本文選題：非局部對稱 + 守恒律　； 參考：《西北大學(xué)》2015年博士論文

【摘要】：如今非線性現(xiàn)象越來越多的出現(xiàn)在自然科學(xué)與社會科學(xué)中,用來描述該現(xiàn)象的微分方程受到相關(guān)數(shù)學(xué)家和物理學(xué)家的關(guān)注.本文主要研究了幾個偏微分方程的非局部對稱和守恒律以及精確解.包括兩分量μ-Camassa-Holm方程,帶有色散項γux的μ-Camassa-Holm方程,Foursov方程,復(fù)Camassa-Holm方程.本文還考慮淺水波方程3×3的譜問題,構(gòu)造了Degasperis-Procesi方程和Novikov方程的無窮多守恒律,證明了Degasperis-Procesi方程和Novikov方程沒有依賴其擬勢函數(shù)的非局部對稱.本文主要內(nèi)容如下：首先,主要介紹了所研究方程的背景,以及研究非局部對稱和求精確解所用的基本理論和方法.其次,我們研究了兩分量μ-Camassa-Holm方程的非局部對稱和守恒律.該方程是兩分量Camassa-Holm方程的一個short-wave極限.先從該方程的幾何可積性出發(fā),得到方程的擬勢函數(shù)進(jìn)而構(gòu)造方程的守恒律,并且得到方程無窮多守恒律.其次,根據(jù)這個方程僅有平凡的李對稱和方程特殊結(jié)構(gòu),我們研究了它的非局部對稱.最后,得到兩分量μ-Camassa-Holm方程依賴其勢函數(shù)的非局部對稱.然后考慮帶有低階色散項γux的μ-Camassa-Holm方程,從該方程Lax對出發(fā),借助其波函數(shù),構(gòu)造方程擬勢函數(shù),利用所得的擬勢函數(shù)得到守恒律以及無窮多守恒量,進(jìn)而運用李對稱方法得到該方程依賴擬勢函數(shù)的非局部對稱.修正的μ-Camassa-Holm方程是作為修正Camassa-Holm方程非局部形式提出來的,從該方程幾何可積性出發(fā),求出擬勢函數(shù),進(jìn)一步構(gòu)造無窮多守恒律,并且證明該方程不具有依賴擬勢函數(shù)的非局部對稱.短脈沖方程是包含一階線性項ux的修正μ-Camassa-Holm方程的伸縮極限方程,本章最后得到該方程的擬勢函數(shù),無窮多守恒律.第三,運用不變子空間的方法結(jié)合廣義條件對稱方法研究b-family系統(tǒng)的精確解.這種解是由三維的不變子空間生成,對應(yīng)于方程的一類廣義條件對稱,其中系數(shù)依賴于時間的函數(shù)滿足Emden方程,通過對Emden方程解的結(jié)構(gòu)進(jìn)行分析得到了方程解的結(jié)構(gòu),給出了解整體存在性和爆破條件,進(jìn)而對解的結(jié)構(gòu)給出完整的刻畫.然后研究對偶Foursov方程的尖峰孤子解.先得到方程的弱形式,然后利用方程的弱形式證明該方程具有尖峰孤子解.最后討論復(fù)Camassa-Holm系統(tǒng),這類系統(tǒng)可以從Mobus幾何曲線流中得到,我們證明了該方程具有尖峰孤子解.第四,考慮Degasperis-Procesi方程和Novikov方程的無窮多守恒律以及非局部對稱,Degasperis-Procesi方程和Novikov方程是典型的具有3×3譜問題的波方程,以構(gòu)造2×2譜問題方程的非局部對稱和守恒律的方法為基礎(chǔ)擴(kuò)展到3×3譜問題的方程.先構(gòu)造出Degasperis-Procesi方程的兩個勢函數(shù)和兩個擬勢函數(shù),進(jìn)一步通過勢函數(shù)做冪級數(shù)展開,比較相同普參數(shù)λ的系數(shù)達(dá)到無窮多守恒律,最終運用李對稱方法得到該方程的超定方程組,對方稱組分析得到Degasperis-Procesi方程不具有依賴其擬勢函數(shù)的非局部對稱.同樣對于Novikov方程,求出其勢函數(shù)和擬勢函數(shù),對勢函數(shù)進(jìn)行同樣處理,得到無窮多守恒量,最后,知道該方程組不具有依賴其勢函數(shù)的非局部對稱.最后,對尚待解決的問題進(jìn)行了討論.
[Abstract]:Nowadays, the nonlinear phenomena appear more and more in the natural and social sciences. The differential equations used to describe the phenomenon are concerned by the relevant mathematicians and physicists. This paper mainly deals with the non local symmetry and conservation laws and exact solutions of several partial differential equations, including the dichotomous -Camassa-Holm equation with the dispersion term. The -Camassa-Holm equation of gamma UX, the Foursov equation, the complex Camassa-Holm equation. This paper also considers the spectral problem of the shallow water wave equation 3 x 3, and constructs the infinitely many conservation laws of the Degasperis-Procesi equation and the Novikov equation. It is proved that the Degasperis-Procesi equation and the Novikov equation do not depend on the non local symmetry of the potential function. First, we introduce the background of the equation and the basic theory and method to study the nonlocal symmetry and precision. Secondly, we study the nonlocal symmetry and conservation law of the two component -Camassa-Holm equation. This equation is a short-wave limit of the two component Camassa-Holm equation. First, the geometry of the equation is derived. Based on the integrability, the quasi potential function of the equation is obtained and the conservation law of the equation is constructed, and the infinitely many conservation laws of the equation are obtained. Secondly, we have studied the nonlocal symmetry of the equation based on the ordinary Lie symmetry and the special structure of the equation. Finally, we get the non local symmetry of the -Camassa-Holm equation of the two points dependent on its potential function. Then we consider the muon -Camassa-Holm equation with the low order dispersion term gamma UX. Starting from the Lax pair of the equation, the pseudo potential function of the equation is constructed by its wave function, and the conservation law and infinitely many conservation quantities are obtained by using the obtained potential function. Then the Lie symmetry method is used to obtain the non local symmetry of the equation depending on the quasi potential function. The modified mu -Camassa-Ho is obtained. The LM equation is derived from the non local form of the modified Camassa-Holm equation. From the geometric integrability of the equation, the pseudo potential function is obtained, and the infinitely many conservation laws are constructed, and it is proved that the equation does not have the non local symmetry dependent on the pseudo potential function. The short pulse equation is a modified mu -Camassa-Holm equation containing the first order linear term UX. In this chapter, the quasi potential function and infinite conservation law of the equation are obtained. Third, the exact solution of the B-family system is studied with the method of invariant subspace combined with the generalized conditional symmetry method. This solution is generated by a three dimensional invariant subspace, corresponding to a class of generalized conditional symmetry of the equation, and the coefficients depend on the time. The function satisfies the Emden equation. By analyzing the structure of the solution of the Emden equation, the structure of the solution of the equation is obtained. The whole existence and blasting conditions are given. Then the structure of the solution is fully depicted. Then the peak soliton solution of the dual Foursov equation is studied. First, the weak form of the square path is obtained, and then the weak form of the equation is proved by the weak form of the equation. The equation has a peak soliton solution. Finally, we discuss the complex Camassa-Holm system, which can be obtained from the Mobus geometric flow. We prove that the equation has a peak soliton solution. Fourth, the infinitely many conservation laws of the Degasperis-Procesi equation and the Novikov equation, and the non local symmetry, the Degasperis-Procesi equation and the Novikov equation are considered. It is a typical wave equation with 3 x 3 spectrum problems, which is extended to the 3 x 3 spectral problem based on the non local symmetry and conservation law of the 2 x 2 spectral equation. It first constructs the two potential functions and two pseudo potential functions of the Degasperis-Procesi equation, and further expands the power series by the potential function number, and compares the same general parameter [lambda]. The coefficients reach infinitely many conservation laws. Finally, we use the Lie symmetry method to get the hyperset equations of the equation. The other is called the group analysis to obtain the non local symmetry of the Degasperis-Procesi equation which does not depend on its potential function. Also, the potential function and potential function are obtained for the Novikov equation, and the potential function is treated equally, and the infinitely many conservation are obtained. Finally, we know that the equations do not have non local symmetry depending on their potential functions. Finally, we discuss the unsolved problems.
【學(xué)位授予單位】：西北大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2015
【分類號】：O175.29

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 宋軍鋒;屈長征;;Geometric Integrability of Two-Component Camassa-Holm and Hunter-Saxton Systems[J];Communications in Theoretical Physics;2011年06期

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本文編號：1969481