论文标题
双曲线和椭圆形正弦和sinh-gordon方程的几何bäcklund转换的叠加公式
Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations
论文作者
论文摘要
我们为六种Bäcklund变换提供了叠加公式，该案例对应于三维伪euclidean空间中的太空状和时间状的表面。在每种情况下，表面都有恒定的阴性或正曲线曲率，它们对应于以下方程之一的溶液：正弦 - 戈登，sinh-gordon，椭圆形的正弦 - 戈登和椭圆形的sinh-gordon方程。叠加公式在Bäcklund转化的第一次整合后，在代数上提供了无限的许多解决方案。这样的转换和相应的叠加公式提供了相同双曲方程的解，而它们显示了椭圆方程的异常特性。 Bäcklund变换将椭圆形SINH-GORDON方程的溶液与椭圆形正弦方程的溶液和叠加公式提供的溶液提供了相同的椭圆方程的溶液。给出了明确的例子和插图。
We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature and they correspond to solutions of one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic sine-Gordon and the elliptic sinh-Gordon equation. The superposition formulae provide infinitely many solutions algebraically after the first integration of the Bäcklund transformation. Such transformations and the corresponding superposition formulae provide solutions of the same hyperbolic equation, while they show an unusual property for the elliptic equations. The Bäcklund transformation alternates solutions of the elliptic sinh-Gordon equation with those of the elliptic sine-Gordon equation and the superposition formulae provide solutions of the same elliptic equation. Explicit examples and illustrations are given.
