation of a first order nonlinear ordinary differential equation with at most an eighth-degree nonlinear term. Furthermore, by performing the auxiliary equation into a more general expansion form, we develop an algebraic method and apply it to the Sharma-Tasso-Olver (STO) equation and the generalized Camassa-Holm (GCH) equation. And some exact solutions to these two NEEs are obtained.
Finally, in Chapter 5, we summarize the conclusions and also discuss the prospect of the future studies.
在本研究中,我们将主要集中于若干复杂非线性系统中的孤子传播问题,其中包括理想化的均匀系统和带有离散缺陷的非均匀系统。对于理想的均匀系统,我们将关注非线性发展方程精确解求解方法的研究,且具体研究其中的辅助方程法；而在带缺陷的非均匀系统中,我们将研究缺陷与孤子的相互作用,且将涉及两种类型的物理系统：连续的流体系统和离散的非线性传输线电路。
在第一章中,我们将对孤子进行简要地介绍,其中包括孤子发现的历史、研究孤子的常用方法,以及缺陷与孤子相互作用的研究现状等,并进一步提出本文的研究内容。
相比于非传播孤子,人们对流体中的传播孤子在离散缺陷系统中的传播问题还缺乏认识。在第二章中,我们将研究流体系统中离散缺陷与传播孤子的相互作用。具体我们将对水槽底部缺陷与水表面传播孤子(Korteweg-de Vries (KdV)孤子)之间的相互作用进行研究。首先,我们利用摄动方法从流体动力学方程中约化出近似的理论模型,然后对该模型进行数值计算。结果表明,根据缺陷极性的不同,其对传播水波孤子将产生减速或加速的影响。这与非传播水波孤子的情形一致。但是,在单底部缺陷对传播水波孤子的作用中,包含有一种偶极子的影响,这在目前人们研究单缺陷情形的非传播孤子中没有发现到。另外,我们还考察了不同强度的缺陷情形。结果表明,缺陷对非传播水波孤子的影响还与缺陷强度有关。
在第三章中,我们将研究非线性传输线电路中缺陷与孤子的相互作用。目前,关于非线性系统中缺陷与孤子的相互作用,在非传播孤子方面,在Frenkel-Kontorova (FK)模型和流体系统中取得丰富而且一致的结果,即根据缺陷极性的不同,缺陷可以吸引或排斥孤子。在传播孤子方面,我们通过第二章流体力学系统中的研究发现,缺陷可以改变孤子的传播速度。具体为,对于不同极性的缺陷,其将加速或减速孤子。在非线性传输线电路中,缺陷与孤子的相互作用有一定的研究成果,但缺少系统的了解。我们通过对一种新类型的非线性传输线电路中电容缺陷与孤子相互作用的研究,达到系统地理解非线性传输线电路系统中缺陷与孤子相互作用的原理。我们通过改变非线性电容器的线性电容来引入单缺陷,并从基于Kirchhoff定律的直接数值模拟和基于它的约化理论模型的数值计算两方面对系统的缺陷与孤子的相互作用进行研究。将我们的结果和已有的研究结果作比较和分析,表明对于非线性传输线电路系统中不同类型的参量缺陷,其与孤子的相互作用具有共同的机理,进而给出了非线性传输线电路中缺陷与孤子相互作用统一的物理机制。同时,我们将非线性传输线电路中的缺陷与孤子相互作用与FK模型和流体系统中的情形进行对比。结果表明,在这些不同的非线性系统中,缺陷与孤子的相互作用具有共同的机理,进而可以用一种统一的方式来进行理解。
另外,我们还研究了非线性传输线电路中线性电容单缺陷的不同强度情形。结果表明,缺陷对非线性传输线电路孤子的影响还与缺陷强度有关。此外,鉴于包含单缺陷的系统只是实际非均匀系统的一种初级近似,我们还尝试在均匀非线性传输线电路中考虑由若干线性电容单缺陷组合而成的对缺陷、偶极子缺陷和四极子缺陷情形,并研究了它们对孤子传播的影响。
在第四章中,我们采用解析的方法,探讨理想非线性系统精确解的求解方法。求解非线性方程的精确解在非线性科学中扮演着重要的作用。目前,人们已经提出许多求解方法,比如,反散射方法(inverse scattering method), Hirota双线性方法,Backlund变换法等。但是,由于线性叠加原理的失效,目前人们还没有办法给出各种非线性方程精确解的一般形式,通常只能针对各种不同的非线性问题而寻求不同的方法。在本章中,我们将集中研究求解非线性发展方程精确解中的辅助方程法。具体地,我们进一步推广Huang方法到一种至多含有8次非线性项的一阶常微分方程的形式,并在函数展开上将这种一阶常微分方程展开到范围更为广泛的展开式形式,从而发展了一种求解非线性发展方程精确解的方法。利用该方法,我们对Sharma-Tasso-Olver(STO)方程和广义的Camassa-Holm(GCH)方程进行求解,并获得了这两个非线性发展方程的一些精确解。
论文的最后,我们在第五章对全文进行了总结,并对后续的工作进行了展望。
It is now commonly acknowledged that nonlinear science is another scientific revolution after the Quantum mechanics and the theory of relativity. One has now gradually recognized that linear system is only approximate to the complex real physical world, but the nonlinear one can be better close to the nature of the world. At present, there are three main nonlinear science research fields:chaos, soliton and fractal. We will focus on the soliton in this study.
Here, we will mainly study the propagation of solitons in some complex nonlinear systems, including the ideal homogeneous systems and the inhomogeneous ones with discrete impurities. As for the homogeneous case, we will focus our interest on the research on developing method for solving exact solutions to nonlinear evolution equations (NEEs), and specifically, we will study the auxiliary equation method; and for the inhomogeneous case, we will study the impurity-soliton interactions (ISIs), and will investigate two types of nonlinear systems:the continuous hydrodynamic system and the discrete nonlinear electrical transmission line (NETL).
In Chapter 1, we will briefly introduce the soliton, including its history and research methods, the developments of ISIs, and so on. Then, we will further present the main research in this study.
At present, the propagation of hydrodynamic solitons influenced by discrete impurities, compared with that of non-propagating ones, still lacks the understanding. In Chapter 2, we will study the interactions between discrete impurity and propagating soliton in the hydrodynamics. Specifically, we will pay our attention to the interactions between depth defects of a trough and surface hydrodynamic propagating solitons (Korteweg-de Vries (KdV) solitons). First, we will use the perturbation method to derive the corresponding theoretical approximate ISI model from the hydrodynamic dynamic equations, and then do the numerical study based on this model. The results show that, the defect can decelerate or accelerate the propagating hydrodynamic solitons depending on its polarity, which is consistent with the case for the non-propagating ones. However, from the present study, we find that a dipole defect-induced effect is involved on propagating hydrodynamic solitons, which is not found in the present non-propagating cases influenced by single impurity. In addition, we also investigate different impurity intensity cases, and find that the ISIs in the hydrodynamics for propagating solitons are related to the impurity intensity.
In Chapter 3, we will study the ISIs in the NETL. As we know, the present ISIs both in the Frenkel-Kontorova (FK) model and hydrodynamics for the non-propagating solitons have got fruitful results and reached a good agreement each other, namely, an impurity or defect can attract or repel non-propagating solitons depending on its polarity. And from Chapter 2, we can get that, a defect can change the propagating speeds of the propagating hydrodynamic solitons, accelerating or decelerating also depending on its polarity. The present ISIs in the NETL lack the systemic understanding though they have also got some results. This Chapter is aimed to systemically understand the ISIs in the NETL by simulating the influence of a new capacitor impurity on the NETL solitons. Here, a capacitor with a slight difference in linear capacitance will serve as the single impurity. We will study the influence of the capacitor impurity on the NETL solitons both by the simulation which is based on the Kirchhoffs laws and by the calculation of the corresponding theoretical ISI model. The results show that, compared with the present ISI results of the NETL, the impurity-induced influence in the NETL for different types of parameter impurities has the same nature and essentially shares the same physical mechanism. At the same time, they also show that, compared with the present ISI results of the FK model and hydrodynamics, the impurity-induced influence in the NETL, FK model, and hydrodynamics essentially shares the same physical mechanism and thus can be understood in a unified way.
In addition, we also study the different intensity cases of the single linear capacitance impurity, and find that, the ISIs in the NETL are related to the impurity intensity. Besides, in view of the fact that single impurity is only a primary approximation to the real inhomogeneous systems, we tentatively consider the pair, dipole and quadrupole impurities in the homogeneous NETL, which are the different combination of the single linear capacitance impurity, to study the impurity-induced influence on the propagation of solitons.
In Chapter 4, we will analytically study the method for solving exact solutions to ideal nonlinear systems. Solving exact solutions to nonlinear equations play an important role in nonlinear science. Up to now, a wealth of powerful methods have been proposed, such as the inverse scattering method, Backlund transformation, the Hirota's bilinear method, and so on. However, because the superposition principle is no longer valid in nonlinear cases, one can not give the general solutions to the different nonlinear equations, and usually, he only can seek the different solving method depending on the different nonlinear problems. In this Chapter, we will focus our interest on studying the auxiliary equation method for solving nonlinear evolution equations (NEEs). Specifically, we will further extend the Huang's method to the form which possesses an auxiliary equation of a first order nonlinear ordinary differential equation with at most an eighth-degree nonlinear term. Furthermore, by performing the auxiliary equation into a more general expansion form, we develop an algebraic method and apply it to the Sharma-Tasso-Olver (STO) equation and the generalized Camassa-Holm (GCH) equation. And some exact solutions to these two NEEs are obtained.
Finally, in Chapter 5, we summarize the conclusions and also discuss the prospect of the future studies.
引文
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