Integrable mappings of the plane preserving biquadratic invariant curves
Integrable mappings of the plane preserving biquadratic invariant curves
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保留双二次不变曲线的平面的可积映射
DOI：
10.1088/0305-4470/34/34/308
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发表时间：
2001
期刊：
影响因子：
--
通讯作者：
J. Roberts
中科院分区：
文献类型：
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作者：
Apostolos Iatrou;J. Roberts
我们提供了一个构造平面的可积映射的一般框架，它保持了双二次不变曲线族B(x，y，K)的单参数族，其中K的参数化是非常一般的。这些映射通过构造是可逆的(即它们是两个对合的组合)，并且可以被证明是保持度量的。它们推广了McMillan和Quispel，Roberts和Thompson以前给出的可积映射。通过考虑对称双二次情形到标准型的变换，我们给出了作用在每条不变曲线上的对称可积映射的标准形。我们给出了对称可积映射的一个大子类的Lax对，其中至少包括12参数对称Quispel-Roberts-Thompson映射族的一个10参数子族。
We provide a general framework to construct integrable mappings of the plane that preserve a one-parameter family B(x,y,K) of biquadratic invariant curves where parametrization by K is very general. These mappings are reversible by construction (i.e. they are the composition of two involutions) and can be shown to be measure preserving. They generalize integrable maps previously given by McMillan and Quispel, Roberts and Thompson. By considering a transformation of the case of the symmetric biquadratic to a canonical form, we provide a normal form for the symmetric integrable map acting on each invariant curve. We give a Lax pair for a large subclass of our symmetric integrable maps, including at least a 10-parameter subfamily of the 12-parameter symmetric Quispel-Roberts-Thompson maps.