本文選題：反散射變換　切入點：指數(shù)函數(shù)法　出處：《渤海大學》2017年碩士論文

【摘要】：反散射變換法和指數(shù)函數(shù)法是孤子理論近些年發(fā)展起來的求解非線性偏微分方程的重要方法.反散射變換法首先通過正散射求出t(28)0時刻的散射數(shù)據(jù),然后利用時間發(fā)展式求出散射數(shù)據(jù)隨時間t的變化規(guī)律,最后通過位勢重構得到非線性偏微分方程的解.指數(shù)函數(shù)法首先設所求非線性偏微分方程的指數(shù)函數(shù)有理擬解,然后通過平衡最高階導數(shù)項和最高次非線性項以及收集指數(shù)函數(shù)同次冪系數(shù)確定擬解中待定參數(shù)的值.本文一方面研究如何將反散射變換法推廣應用于求解譜參數(shù)分別按照正弦函數(shù)和有理式發(fā)展的兩個新非等譜AKNS方程組的問題.另一方面研究如何解決指數(shù)函數(shù)法在運算過程中出現(xiàn)的“中間表達式膨脹”問題和如何確定指數(shù)函數(shù)法求解非線性晶格方程的最簡擬解問題.本文的主要工作有:首先,通過推廣AKNS線性譜問題及其時間發(fā)展式推導出譜參數(shù)按照正弦函數(shù)發(fā)展以及按照有理式發(fā)展的兩個新非等譜AKNS方程組,然后推廣反散射變換法分別對其求解,結果得到這兩個非等譜方程組的新精確解和新n孤子解,并對所得部分解的局域空間結構和動力演化行為進行模擬.其次,通過給指數(shù)函數(shù)法擬解的新形式提出指數(shù)函數(shù)法的一個直接算法,作為算法的兩個例子,我們將其應用于KdV方程和Jimbo-Miwa方程.算例表明我們的算法能在較大程度上解決指數(shù)函數(shù)法的“中間表達式膨脹”問題.最后,通過定義有理指數(shù)函數(shù)擬解的正負方冪給出指數(shù)函數(shù)法在求解一類變系數(shù)非線性晶格方程時最簡擬解的一個定理及其證明,應用我們所給定理可以省略利用平衡方程中最高階導數(shù)項和最高次非線性項的方式確定擬解的過程,從而將求解這類非線性晶格方程的指數(shù)函數(shù)法進行改進.作為算例,我們利用最簡擬解求解了變系數(shù)mKdV晶格方程,從中展示出最簡擬解的有效性.
[Abstract]:Inverse scattering transform method and exponential function method are important methods for solving nonlinear partial differential equations developed in recent years. Then the law of scattering data with time t is obtained by time evolution. Finally, the solution of nonlinear partial differential equation is obtained by potential reconstruction. The exponential function method first establishes the rational quasi-solution of exponential function of nonlinear partial differential equation. Then, by balancing the highest derivative term and the highest order nonlinear term and collecting the same power coefficient of the exponential function, the value of the parameters to be determined in the quasi solution is determined. On the one hand, this paper studies how to extend the backscattering transformation method to solve the spectral parameters. The problems of two new nonisospectral AKNS equations developed according to sinusoidal function and rational formula respectively are discussed. On the other hand, how to solve the problem of "intermediate expression expansion" in the operation of exponential function method and how to determine it are studied. Exponential function method is used to solve the most simple quasi solution problem of nonlinear lattice equation. The main work of this paper is as follows: first of all, By extending the AKNS linear spectrum problem and its time evolution, two new nonisospectral AKNS equations with spectral parameters developed according to sinusoidal function and rational formula are derived, and then the generalized inverse scattering transformation method is used to solve them respectively. Results the new exact solutions and new n-soliton solutions of these two nonisospectral equations are obtained, and the local spatial structure and dynamic evolution behavior of the obtained partial solutions are simulated. This paper presents a direct algorithm of exponential function method by giving a new form of solution to exponential function method, as two examples of the algorithm. We apply it to KdV equation and Jimbo-Miwa equation. The example shows that our algorithm can solve the problem of "intermediate expression expansion" in exponential function method to a large extent. Finally, By defining the positive and negative power of the quasi-solution of rational exponential function, a theorem and proof of the simplest quasi-solution of exponential function method for a class of nonlinear lattice equations with variable coefficients are given. By using the theorem we give, we can omit the process of determining the quasi solution by means of the highest derivative term and the highest subnonlinearity term in the equilibrium equation, and then improve the exponential function method for solving the nonlinear lattice equation of this kind of equation. In this paper, we solve the mKdV lattice equation with variable coefficients by using the simplest quasi solution, which shows the validity of the simplest quasi solution.
【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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本文編號：1697849