孤子方程的数值解和混沌系统的函数级联同步
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摘要
随着科学技术的发展，非线性科学得到了莲勃的发展它们已经涉及几乎所有的科学领域孤子、混沌和分形作为非线性科学的三大重要分支，对它们的研究有着重大的理论意义和应用价值，如光孤子通信、混沌保密通信和海岸线的长度等
本文主要研究孤立子的数值解和混沌同步这两方面的问题虽然它们表面上看是两种不同的现象，但它们却具有很多相似的特点例如：它们都与非线性系统有关：孤立子与非线性偏微分、常微分、微分-积分方程(组)等相关，而混沌与非线性常微分、差分方程(组)等有联系
本文以符号计算软件Maple为工具，拓展了Adomian分解法和同伦摄动法，研究了一些有重要物理意义的非线性孤子方程，并获得了一些有用的数值解(逼近解)；同时对混沌系统的同步问题做了进一步的探究，提出了时间连续和时间离散混沌系统的函数级联同步法，给出了它们的自动推理格式，并研究了一些连续和离散、带有和不带有不定参数的混沌系统的函数级联同步
本文分四章来介绍：
第一章介绍了孤立子研究的历史和发展；Adomian分解法和同伦摄动法的发展；混沌及混沌同步的发展历史和现状，同时介绍了了国内外学者这些领域所取得的成果
第二章将原本用于求解整数阶微分方程精确解及数值解的Adomian分解法和同伦摄动法推广到了一些重要的分数阶非线性孤子方程中例如：研究了一个数学物理上重要的、具有任意阶非线性项的非线性发展方程，获得了一些广义数值解；研究了一系列非线性分数阶耦合微分方程，获得了一些具有实际物理意义的数值解；研究了复KdV方程，获得了它的数值positon、negaton以及数值complexiton解
第三章给出了混沌系统的时间连续和离散型的函数级联同步法的自动推理格式，研究统一混沌系统、带不定参数的Lorenz系统、超混沌Lü系统和、离散的广义Hénon映射等混沌系统的函数级联同步问题，并利用数值模拟验证该算法的有效性
第四章给出了本文的总结和展望
With the development of science and technology, nonlinear science has developed rapidly and involved in almost all the scientific fields. Chaos, soliton and fractal consist of the main three branches of nonlinear science. There are many theoretical significance and applicable value to investigate them, such as optical soliton communication, chaos secure communication and the length of the coastline et al.
This dissertation investigates two problems of numerical solutions of soliton equations and chaos synchronization. Although different with each other, in fact, they have many common characteristics. For example, both of them are relevant to nonlinear system: soliton is associated with nonlinear partial differential、ordinary differential、differential-integral equation (system) et al and chaos relevant with nonlinear ordinary differential、difference one (ones) et al.
In this dissertation, based on symbolic-numeric computation software, the applications of Adomian decomposition method and homotopy perturbation method are extended to a number of nonlinear soliton equations owning important physical significance and some useful general numerical (approximate) solutions are obtained. The chaos synchronization is also further investigated: the function cascade synchronization method is proposed for both continuous-time and discrete-time systems and its automatic reasoning scheme is given; the function cascade synchronization is realized for some chaotic systems including continuous, discrete, with and without unknown parameters systems.
It is organized as follows:
Chapter 1 briefly reviews the history and progress of soliton, Adomian decomposition method, homoptopy perturbation method as well as chaos and chaos synchronization. Some achievements on these subjects involved in this dissertation are presented at home and abroad.
Chapter 2 directly extends the Adomian decomposition method and homotopy perturbation method to study some nonlinear soliton equations with physical significance. These two methods were used for differential equations of integer order traditionally. We investigate and obtain some general numerical solutions of the nonlinear evolution equations with nonlinear terms of any order; a type of nonlinear fractional coupled differential equations and some numerical solutions owning actual physical meaning; the complex KdV equation and the numerical positon、negaton solutions as well as numerical complexiton solutions.
Chapter 3 gives the automatic reasoning scheme of the function cascade synchronization method for both continuous-time and discrete-time chaotic system. The function cascade syn- chronization of chaotic system is investigated, which includes the unified chaotic system、Lorenz system with unknown parameters、hyperchaotic Lüsystem、discrete-time generalized Hénon map and so on. Numerical simulations are used to verify the effectiveness of the proposed scheme.
Chapter 4 is the summary and outlook of the dissertation.
本文主要研究孤立子的数值解和混沌同步这两方面的问题虽然它们表面上看是两种不同的现象，但它们却具有很多相似的特点例如：它们都与非线性系统有关：孤立子与非线性偏微分、常微分、微分-积分方程(组)等相关，而混沌与非线性常微分、差分方程(组)等有联系
本文以符号计算软件Maple为工具，拓展了Adomian分解法和同伦摄动法，研究了一些有重要物理意义的非线性孤子方程，并获得了一些有用的数值解(逼近解)；同时对混沌系统的同步问题做了进一步的探究，提出了时间连续和时间离散混沌系统的函数级联同步法，给出了它们的自动推理格式，并研究了一些连续和离散、带有和不带有不定参数的混沌系统的函数级联同步
本文分四章来介绍：
第一章介绍了孤立子研究的历史和发展；Adomian分解法和同伦摄动法的发展；混沌及混沌同步的发展历史和现状，同时介绍了了国内外学者这些领域所取得的成果
第二章将原本用于求解整数阶微分方程精确解及数值解的Adomian分解法和同伦摄动法推广到了一些重要的分数阶非线性孤子方程中例如：研究了一个数学物理上重要的、具有任意阶非线性项的非线性发展方程，获得了一些广义数值解；研究了一系列非线性分数阶耦合微分方程，获得了一些具有实际物理意义的数值解；研究了复KdV方程，获得了它的数值positon、negaton以及数值complexiton解
第三章给出了混沌系统的时间连续和离散型的函数级联同步法的自动推理格式，研究统一混沌系统、带不定参数的Lorenz系统、超混沌Lü系统和、离散的广义Hénon映射等混沌系统的函数级联同步问题，并利用数值模拟验证该算法的有效性
第四章给出了本文的总结和展望
With the development of science and technology, nonlinear science has developed rapidly and involved in almost all the scientific fields. Chaos, soliton and fractal consist of the main three branches of nonlinear science. There are many theoretical significance and applicable value to investigate them, such as optical soliton communication, chaos secure communication and the length of the coastline et al.
This dissertation investigates two problems of numerical solutions of soliton equations and chaos synchronization. Although different with each other, in fact, they have many common characteristics. For example, both of them are relevant to nonlinear system: soliton is associated with nonlinear partial differential、ordinary differential、differential-integral equation (system) et al and chaos relevant with nonlinear ordinary differential、difference one (ones) et al.
In this dissertation, based on symbolic-numeric computation software, the applications of Adomian decomposition method and homotopy perturbation method are extended to a number of nonlinear soliton equations owning important physical significance and some useful general numerical (approximate) solutions are obtained. The chaos synchronization is also further investigated: the function cascade synchronization method is proposed for both continuous-time and discrete-time systems and its automatic reasoning scheme is given; the function cascade synchronization is realized for some chaotic systems including continuous, discrete, with and without unknown parameters systems.
It is organized as follows:
Chapter 1 briefly reviews the history and progress of soliton, Adomian decomposition method, homoptopy perturbation method as well as chaos and chaos synchronization. Some achievements on these subjects involved in this dissertation are presented at home and abroad.
Chapter 2 directly extends the Adomian decomposition method and homotopy perturbation method to study some nonlinear soliton equations with physical significance. These two methods were used for differential equations of integer order traditionally. We investigate and obtain some general numerical solutions of the nonlinear evolution equations with nonlinear terms of any order; a type of nonlinear fractional coupled differential equations and some numerical solutions owning actual physical meaning; the complex KdV equation and the numerical positon、negaton solutions as well as numerical complexiton solutions.
Chapter 3 gives the automatic reasoning scheme of the function cascade synchronization method for both continuous-time and discrete-time chaotic system. The function cascade syn- chronization of chaotic system is investigated, which includes the unified chaotic system、Lorenz system with unknown parameters、hyperchaotic Lüsystem、discrete-time generalized Hénon map and so on. Numerical simulations are used to verify the effectiveness of the proposed scheme.
Chapter 4 is the summary and outlook of the dissertation.
引文
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[25]Liu Q P，Hu X B and Zhang M X Supersymmetric modified Korteweg-de Vries equation：bilinear approach Nonlinearity，2005，18：1597-1603
[26]Lou S Y and Lu J Z Special solutions from variable separation approach：Davey—Stewartsonequation J Phys A：Math Gen 1996．29：4209-4215
[27]Wang M L Solitary wave solutions for variant Boussinesq equations Phys Lett A 1995，199：169-17228 Tian B and Gao Y T Variable-coefficient balancing-act method and variable-coefficientKdV equation from fluid dynamics and plasma physics Eur Phys J B 2001．22：351-360
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[2]Russell J S Report on waves Fourteen meeting of the British association for the advance-ment of science，John Murray，London，1844，311-390
[3]Korteweg D J and deVries G On the change of form of long waves advancing in a rectangularcanal and on a new type of long stationary waves Phil Mag 1895，39：422-433
[4]Fermi A，Pasta J and Ulam I Studies of Nonlinear Problems Los Alamos Rep LA 1955
[5]Perring J K and Skyrme T H A model unified field equation Nucl Phys 1962，31：550-555
[6]Zabusky N J and Kruskal M D Interaction of solitons in a collisionless plasma and therecurrence of initial states Phys Rev Lett 1965．15：240-243
[7]Ablowitz M J and Clarkson P A Solitons，nonlinear evolution equations and inverse scat-tering Cambridge：Cambridge University Press，1991
[8]Matveev V B and Salle M A Darboux transformation and Solitons Berlin：Springer-Verlag，Heidelberg．1991
[9]郭柏灵，庞小峰孤立子北京：科学出版社，1987
[10]谷超豪等孤立子理论中的Darboux变换及其几何应用上海：上海科技出版社，1999
[11]李翊神孤子与可积系统上海：上海科技出版社，1999
[12]Hirota R The direct method in soliton theory New York：Cambridge University Press，2004
[13]Ablowitz M J，Kaup D J，Newell C and Segur H Method for solving the sine-Gordonequation Phys Rev Lett 1973．30：1262-1264
[14]Gardner C S，Greene J M，Kruskal M D and Miura R M Method for solving the Korteweg-deVries equation Phys Rev Lett 1967．19：1095-1097
[15]Wahlquist H D and Estabrook F B Backlund Transformation for Solutions of the Korteweg-de Vries Equation Phys Rev Lett 1973，31：1386-1390
[16]Darboux G Sur une proposition relative auxéquation linéaires Compts Rendus Hebdo madaires des Seance de tAcademie des Sciences，Paris，1882，94：1456-1459
[17]Bluman G W and Cole J D Similarity method of differential equations New York：Springer-Verlag，1974
[18]Hirota R Exact solution of the KdV equation fornmltiple collisions of solitons Phys Rev Lett 1971，27：1192-1194
[19]Weiss J，Tabor M and Carnvale G The Painlevéproperty for partial differential equationsJ Math Phys 1983．24：522-526
[20] GU C H and Zhou Z X On the Darboux nlatrices 0f Bācklund transformations for AKNSsystems Lett Math Phys 1987．13：179-187
[21]Cao C W Nonlinearization of the Lax System for AKNS Hierarchy Sci China A 1990，33：528-536
[22]Cheng Y and Li Y S The constraint of the Kadomtsev-Petviashvili equation and its specialsolutions Phys Lett A 1991．175：22-26
[23]HU X B and Zhu Z N Some new results on the Blaszak-Marciniak lattice：Backlundtransformation and nonlinear superposition formula J Math Phys 1998．39：4766-4772
[24]Hu X B and wu Y T A new integrable differential-difference system and its explicit solu-tions J Phys A：Math Gen 1999．32：1515-1521
[25]Liu Q P，Hu X B and Zhang M X Supersymmetric modified Korteweg-de Vries equation：bilinear approach Nonlinearity，2005，18：1597-1603
[26]Lou S Y and Lu J Z Special solutions from variable separation approach：Davey—Stewartsonequation J Phys A：Math Gen 1996．29：4209-4215
[27]Wang M L Solitary wave solutions for variant Boussinesq equations Phys Lett A 1995，199：169-17228 Tian B and Gao Y T Variable-coefficient balancing-act method and variable-coefficientKdV equation from fluid dynamics and plasma physics Eur Phys J B 2001．22：351-360
[29]楼森岳，唐晓艳非线性数学物理方法北京：科学出版社，2006
[30]Fan E G Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and acoupled mKdV equation Phys Lett A 2001．282：18-22
[31]闫振亚大连理工大学博士学位论文大连：大连理工大学，2003
[32]陈勇大连理工大学博士学位论文大连：大连理工大学，2003
[33]李彪大连理工大学博士学位论文大连：大连理工大学，2005
[34]谢福鼎大连理工大学博士学位论文大连：大连理工大学，2003
[35]王琪大连理工大学博士学位论文大连：大连理工大学，2006
[36]Adomian G Stochastic systems New York：Academic Press，1983
[37]Adomian G and Rach R On the solution of algebraic equations by the decompositionmethod Math Anal Appl 1985．105：141-166
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