Lie and point symmetries of Nyzhnyk models
Abstract
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.
Related
- 32-point DFT Approximations Based on Minimal Frobenius Error and DFT Symmetries
- Data-driven discovery of intrinsic Lie point symmetries of differential equation systems
- Symmetries for the gKPZ equation via multi-indices
- Symmetries of colored Khovanov-Rozansky homology
- Logarithmic Brunn--Minkowski Inequality under n−2 Reflection Symmetries
BibTeX
@misc{vinnichenko2026point,
title = {{Lie and point symmetries of Nyzhnyk models}},
author = {Oleksandra O. Vinnichenko and Vyacheslav M. Boyko and Roman O. Popovych},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29353},
eprint = {2609.29353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29353},
}
RIS
TY - UNPB
TI - Lie and point symmetries of Nyzhnyk models
AU - Oleksandra O. Vinnichenko
AU - Vyacheslav M. Boyko
AU - Roman O. Popovych
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29353
UR - https://arxiv.org/abs/2609.29353
AB - 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.
ER -