【摘要】：反散射作為重要的數(shù)學(xué)物理方法,主要用于研究非線(xiàn)性可積偏微分方程.2005年,Manakov和Santini提出了一種新的反散射方法,通過(guò)研究與單參數(shù)向量場(chǎng)形式的Lax對(duì)相聯(lián)系的正問(wèn)題和反問(wèn)題,求解流體力學(xué)型非線(xiàn)性可積偏微分方程.本文主要基于這種新的反散射方法討論(3+1)維的Dunajski方程的精確解.該方程與來(lái)源于Einstein(反)自對(duì)偶引力場(chǎng)方程,在數(shù)學(xué)和物理中具有重要的研究意義.本文通過(guò)構(gòu)造與雙曲函數(shù)有關(guān)的可解非線(xiàn)性Riemann-Hilbert問(wèn)題,以及通過(guò)動(dòng)力系統(tǒng)來(lái)構(gòu)造滿(mǎn)足條件的非線(xiàn)性Riemann-Hilbert問(wèn)題來(lái)研究Dunajski方程的解,同時(shí)考慮線(xiàn)性化的Dunajski方程的解的長(zhǎng)時(shí)間演化性。
[Abstract]:Backscattering, as an important mathematical and physical method, is mainly used to study nonlinear integrable partial differential equations. In 2005, Manakov and Santini proposed a new backscattering method. By studying the positive and inverse problems related to Lax pairs in the form of single parameter vector fields, the nonlinear integrable partial differential equations of fluid mechanics are solved. In this paper, the exact solution of (31) dimensional Dunajski equation is discussed based on this new backscattering method. The equation and the equation derived from Einstein (inverse) self-dual gravitational field are of great significance in mathematics and physics. In this paper, the solution of Dunajski equation is studied by constructing the solvable nonlinear Riemann-Hilbert problem related to hyperbolic function and constructing the nonlinear Riemann-Hilbert problem satisfying the condition by dynamic system. At the same time, the long time evolution of the solution of the linear Dunajski equation is considered.
【學(xué)位授予單位】：華僑大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O175

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1 郭吉?jiǎng)?Dunajski方程的精確解及其線(xiàn)性系統(tǒng)的長(zhǎng)時(shí)間演化性[D];華僑大學(xué);2017年

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本文編號(hào)：2484875