本文選題：Hirota雙線性方法　切入點：廣義WBK方程組　出處：《渤海大學》2017年碩士論文　論文類型：學位論文

【摘要】：隨著孤子理論和科學技術的不斷發(fā)展,近年來求解孤子方程的精確解已成為孤子理論研究中的重中之重.在孤子方程的諸多求解方法中,Hirota雙線性方法起著舉足輕重的作用,是眾多學者的研究焦點,Hirota雙線性方法屬于構造性求解,這種構造性求解法較于其他求解法的優(yōu)勢在于其不依賴于方程的Lax對或者譜問題,正是由于這種構造性求解法簡捷、直觀的特點,激發(fā)了學者們的研究熱潮.近幾年來分數(shù)階問題也引起了人們的廣泛關注,分數(shù)階非線性偏微分方程成為研究熱題.本文一方面利用Hirota雙線性方法來分別構造廣義WBK方程組和廣義AKNS方程族的單孤子解、雙孤子解以及N-孤子解的表達式.另一方面緊緊圍繞分數(shù)階微積分、分數(shù)階導數(shù)的相關知識,構造帶有初邊值條件的變系數(shù)時間分數(shù)階對流擴散方程的新的精確解.本文的主要工作概括如下:首先,在第三章和第四章中推廣并應用Hirota雙線性方法構造廣義WBK方程組和廣義AKNS方程族的單孤子解、雙孤子解、三孤子解,并歸納出N-孤子解的表達式,這得益于成功將廣義WBK方程組和廣義AKNS方程族分別進行轉化,在廣義WBK方程組和廣義AKNS方程族的求解中,關鍵是通過一系列有效變換找到它們的雙線性形式,從而獲得新的孤子解.其次,在第五章中運用分離變量法和Mittag-Leffler函數(shù)的性質獲得一類變系數(shù)時間分數(shù)階對流擴散方程的滿足一定初邊值條件的精確解的統(tǒng)一表達式,進而通過考慮這類對流擴散方程的具體實例和具體初邊值條件得到新的分離變量解,這為求解分數(shù)階的非線性偏微分方程提供重要參考價值.
[Abstract]:With the development of soliton theory and science and technology, the exact solution of soliton equation has become the most important part in the study of soliton theory in recent years, and Hirota bilinear method plays an important role in solving soliton equation. Hirota bilinear method is the focus of many scholars. The advantage of this method over other methods is that it does not depend on the Lax pair of equations or spectral problems. In recent years, the problem of fractional order has also aroused widespread concern. The fractional order nonlinear partial differential equation has become a hot topic. In this paper, on the one hand, we use Hirota bilinear method to construct the single soliton solutions of generalized WBK equations and generalized AKNS equations, respectively. Expressions of double soliton solutions and N-soliton solutions. On the other hand, the knowledge of fractional calculus, fractional derivative, A new exact solution of time fractional convection-diffusion equation with variable coefficients with initial boundary value condition is constructed. The main work of this paper is summarized as follows: first of all, In chapter 3 and chapter 4th, we generalize and apply Hirota bilinear method to construct the single soliton solution, double soliton solution, three-soliton solution of generalized WBK equations and generalized AKNS equation family, and generalize the expression of N-soliton solution. This is due to the successful transformation of the family of generalized WBK equations and generalized AKNS equations. In the solution of generalized WBK equations and generalized AKNS equations, the key is to find their bilinear forms through a series of effective transformations. A new soliton solution is obtained. Secondly, in Chapter 5th, by using the method of separating variables and the properties of Mittag-Leffler function, a unified expression of exact solutions of a class of fractional convection-diffusion equations with variable coefficients is obtained, which satisfies certain initial boundary value conditions. By considering the concrete examples and the concrete initial boundary value conditions of this kind of convection-diffusion equation, a new solution of separated variables is obtained, which provides an important reference value for solving fractional nonlinear partial differential equations.
【學位授予單位】：渤海大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O175.29

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