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反散射變換與指數(shù)函數(shù)法的三個(gè)問(wèn)題研究

發(fā)布時(shí)間:2018-04-01 23:38

  本文選題:反散射變換 切入點(diǎn):指數(shù)函數(shù)法 出處:《渤海大學(xué)》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(lái)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(guò)正散射求出t(28)0時(shí)刻的散射數(shù)據(jù),然后利用時(shí)間發(fā)展式求出散射數(shù)據(jù)隨時(shí)間t的變化規(guī)律,最后通過(guò)位勢(shì)重構(gòu)得到非線性偏微分方程的解.指數(shù)函數(shù)法首先設(shè)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(guò)平衡最高階導(dǎo)數(shù)項(xiàng)和最高次非線性項(xiàng)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(yīng)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(gè)新非等譜AKNS方程組的問(wèn)題.另一方面研究如何解決指數(shù)函數(shù)法在運(yùn)算過(guò)程中出現(xiàn)的“中間表達(dá)式膨脹”問(wèn)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(jiǎn)擬解問(wèn)題.本文的主要工作有:首先,通過(guò)推廣AKNS線性譜問(wèn)題及其時(shí)間發(fā)展式推導(dǎo)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(gè)新非等譜AKNS方程組,然后推廣反散射變換法分別對(duì)其求解,結(jié)果得到這兩個(gè)非等譜方程組的新精確解和新n孤子解,并對(duì)所得部分解的局域空間結(jié)構(gòu)和動(dòng)力演化行為進(jìn)行模擬.其次,通過(guò)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(gè)直接算法,作為算法的兩個(gè)例子,我們將其應(yīng)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(dá)式膨脹”問(wèn)題.最后,通過(guò)定義有理指數(shù)函數(shù)擬解的正負(fù)方冪給出指數(shù)函數(shù)法在求解一類(lèi)變系數(shù)非線性晶格方程時(shí)最簡(jiǎn)擬解的一個(gè)定理及其證明,應(yīng)用我們所給定理可以省略利用平衡方程中最高階導(dǎo)數(shù)項(xiàng)和最高次非線性項(xiàng)的方式確定擬解的過(guò)程,從而將求解這類(lèi)非線性晶格方程的指數(shù)函數(shù)法進(jìn)行改進(jìn).作為算例,我們利用最簡(jiǎn)擬解求解了變系數(shù)mKdV晶格方程,從中展示出最簡(jiǎn)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(xué)位授予單位】:渤海大學(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)期刊論文 前2條

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