A direct method for solving the generalized sine-Gordon equation II
A direct method for solving the generalized sine-Gordon equation II
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DOI：
10.1088/1751-8113/43/37/375201
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Y. Matsuno
中科院分区：
文献类型：
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作者：
Y. Matsuno
导出了广义Sine-Gordon(SG)方程Utx=(1+ν∂2x)Sin u，它是SG方程的一个可积推广。在之前的一篇论文中(Matsuno 2010 J.Phys.答：数学。西奥。43 105204)，我们发展了一种系统的方法来求解ν=−=1的广义SG方程。在这里，我们处理ν=1的方程。通过对方程的解析解，我们发现解的结构与以前的方程有很大的不同。特别地，我们证明了该方程存在扭结解和呼吸解，并且不允许存在像I中所得到的回路孤子那样的多值解。我们还证明了在适当的标度范围内，该方程退化为短脉冲和SG方程。给出了多孤子解的极限形式。最后，我们给出了利用一种新的Bäck lund变换将SG方程和广义SG方程的解联系起来，从而导出无穷多个守恒律的方法。
The generalized sine-Gordon (sG) equation utx = (1 + ν∂2x)sin u was derived as an integrable generalization of the sG equation. In a previous paper (Matsuno 2010 J. Phys. A: Math. Theor. 43 105204) which is referred to as I, we developed a systematic method for solving the generalized sG equation with ν = −1. Here, we address the equation with ν = 1. By solving the equation analytically, we find that the structure of solutions differs substantially from that of the former equation. In particular, we show that the equation exhibits kink and breather solutions and does not admit multi-valued solutions like loop solitons as obtained in I. We also demonstrate that the equation reduces to the short pulse and sG equations in appropriate scaling limits. The limiting forms of the multisoliton solutions are also presented. At last, we provide a recipe for deriving an infinite number of conservation laws by using a novel Bäcklund transformation connecting solutions of the sG and generalized sG equations.