Wadati-Konno-Ichikawa方程的精确解

发布时间：2018-03-30 16:45

本文选题：反散射方法　切入点：GLM方程　出处：《中国科学技术大学》2017年博士论文

【摘要】：本文使用反散射方法、Riemann-Hilbert方法和达布变换方法研究Wadati-Konno-Ichikawa(WKI)方程的孤立子解、呼吸子解和怪波解.第一章主要介绍本文的研究对象——WKI方程,以及本文所使用的三种方法.第二章给出反散射方法求解WKI方程的N阶孤立子解.WKI方程的约斯特解比较特殊,当谱参数趋向于无穷时,约斯特解的两个分量并不是趋向于1或者0的情形,而是关于势函数的变量.文章将原来约斯特解的第二个分量替换为该分量与谱参数的商,化简约斯特解的渐近行为,并使用变换之后约斯特解的三角核(triangular kernel)表示构造 Gel'fand-Levitan-Marchenko(GLM)方程,求解GLM方程,从而得到WKI方程的N阶孤立子解.第三章给出Riemann-Hilbert方法求解WKI方程的N阶孤立子解.第二章约斯特解的渐近行为表明WKI方程的约斯特解不能直接用来构造Riemann-Hilbert问题,否则将会导致Riemann-Hilbert问题不可解.为解决此困难,文中引入变换矩阵得到合适的约斯特解,并对谱参数在原点和无穷远处进行谱分析构造可解的Riemann-Hilbert问题,从而求解WKI方程的N阶孤立子解.第四章给出Darboux变换方法求解WKI方程的N阶解.由于WKI方程拉克斯对形式的特殊型,文中首先引入hodograph变换得到WKI方程的等价形式——广义WKI方程,然后研究广义WKI方程的达布变换,最后将广义WKI方程的解转化为WKI方程的解.此外,我们还研究广义WKI方程N阶孤立子解的渐近形式,怪波的解析性以及转化为WKI方程的圈形孤立子解和圈形怪波解.
[Abstract]:In this paper, the Riemann-Hilbert method and Darboux transform method are used to study the soliton solution, respiration solution and odd wave solution of the Wadati-Konno-Ichikawa WKI equation. In chapter 2, the inverse scattering method is given to solve the N-order soliton solution of the WKI equation. The Yost solution of the WKI equation is quite special, when the spectral parameters tend to be infinite, The two components of the Jost solution are not the case that tends to 1 or 0, but the variables of the potential function. In this paper, the second component of the original solution is replaced by the quotient of the component and the spectral parameter, and the asymptotic behavior of the solution is simplified. The GLM equation is constructed by using the triangular kernels of the Jost's solution after the transformation, and the Gelfand-Levitan-Marchenkoan GLM equation is constructed, and the GLM equation is solved. In chapter 3, the Riemann-Hilbert method is given to solve the N-order soliton solution of the WKI equation. In chapter 2, the asymptotic behavior of the Jost solution shows that the WKI equation solution cannot be directly used to construct the Riemann-Hilbert problem. Otherwise, the Riemann-Hilbert problem will not be solvable. In order to solve this difficulty, the transformation matrix is introduced to obtain the suitable solution of the Riemann-Hilbert problem. The spectral parameters are analyzed at the origin and infinity to construct the solvable Riemann-Hilbert problem. In chapter 4, the Darboux transformation method is given to solve the N-order solution of WKI equation. Because of the special form of the WKI equation, the hodograph transformation is introduced to obtain the equivalent form of WKI equation, the generalized WKI equation. Finally, the solution of the generalized WKI equation is transformed into the solution of the WKI equation. In addition, we also study the asymptotic form of the N-order soliton solution of the generalized WKI equation. The analytic properties of strange waves and the solutions of circular solitons and circular strange waves transformed into WKI equations.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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