【作者】 李丽莉；

【导师】 田播；

【作者基本信息】 北京邮电大学 ， 计算机软件与理论， 2009， 博士

【摘要】 随着信息技术的迅速发展,计算机符号计算作为人工智能的新分支学科之一,也逐渐成熟和完善,并被应用到非线性科学的研究中来。目前,计算机符号计算因其强大而精确的符号计算能力和操作简便、直观,易于实现等特点,已经逐渐发展成为人们从事非线性科学研究必不可少的计算机辅助工具之一,并在此基础上取得了丰硕的研究成果。尤其是近十几年间,基于计算机符号计算的孤子理论研究越来越成熟和发展起来。作为非线性科学的核心问题之一,孤子理论一直是国内外众多学者关注的一个热点领域,其中对于非线性模型孤子解的研究,无论在理论研究还是在实际应用上一直都是重要而又难以解决的问题,几乎在各个学科领域中都会遇到。在本论文中,作者结合本研究领域的发展现状,基于计算机符号计算对若干有着实际物理应用的非线性模型孤子解进行了研究,主要内容包括两个方面,研究构造非线性模型精确孤子解的方法;研究非线性模型孤子解的时间和空间特性。具体包括如下几个方面:①基于计算机符号计算将传统的双线性求解方法进行推广,使其适合于求解更复杂的高维和高阶非线性模型。推广的双线性方法借助于齐次平衡思想推导出非线性模型的双线性变换,将目标模型转化成易于求解的齐次方程。同时在求解过程中略去了将齐次方程分块和双线性化的过程,这样就避免了不适当的分块可能产生的对于模型孤子解的结构的限制。同时,利用计算机符号计算给出双线性算子的计算式,大大方便了我们利用双线性方法处理相关问题。②一般来讲,广义的时间变系数模型大多是不可积的,但是,当各个系数函数满足了一定的约束条件时却是可积的,也就是存在多孤子解,B(a｜¨)cklund变换,Lax对等。本文将推广的双线性方法应用于求解广义的变系数Korteweg-de Vries模型和变系数Kadomtsev-Petviashvili模型的精确多孤子解;基于这些方程的双线性形式,利用双线性算子的性质,推导出其对应解的自B(a｜¨)cklund变换,包括双线性形式和Lax形式的自B(a｜¨)cklund变换;进一步地,从解的双线性形式自B(a｜¨)cklund变换出发,可以得到模型的Lax对,或者解的非线性叠加公式;对于求得的多孤子解,通过绘图分析研究模型的各个变系数对于解的稳定性和传输方式的影响。③近年来,对于非线性Schr(o｜¨)dinger模型多孤子间的部分相干相互作用的研究引起了人们的兴趣。本文我们借助于计算机符号计算和推广的双线性方法,研究一个在现代光纤通信领域中有着重要应用的广义(1+1)维耦合非线性Schr(o｜¨)dinger模型的孤子解。并从得到的解出发,研究随着时间的发展各个变系数函数对于孤子稳定性的影响,包括传播速度和振幅,重点研究孤子间的相互作用,包括部分相干相互作用和孤波间的成对碰撞。④部分非线性Schr(o｜¨)dinger模型的孤子间存在一种特殊的形变碰撞,由于在这种碰撞过程中发生了能量再分配,而越来越受到人们的关注。借助于计算机符号计算,我们将研究一个广义耦合高阶非线性Schr(o｜¨)dinger模型间的这种奇妙的非弹性相互作用。该研究在光纤中的一个重要应用就是能量放大器,由于这种放大过程不需要外界的干扰和新能量的补充,因而很稳定,拥有广阔的实际应用前景。⑤对于非线性模型局域孤子的研究越来越积极和发展起来,各种局域相干结构也被不断地发现。同时,做为孤子间的一种非弹性相互作用,孤子的裂变和聚变也开始引起人们的关注。我们对于(1+1)维孤子解的相互作用已经研究的很深,并取得了很大的成就,但是高维方程的孤子解之间的相互作用确实很复杂的,而且维数越高,相互作用就越复杂。本文借助于计算机符号计算和计算机模拟,对于色散长波模型,在多孤子解的基础上,通过计算机软件模拟绘图的方式集中分析研究不同的局域相干结构间的这种裂变型非弹性相互作用,并且从数学角度简要分析裂变现象产生的原因。综上所述,本文将计算机符号计算与传统的双线性方法相结合,并将其推广应用于求解复杂的高维、高阶非线性模型中。对于几个有着典型物理意义的非线性模型进行了解析研究,包括构造多孤子解,B(a｜¨)cklund变换,Lax对等;并利用Mathematica计算软件的模拟绘图功能对求得孤子解的时间发展特性,相互之间的弹性和非弹性相互作用,以及它们潜在的物理应用进行了深入分析。希望本文的理论研究结果在将来的实验观察中能够得到证实,并对于相关研究有所帮助。更多还原

【Abstract】 With the quick development of information technology, symbolic computation, as a new branch of artificial intelligence, has become perfect and ripe gradually, and has been applied to the research of nonlinear science. Nowadays, because of its powerful ability in exactly dealing with the complicated and tedious calculations, and its convenient and direct practicability, symbolic computation has grown into a necessary and wonderful assistant tool for the study of nonlinear science. Especially in the latest decade, much progress has already been made in the further development of soliton theory based on symbolic computation. As a core issue of nonlinear science, the soliton theory has always been the hot research topic, and absorbed the close attention of numerous researchers in the world. Both in theoretical research and in pratical application, the study on soliton solutions of nonlinear models has been an important and difficult issue, which may be encountered in every field of science.Based on symbolic computation and the development status of this area, the present dissertation is carried out to analytically investigate the soliton solutions of some physically important nonlinear models, such as generalizing the known methods that are used to construct the exact soliton solutions, analyzing the stabilities and structures of solitons, studying the interactions between solitonic waves, including elastic and inelastic ones. The research work of the dissertation mainly includes the following aspects:①With the help of symbolic computation, the traditional bilinear method is extended to dealing with the more complicated higher-order and higher-dimensional nonlinear models. The main idea of our extended bilinear method is that, through the bilinear transformation derived by employing the balancing-act method, the original model is transformed into a homogeneous equation, which is then solved directly by the format-parameter expansion method. One key point of the extended method is that the unimaginable complicated calculations involved in the solving process with the increase of the order of solution, are easily done via symbolic computation. This extended procedure skips the process of equation splitting and bilinearization so as to avoid the possibility of introducing additional limitation on the structure of the soliton solutions. Besides, the computerized function of bilinear operation is defined in software Mathematica, and this greatly conveniences our calculation.②Generally speaking, most of the generalized variable-coefficient models are not completely integrable. But when some constraints on the variable-coefficient functions are satisfied, the variable-coefficient models may possess some integrabel properties, such as multi-soliton solutions, Backlund transformation, and Lax pair. With the help of symbolic computation, the multi-soliton solutions of a nonisospectral and variable-coefficient Korteweg-de Vries model and a generalized Kadomtsev-Petviashvili model with variable coefficients are investigated by using the extended bilinear method. Based on their bilinear forms and by means of properties of the bilinear operator, the Backlund transformations in both bilinear form and Lax pair form are derived. Furthermore, starting from the Backlund transformations, the Lax pairs and the nonlinear superposition formula are obtained. Finally, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.③In recent years, more and more interests have been attracted on the research of partially coherent interactions of multi-structures for nonlinear Schrodinger models. Hereby, via symbolic computation, a generalized (1+1)-dimensional coupled nonlinear Schrodinger model is investigated. With the obtained multi-soliton solutions, some propagation and interaction properties of the solitons are discussed simultaneously. Moreover, some figures are plotted to graphically analyze the pairwise collisions and partially coherent interactions among three solitons.④Recent developments have already revealed that soliton collisions encountered in some coupled nonlinear Schrodinger models possess novel features, such as the shape changes with intensity redistributions or energy exchanges. It is a desirable feature of such collisions that solitons during interaction can transform energy, in other words, solitons can be amplified without using other techniques. This type of collision-based amplification has aroused increasing interests because it does not require any external amplification medium, neither does it induce any noise. Motivated by this, the shape-changing collision of solitons in a generalized coupled higher-order nonlinear Schrodinger model is investigated through asymptotic analysis on the various features underlying the desirable collision between two solitons.⑤It is of great significance to study localized coherent structures and their interaction behavior in (2+1)-dimensional integrable nonlinear systems. With the aid of symbolic computation, under investigation is the (2+1)-dimensional dispersive long wave equations. Based on the analytic multi-soliton solutions derived by the extended bilinear method, some novel soliton interaction behaviors for this model are revealed. At the same time, the main propagation properties are studied through the graphical illustration. Our analysis shows that the novel interactions include the fission phenomenon, which is a kind of non-elastic behavior. Compared with the completely elastic interaction of solitons, a simple answer to why the fission phenomenon exists is given from the viewpoint of mathematics.In conclusion, combined with the technique of symbolic computation, the bilinear method is extended to investigate some complicated higher-order or higher-dimensional models. These models are of practical importance in various branches of physics like fluid dynamics, fiber communications, superconductors, Bose-Einstein condensates, plasma physics, atmosphere and oceans. The study of the models includes deriving soliton solutions, B(a|¨)cklund transformations and Lax pairs, where symbolic computation becomes a necessary and helpful implement. Furthermore, through computer simulation in software Mathematica, the stabilities, propagation characteristics, and the elastic and inelastic interaction properties of the solitonic waves are all graphically analyzed. Moreover, their underlying physical applications are discussed in detail. It is expected that the analytic results and relevant discussions in this dissertation will be observed in the future laboratory experiments, and will be helpful to the future studies.更多还原