论文标题:Dullin-Gottwald-Holm方程的双线性问题及广义Chebyshev映射的谱分解
Bilinear Problem of Dullin-Gottwald-Holm Equation and Spectral Decomposition of a General Chebyshev Map
论文作者
论文导师 田立新,论文学位 硕士,论文专业 应用数学
论文单位 江苏大学,点击次数 100,论文页数 60页File Size2575K
2007-12-01论文网 http://www.lw23.com/lunwen_231226072/
DGH equation;; Horita method;; Bilinear form;; 1-soliton;; general Chebyshev maps;; Spectral decomposition;; eigenvalues;; eigenvectors
本文研究内容主要分为两部分:第一部分利用Hirota法研究了一类含线性色散项和非线性色散项的新型非线性浅水波方程即Dullin-Gottwald-Holm方程(简称为DGH方程)的双线性问题。DGH方程是Dullin,Gottwald和Holm从Euler方程出发,利用渐近扩张思想研究无旋不可压缩无粘浅层受地球重力和流体自身表面张力影响的运动规律,得到的一类1+1维新型单向浅水波方程,它是一类完全可积型方程。第二部分研究了一种广义Chebyshev映射的谱分解问题。 首先,我们利用倒数变换把DGH方程映射到一个与它关联的双系统方程,即联合DGH方程。把类似的倒数变换应用到DGH方程的Lax对中,通过引入恰当的势函数得到了一组新的可积系统。这组可积系统中包含了DGH方程的变量u(x,t)。通过消掉该可积系统的一个变量得到一个(y,t)空间上的可积系统,该可积系统与BBM方程存在一定的联系。BBM方程的双线性已知,通过恰当的设法可以类似得到关联DGH方程的双线性形式,即得到了DGH方程的双线性形式,并由此得到了DGH方程的1—孤子解。 其次,讨论了一类广义Chebyshev映射的谱分解问题。为了构建广义Chebyshev映射在Frobenius-Perron算子下的一种谱分解。我们定义一种恰当的对偶对或是装备Hilbert空间,这给谱分解提供了一种数学上的意义。构建帐篷映射在Frobenius-Perron算子下的一种谱分解,得到帐篷映射的特征值和特征向量。利用拓扑代换把广义Chebyshev映射拓扑等价为帐篷映射,从而得到广义Chebyshev映射的特征值和特征向量。
In this paper, there is two parts. The first part: we study the Hirota problem for a new nonlinar dispersive shallow water wave equations, named Dullin-Gottwald-Holm (i.e. DGH equation). DGH equation is the 1+1 quadratically nonlinear equation for unidirectional water waves,which was derived by Dullin,Gottwald and Holm,by using asymptotic expansions direcly in the Hamiltonian for Euler"s equations in the irrotational incompressible flow of a shallow layer of inviscid fluid moving under the in fluence of gravity as well as surface tension. It is a completely-integrable equation. The second part: we study the spectral decomposition of a general Chebyshev maps. To begin with, we use reciprocal transformation to map DGH equation to associaed DGH equation. Use samiliar reciprocal transformation on Lax pairs of DGH equation, we obtain a new integrable system through introducing appropriate potential function. This integrable system includes variable u(x,t) .We get a integrable system of (y,t) space through eliminating the variable of the integrable system. The integrable system has some relation with BBM equation. The bilinear form of BBM equation has known, so we can similiarly get the bilinear form of associated DGH equation, then we can get the bilinear form of DGH equation and 1-soliton by a approciate method. In the following, we discuss the spectral decomposition of a general Chebyshev maps. In order to construct the spectral decomposition of the Frobenius-Perron operator for a general Chebyshev maps We define a suitable dual pairs or rigged Hilbert space, which provides mathematical meaning for the spectral decomposition. Construct the spectral decompositions of the family of Tent maps under the Frobenius-Perron operator. We can get the eigenvalues and eigenvectors of the Tent maps. Through the topological equivalence of transformations, we can take the general Chebyshev maps into the Tent maps, then we can get the eigenvalues and eigenvectors of the general Chebyshev maps.
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