Integrability properties of the differential-difference Kadomtsev–Petviashvili hierarchy and continuum limits
Integrability properties of the differential-difference Kadomtsev–Petviashvili hierarchy and continuum limits
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DOI：
10.1088/0951-7715/26/12/3197
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发表时间：
2012-11
期刊：
影响因子：
1.7
通讯作者：
Wei-jie Fu;Lin Huang;K M Tamizhmani;Da‐jun Zhang
中科院分区：
文献类型：
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作者：
Wei-jie Fu;Lin Huang;K M Tamizhmani;Da‐jun Zhang
本文揭示了差分Kadomtsev-Petviashvili系统与(连续)Kadomtsev-Petviashvili系统之间的明确联系。利用Lax三联法得到了等谱差分Kadomtsev-Petviashvili流和非等谱差分Kadomtsev-Petviashvili流。Lax三联体还为所获得的流提供了简单的零曲率表示。非等谱流动充当主对称，为所获得的流动提供递归关系。这些流产生一个李代数，它是研究更多可积性性质的起点。我们得到了等谱差分Kadomtsev-Petviashvili方程的对称性、哈密顿量和守恒量。分别由流、对称性、哈密顿量和守恒量生成的李代数具有相同的结构。最后，我们给出了一个不同于Miwa变换的一致连续极限。通过定义某些元素的次数w.r.t.在连续统极限的基础上，证明了在一致连续统极限下，微分-差分Kadomtsev-Petviashvili方程及其Lax三元组、零曲率表示和可积性特征都是连续的。还解释了李代数在连续统极限下的结构变形。
The paper reveals clear links between the differential-difference Kadomtsev–Petviashvili hierarchy and the (continuous) Kadomtsev–Petviashvili hierarchy. Isospectral differential-difference Kadomtsev–Petviashvili flows and non-isospectral differential-difference Kadomtsev–Petviashvili flows are derived through Lax triad approach. The Lax triads also provide simple zero-curvature representations for the obtained flows. The non-isospectral flow acts as a master symmetry to provide recursive relations for the obtained flows. These flows generate a Lie algebra, which is a starting point for investigating more integrability properties. We derive symmetries, Hamiltonians and conserved quantities for the isospectral differential-difference Kadomtsev–Petviashvili hierarchy. The Lie algebras generated respectively by the flows, symmetries, Hamiltonians and conserved quantities have same structures. Finally, we provide a uniform continuum limit which is different from Miwa's transformation. By means of defining degrees of some elements w.r.t. the continuum limit, we prove that in the uniform continuum limit the differential-difference Kadomtsev–Petviashvili hierarchies together with their Lax triads, zero-curvature representations and integrability characteristics go to their continuous counterparts. Structure deformation of Lie algebras in the continuum limit is also explained.