本文選題：修正的簡單方程法 + 精確解�。� 參考：《內(nèi)蒙古師范大學(xué)》2017年碩士論文

【摘要】：眾多學(xué)科領(lǐng)域中出現(xiàn)的大量非線性現(xiàn)象一般都可以用非線性發(fā)展方程來刻畫,因而用非線性模型來反映客觀世界成為非線性科學(xué)研究的一股主流。于是尋找非線性發(fā)展方程精確解的問題自然成為直觀深刻地去通曉非線性模型的物理意義及其性質(zhì)所不可或缺的途徑之一。正因?yàn)槿绱?建立尋找非線性發(fā)展方程精確解的新方法,借助或改進(jìn)原有方法而給出非線性發(fā)展方程的新的精確解等問題能夠在理論上幫助人們理解實(shí)際物理問題所蘊(yùn)含的性質(zhì)和特性,在應(yīng)用方面能夠提供新的方法和不同技巧等意義.此外,非線性模型的可積性問題是孤立子理論研究的另一重要課題,與此聯(lián)系的可積系統(tǒng)的Lax對是將非線性方程的求解問題轉(zhuǎn)化為線性方程的求解問題的橋梁,而Backlund變換則是構(gòu)造非線性方程精確解的有效工具。因此,對于具體的非線性方程給出它的Lax對與Backlund變換不僅為可積系統(tǒng)各種性質(zhì)的研究奠定理論基礎(chǔ),同時(shí)也對構(gòu)造非線性方程精確解提供工具的作用.本文工作將圍繞上述課題主要研究非線性發(fā)展方程(組)的求解問題并給出sine-Gordon方程、廣義的變系數(shù)KdV-mKdV方程以及(2+1)維色散長波方程組的精確解,同時(shí)考慮變系數(shù)李方程組的Painleve性質(zhì)、Lax對、Backlund變換與精確解的構(gòu)造等問題。本文具體內(nèi)容安排如下第一章為緒論,分別對Painleve分析、變系數(shù)非線性發(fā)展方程的研究現(xiàn)狀、修正的簡單方程法、輔助方程法作一簡單介紹,并簡短介紹本文的主要研究工作.第二章將借助修正的簡單方程法給出sine-Gordon方程、廣義的變系數(shù)KdV-mKdV方程以及(2+1)維色散長波方程組的精確解,得到其對應(yīng)的精確孤波解.第三章研究變系數(shù)李方程組,證明變系數(shù)李方程組具有Painleve性質(zhì),給出它的Lax對、自Backlund變換以及精確解.第四章將分別借助Raccati輔助方程法和擴(kuò)展的G'/G展開法給出(2+1)維AKNS方程以及高階色散NLS方程的精確解,包括周期解、孤波解以及有理解.第五章,對全文的具體工作進(jìn)行總結(jié),并對今后開展的工作提出了詳細(xì)的計(jì)劃.
[Abstract]:A large number of nonlinear phenomena in many disciplines can be described by nonlinear evolution equations, so the nonlinear model to reflect the objective world has become the mainstream of nonlinear science research. Therefore, finding the exact solution of nonlinear evolution equation is naturally one of the indispensable ways to understand the physical meaning and properties of nonlinear model directly and profoundly. Because of this, a new method for finding exact solutions of nonlinear evolution equations is established. The new exact solutions of nonlinear evolution equations can help people understand the properties and characteristics of practical physical problems in theory by using or improving the original methods. It can provide new methods and different skills in application. In addition, the integrability problem of nonlinear model is another important subject of soliton theory. The Lax pair of integrable system connected with this problem is a bridge to transform the solving problem of nonlinear equation into the solving problem of linear equation. Backlund transformation is an effective tool for constructing exact solutions of nonlinear equations. Therefore, for specific nonlinear equations, the Lax pair and Backlund transformation not only lay a theoretical foundation for the study of various properties of integrable systems, but also provide a tool for constructing exact solutions of nonlinear equations. In this paper, the solution of nonlinear evolution equations (systems) is studied and the exact solutions of the sine-Gordon equation, the generalized KdV-mKdV equation with variable coefficients and the long-wave equations of dispersion in the first dimension are given. At the same time, we consider the Painleve property of lie equations with variable coefficients and the construction of exact solutions and so on. The main contents of this paper are as follows: the first chapter is the introduction, which gives a brief introduction to Painleve analysis, variable coefficient nonlinear evolution equation, modified simple equation method and auxiliary equation method, and briefly introduces the main research work of this paper. In the second chapter, the exact solutions of the sine-Gordon equation, the generalized KdV-mKdV equation with variable coefficients and the long-wave equations with dispersion of 21) dimension are given by using the modified simple equation method, and the corresponding exact solitary wave solutions are obtained. In chapter 3, we study the lie equations with variable coefficients. We prove that the lie equations with variable coefficients have Painleve properties, and give their Lax pairs, self- transformations and exact solutions. In chapter 4, the exact solutions of AKNS equation and higher order dispersive NLS equation, including periodic solution, solitary wave solution and understanding, are obtained by using the Raccati auxiliary equation method and the extended G / G expansion method, respectively. Chapter five summarizes the specific work of the paper and puts forward a detailed plan for the future work.
【學(xué)位授予單位】：內(nèi)蒙古師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175

【參考文獻(xiàn)】

相關(guān)期刊論文 前10條

1 smail Aslan;;Traveling Wave Solutions for Nonlinear Differential-Difference Equations of Rational Types[J];Communications in Theoretical Physics;2016年01期

2 那仁滿都拉;鐘鳴華;斯仁道爾吉;;李方程組的顯式精確行波解[J];高師理科學(xué)刊;2015年05期

3 E.M.E.Zayed;S.A.Hoda Ibrahim;;Exact Solutions of Kolmogorov-Petrovskii-Piskunov Equation Using the Modified Simple Equation Method[J];Acta Mathematicae Applicatae Sinica(English Series);2014年03期

4 馮青華;;A New Fractional Projective Riccati Equation Method for Solving Fractional Partial Differential Equations[J];Communications in Theoretical Physics;2014年08期

5 M.Ali Akbar;Norhashidah Hj.Mohd.Ali;E.M.E.Zayed;;Generalized and Improved(G′/G)-Expansion Method Combined with Jacobi Elliptic Equation[J];Communications in Theoretical Physics;2014年06期

6 Md.Nur Alam;Md.Ali Akbar;Syed Tauseef Mohyud-Din;;A novel (G'/G)-expansion method and its application to the Boussinesq equation[J];Chinese Physics B;2014年02期

7 張哲;李德生;;修正的BBM方程新的精確解[J];原子與分子物理學(xué)報(bào);2013年05期

8 馮青華;;Exact Solutions for Fractional Differential-Difference Equations by an Extended Riccati Sub-ODE Method[J];Communications in Theoretical Physics;2013年05期

9 ;Explicit Solutions of (2+1)-Dimensional Canonical Generalized KP, KdV, and (2+1)-Dimensional Burgers Equations with Variable Coefficients[J];Communications in Theoretical Physics;2009年11期

10 施業(yè)瓊;;應(yīng)用(G′/G)—展開法求解高階非線性薛定諤方程[J];廣西工學(xué)院學(xué)報(bào);2009年03期

相關(guān)碩士學(xué)位論文 前1條

1 賀嵐云;某些非線性方程的精確解[D];內(nèi)蒙古師范大學(xué);2008年

，

本文編號：1860773