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首页> 外文期刊>The journal of high energy physics >Duality of positive and negative integrable hierarchies via relativistically invariant fields
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Duality of positive and negative integrable hierarchies via relativistically invariant fields

机译：通过相对义本的字段，通过相对义的字段对正极和负整个层次结构的二元性

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A bstract It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.
机译：一个bstract结果表明，相对论不变性起着积系统的研究了关键作用。利用相对论不变正弦戈登方程，特里忒蔡卡方程，托达场和第二天上方程双重关系，一些连续和离散积积极层次势如修改Korteweg - 德弗里斯层次，潜在的福地 - 吉本斯的层次结构，潜在的无散射Kadomtsev-Petviashvili样（dKPL）的层次结构，差动差分dKPL层次和第二天上层次结构被转换为积的负层次结构包括SG层次结构和Tzitzeica层次结构中，该二维无散射户田层次结构中，两个维户层次和负天上的层次结构。在（1 + 1）维情况下，正/负的层次结构的二元性是由递归运算符和它们的逆之间的二元性保证。在（2 + 1）维的情况下，正/负的层次结构对偶被明确地通过使用正式系列对称性的方法，所述方法mastersymmetry和对偶关系的相对论不变性所示。对于4维天朝制度，二元性问题是由正规的一系列对称的方式首先研究。天上方程的两个优雅通勤递归运营商从正规一系列对称的做法自然会出现，这样的两重性问题也可以由运营商递归的方式进行研究。

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