The sine‐Gordon equations: Complete and partial integrability
The sine‐Gordon equations: Complete and partial integrability
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DOI：
10.1063/1.526415
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发表时间：
1984-07
影响因子：
1.3
通讯作者：
J. Weiss
中科院分区：
文献类型：
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作者：
J. Weiss
已知一时空维度中的正弦-戈登方程具有 Painleve 性质并且是完全可积的。展示了“奇异流形”分析方法如何获得该方程的 Backlund 变换和 Lax 对。发现了与高阶 KdV 方程序列的联系。 “修改的”正弦-戈登方程是根据奇异流形定义的。这些方程被证明是相同的 Painleve。此外，某些“理性”解决方案是迭代构建的。双正弦-戈登方程被证明不具备 Painleve 性质。然而，如果奇异流形定义了“仿射最小曲面”，则方程具有可积解。这种限制被称为“部分可积性”。 (N+1) 个变量（N 个空间，1 个时间）中的正弦-戈登方程（其中 N 大于 1）不具备 Painleve 性质。部分可积的条件要求奇异流形是一个“爱因斯坦空间……”
The sine–Gordon equation in one space‐one time dimension is known to possess the Painleve property and to be completely integrable. It is shown how the method of ‘‘singular manifold’’ analysis obtains the Backlund transform and the Lax pair for this equation. A connection with the sequence of higher‐order KdV equations is found. The ‘‘modified’’ sine–Gordon equations are defined in terms of the singular manifold. These equations are shown to be identically Painleve. Also, certain ‘‘rational’’ solutions are constructed iteratively. The double sine–Gordon equation is shown not to possess the Painleve property. However, if the singular manifold defines an ‘‘affine minimal surface,’’ then the equation has integrable solutions. This restriction is termed ‘‘partial integrability.’’ The sine–Gordon equation in (N+1) variables (N space, 1 time) where N is greater than one is shown not to possess the Painleve property. The condition of partial integrability requires the singular manifold to be an ‘‘Einstein space ...