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 本文中介紹6種方法求解KdV方程式及Burgers方程式的精確孤波解，其所有解皆與非線性項係數成反比，而使得解本身無法與方程式同步線性化。經深入研究並分析各種方法後，發現可利用simplest equation method (SEM) with Bernoulli equation as the simplest equation求得KdV 方程式及Burgers方程式的可線性化精確孤波解。新解克服了存在已久的非線性項係數趨近於零而解不連續的問題。藉由與文獻中的精確解比較，可發現新解呈現出新的物理現象，吾人暫稱為”孤波滑移”。除此之外，我們延伸了SEM的運用層面。我們利用Burgers equation屬於完全可積分方程式的性質，將它設定為simplest equation來求解其他完全可積分方程式。經由KdV及potential KdV 兩方程式的驗證，該方法確實可行。有別於 Cole-Hopf transformation 及 Hirota’s method兩種方法，我們所提出的方法更簡單、直接且效率更高。因此，我們相信對於具有孤波及多重孤波解的非線性物理模態皆可利用SEM及延伸型SEM求解。
 In this paper, six different exact solution methods have been presented and applied to solve Korteweg and de Vries (KdV) and Burgers equations, whose solutions are proportional to the nonlinear term coefficients. The new exact solutions of KdV and Burgers equations, with the nonlinear term coefficients being arbitrary constants, are derived by the simplest equation method (SEM) with Bernoulli equation as the simplest equation. It is shown that the proposed exact solutions overcome the long existing problem of discontinuity and can be successfully reduced to linearity when the nonlinear term coefficients approach to zero. Comparison of the existing and new soliton solutions is presented. A new phenomenon, named soliton sliding, is observed. Moreover, we extend the SEM by choosing Burgers equation as the simplest equation. The reason of setting Burgers equation as the simplest equation is due it being completely integrable equation. The extended SEM is applied to handle two completely integrable equations, KdV and the potential KdV equations. The general forms of the multi-soliton solutions are formally established. Unlike Hirota’s method, the results confirm the extended SEM is concise and effective for constructing multi-soliton solutions. Accordingly, we believe that solitary solutions and multi-soliton solutions existing for other classes of nonlinear mathematic physics models can be easily solved by the SEM and the extended SEM. Further work on these aspects is recommended.
 摘 要. …………………IAbstract IIAcknowledgments IVContents VList of Tables VIIIList of Figures IXNomenclature XIIChapter1 Introduction 11.1 Introduction 11.2 Literature review 31.2.1 KdV equation 31.2.2 Burgers equation 71.3 Purpose of present study 81.4 Scope 92 Six exact solution methods 122.1 Analysis of the methods 122.2 The tanh-coth method 122.3 The sine-cosine method 142.4 The simplest equation method 152.5 The Exp-function method 162.6 The Cole-Hopf transformation 172.7 Hirota’s method 183 Exact solutions 203.1 KdV equation 203.2 Existing exact solutions of KdV equation 203.2.1 By the tanh-coth method 203.2.2 By the sine-cosine method 223.2.3 By the Exp-function method 233.2.4 By Hirota’s method 263.3 Existing exact solutions of Burgers equation 293.3.1 By the tanh-coth method 293.3.2 By the SEM with Riccati as the simplest equation 303.3.3 By the Cole-Hopf transformation 313.4 The linearized solutions by the SEM 333.4.1 Linearized solutions of KdV equation 333.4.2 The linearized solution of Burgers equation 363.5 Numerical results and discussions 373.6 Conclusions 424 The general forms of the multi-soliton solutions for the completely integrable equations by the extended SEM 614.1 The extended SEM 614.2 Multi-soliton solutions of KdV equation 634.3 Multi-soliton solutions of the potential KdV equation 654.4 Conclusions 675 Summary and Future Prospects 705.1 General Conclusions 705.2 Future Prospects 71References 72Vita 84
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