发表机构
Institute of Mathematics of NAS of Ukraine; Department of Mathematics, Kyiv Academic University; Mathematical Institute, Silesian University in Opava（乌克兰国家科学院数学研究所; 基辅学术大学数学系; 俄帕瓦西里西亚大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对由Nyzhnyk系统衍生的模型层级，计算其最大李不变伪代数及对称伪群，并基于色散与无色散模型的对应关系，完整分类了Nyzhnyk方程及其Lax表示的子代数。
AI 中文摘要
我们考虑一类模型层级，这些模型可通过以下方式从原始Nyzhnyk系统获得：对参数施加条件、引入势或伪势、关于缩放参数执行极限过程、应用微分替换以及将部分独立和/或因变量解释为复数或实数。因此，在Nyzhnyk模型中，可以区分对称与不对称、色散与无色散、标准与修正，以及实数、复数、混合和特定模型。还可以区分单个偏微分方程或此类方程组，以及分别在色散或无色散情形下的线性或非线性Lax表示。对于每个指定模型，我们计算最大李不变伪代数，并在对称情形下，利用基于megaideal的代数方法版本求出点对称和接触对称伪群。结果表明，这些伪代数之间以及这些伪群之间的关系由相应模型之间的关系诱导。在对称和特定情形下，所有这些伪代数中均被单列出定义（有限维）子代数。基于无色散与色散模型之间已建立的对应关系，我们完全分类了（色散对称势）Nyzhnyk方程的最大李不变伪代数的一维和二维子代数，以及其线性Lax表示的最大李不变伪代数的一维子代数。
英文摘要
We consider a hierarchy of models that can be obtained from the original Nyzhnyk system by imposing conditions on parameters, introducing potentials or pseudopotentials, performing limiting processes with respect to a scaling parameter, applying differential substitutions and interpreting parts of independent and/or dependent variables as complex or real. Therefore, among the Nyzhnyk models, one can distinguish between symmetric and asymmetric, dispersive and dispersionless, standard and modified, as well as real, complex, mixed and specific models. One can also distinguish single partial differential equations or systems of such equations, as well as linear or nonlinear Lax representations in the dispersive or dispersionless cases, respectively. For each specified model, we compute the maximal Lie invariance pseudoalgebra and, in the symmetric case, find the point- and contact-symmetry pseudogroups using the megaideal-based version of the algebraic method. It is shown that relations between these pseudoalgebras and between these pseudogroups are induced by relations between the corresponding models. Defining (finite-dimensional) subalgebras are singled out in all these pseudoalgebras in the symmetric and specific cases. Based on the established correspondences between the dispersionless and dispersive models, we completely classify one- and two-dimensional subalgebras of the maximal Lie invariance pseudoalgebra of the (dispersive symmetric potential) Nyzhnyk equation and one-dimensional subalgebras of the maximal Lie invariance pseudoalgebra of its linear Lax representation.
Comments44 pages, 1 table, further study of the Nyzhnyk models, initiated in arXiv:2211.09759 and arXiv:2308.03744
