本文選題：李對(duì)稱(chēng)分析 + 解析解��； 參考：《中國(guó)礦業(yè)大學(xué)》2017年碩士論文

【摘要】：本文主要研究幾類(lèi)非線(xiàn)性微分方程的對(duì)稱(chēng),守恒律與解析解.首先簡(jiǎn)單介紹了相關(guān)的研究背景和本文的主要工作.然后,將李對(duì)稱(chēng)方法推廣到一種壓力波Kudryashov-Sinelshchikov方程上,求出其無(wú)窮小對(duì)稱(chēng)的向量場(chǎng)與群不變解,并在此基礎(chǔ)上得到了該方程的精確解析冪級(jí)數(shù)解.第三章和第四章,對(duì)廣義的Korteweg-de Vries-Fischer方程進(jìn)行詳細(xì)的對(duì)稱(chēng)分類(lèi)分析,分析出該方程無(wú)窮小對(duì)稱(chēng)的向量場(chǎng).在此基礎(chǔ)上研究其自伴隨性質(zhì),最后基于伴隨方程法和守恒乘子直接構(gòu)造法的相關(guān)理論,系統(tǒng)地研究該方程的守恒律.第五章和第六章,基于Hirota雙線(xiàn)性法,將Bell多項(xiàng)式與黎曼theta函數(shù)推廣到(3+1)-維廣義的B-type Kadomtsev-Petviashvili方程中,得到了該方程的雙線(xiàn)性形式和解析解,包括孤子解和周期波解.并進(jìn)一步研究所求出周期波解的漸近性質(zhì),證明了滿(mǎn)足某種極限條件下,周期波解退化成孤子解.并對(duì)方程的孤子解,周期波解以及周期波解的漸近情況進(jìn)行圖形模擬與分析.接下來(lái),基于可積離散化理論,研究了薛定諤型方程Eckhaus-Kundu方程的半離散和全離散,并給出相應(yīng)的孤子解以及孤子解的圖形模擬.最后對(duì)全文進(jìn)行簡(jiǎn)單的總結(jié)和展望.
[Abstract]:In this paper, the symmetry, conservation laws and analytical solutions of some nonlinear differential equations are studied.Firstly, the research background and the main work of this paper are briefly introduced.Then, the lie symmetry method is extended to a pressure wave Kudryashov-Sinelshchikov equation, and its infinitesimal symmetric vector field and group invariant solutions are obtained, and the exact analytic power series solutions of the equation are obtained.In the third and fourth chapters, the generalized Korteweg-de Vries-Fischer equation is analyzed in detail, and the vector fields of the equation are analyzed.On this basis, the self-adjoint property of the equation is studied. Finally, the conservation law of the equation is studied systematically based on the related theories of the adjoint equation method and the direct construction method of the conservation multiplier.In the fifth and sixth chapters, based on the Hirota bilinear method, the Bell polynomial and Riemannian theta function are extended to the generalized B-type Kadomtsev-Petviashvili equation with 31- dimension. The bilinear form and analytic solution of the equation are obtained, including soliton solution and periodic wave solution.Furthermore, the asymptotic properties of periodic wave solutions are studied, and it is proved that the periodic wave solutions degenerate into soliton solutions under certain limit conditions.The asymptotic behavior of soliton solution, periodic wave solution and periodic wave solution of the equation are simulated and analyzed graphically.Then, based on the integrable discretization theory, the semi-discrete and fully discrete Schrodinger equation Eckhaus-Kundu equations are studied, and the corresponding soliton solutions and the graphical simulation of the soliton solutions are given.At last, the paper makes a brief summary and prospect.
【學(xué)位授予單位】：中國(guó)礦業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O175.29

【參考文獻(xiàn)】

相關(guān)博士學(xué)位論文 前1條

1 田守富;非線(xiàn)性微分方程的若干解析解方法與可積系統(tǒng)[D];大連理工大學(xué);2012年

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本文編號(hào)：1760559