Sine-Gorden方程的某些定解问题解的存在性及其数值分析方法(一)

有许多作者,从物理,几何和分析的角度,对Sine-Gorden方程(1)解的性质以及更广泛的方程作了研究。 Montel将常微分方程中Euler折线法推广到拟线性偏微分方程中,证明了方程(2)的特征问題(即第一问題)解的存在性(“分区”考虑),但此方程的第三四五问题没有解决。当定解条件特殊时,我们研究了方程(1)的第三四和五问题解在全平面上存在。本文仅讨论了第五问题(支柱对称)解的存在性,其方法采用Montel差分方法的思想,提出了整体构造近似解的计算方法(即“转圈构造法”),克服了困难,得到了近似解,然后应用Arzela定理,证明了解的存在性,从近似解构造本身,实际上给出了数值分析方法,有助于实际应用。对方程和条件的右端可适当推广。对于方程(1)的第三四问题以及第五问题(支柱不对称)较复杂,但皆可化为支柱对称情况解决(待续)。

The Existence of Solution of Some Definite Problems about the Sine-Gorden Equation and the Method of Numerical Analysis

Many authors have studied the properties of solutions of the Sine-Gorden equation =sin(u) and more equations from physical, geometrical and and analytical angles. Montel has generalized Euler's method of polygonal arc which applies to ordinary differential equations, to quasiliner partial differetial equations. He has grven the existence proof of solution of characteristic problem (i.e. the first problem) about the equation =f(x, y, u). But the third, the forth and the fifth problems has have not been solved. We have found that undercertain conditions, there exist solutions of all three problems on the whole plane. But in this paper, we oniy discuss the existence of solution of the fifth problem with support and symmetric conditieon. The adopted method is from the thought of Montel's difference method. Also we give a computational method of wholy constructing the approximate solution, i.e. the circular construction. We show the approximate solution and then prove the existence of solution by using Arzela therom. In the process of constructing the approximate solution, we actually show the method of numerical analysis. Also, the founction on the right hand of the equation and the condition can be genralized properly. It's complicated to solve the third, the forth and the fifth problems about the Sine-Gorden equation (with support and non-symmetric condition). But all them can be solved by being transformed into those with support and symmetric condition. (on continue)