本文選題：非線性發(fā)展方程 + 精確解�。� 參考：《內(nèi)蒙古師范大學(xué)》2017年碩士論文

【摘要】：在研究非線性自然現(xiàn)象時,非線性發(fā)展方程扮演著至關(guān)重要的角色,其求解方法也在不斷完善中.本文主要用拓展同宿試驗法來求解幾類非線性發(fā)展方程,主要有五章內(nèi)容構(gòu)成.第一章是緒論部分,主要介紹非線性發(fā)展方程的研究意義和拓展同宿試驗法的研究現(xiàn)狀,以及離散方程的簡單說明,最后介紹本文主要工作.第二章將拓展同宿試驗法應(yīng)用于求解常系數(shù)非線性方程.由(2+1)維耗散Zabolotskaya-Khokhlov方程的常數(shù)平衡解出發(fā),應(yīng)用拓展的同宿試驗法給出該方程的新的孤立波解,同時對得到的二孤波解進行分析,得出新的力學(xué)特征.通過拓展同宿試驗法,構(gòu)造測試函數(shù)給出Boussinesq方程的精確解.應(yīng)用同宿呼吸子極限方法,找出Boussinesq方程的呼吸子孤立波解和有理呼吸波解,并發(fā)現(xiàn)有理呼吸波解恰好是Boussinesq方程的怪波解.第三章應(yīng)用拓展同宿試驗法求解變系數(shù)非線性方程.具體求解(2+1)維變系數(shù)Zakharov-Kuznetsov方程和(2+1)維BKP方程.借助Maple符號計算系統(tǒng),在(2+1)維變系數(shù)Zakharov-Kuznetsov方程和(2+1)維BKP方程雙線性形式的基礎(chǔ)上,引入新的測試函數(shù)推廣拓展同宿試驗法而給出(2+1)維變系數(shù)Zakharov-Kuznetsov方程和(2+1)維BKP方程的幾種精確解,其中包含類周期孤波解、類孤波解和類周期波解.第四章應(yīng)用拓展同宿試驗法求解非線性差分微分方程和分?jǐn)?shù)階非線性偏微分方程.第二節(jié)取新的測試函數(shù)并借助Maple軟件給出離散KdV方程和Toda鏈方程的精確解.第三節(jié)通過變換將分?jǐn)?shù)階偏微分方程化成整數(shù)階偏微分方程,再運用拓展同宿試驗法求得分?jǐn)?shù)階KdV方程的周期孤波解,二孤波解.第五章是結(jié)論與展望部分.
[Abstract]:Nonlinear evolution equations play an important role in the study of nonlinear natural phenomena, and their solving methods are being improved.In this paper, the extended homoclinic test method is used to solve several kinds of nonlinear evolution equations, which consists of five chapters.The first chapter is the introduction, which mainly introduces the research significance of nonlinear evolution equation and the present situation of the extended homoclinic test method, as well as the simple explanation of the discrete equation. Finally, the main work of this paper is introduced.In the second chapter, the extended homoclinic test method is applied to solve the nonlinear equations with constant coefficients.Starting from the constant equilibrium solution of the dissipative Zabolotskaya-Khokhlov equation, a new solitary wave solution of the equation is obtained by using the extended homoclinic test method. The obtained second solitary wave solution is analyzed and the new mechanical characteristics are obtained.By extending the homoclinic test method, the exact solution of the Boussinesq equation is obtained by constructing the test function.By using homoclinic respiratory limit method, the solitary wave solution and rational respiratory wave solution of Boussinesq equation are found, and it is found that the rational respiratory wave solution is the odd wave solution of Boussinesq equation.In chapter 3, the extended homoclinic test method is used to solve the nonlinear equations with variable coefficients.The Zakharov-Kuznetsov equation with variable coefficients and the BKP equation with 21) dimension are solved.With the aid of Maple symbolic computing system, based on the bilinear form of the Zakharov-Kuznetsov equation with variable coefficients and the BKP equation of 21) dimension,This paper introduces a new test function to extend the homoclinic test method and gives some exact solutions of the Zakharov-Kuznetsov equation with variable coefficients and the BKP equation of 21) dimension, including the quasi-periodic solitary wave solution, the similar solitary wave solution and the quasi-periodic wave solution.In chapter 4, the extended homoclinic test method is used to solve nonlinear difference differential equations and fractional nonlinear partial differential equations.In the second section, the exact solutions of the discrete KdV equation and the Toda chain equation are obtained by using the new test function and the Maple software.In the third section, fractional partial differential equations are transformed into integer partial differential equations, and then the periodic solitary wave solutions and the second solitary wave solutions of fractional KdV equations are obtained by extended homoclinic test.The fifth chapter is the conclusion and prospect part.
【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】：O175.29

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