Group Analysis of Differential Equations and Integrable Systems − 2018

Stanislav Opanasenko (Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine & Memorial University of Newfoundland, Canada)

Effective generalized equivalence groups

Given a class $\mathcal C$ of differential equations parameterized by arbitrary elements, it is quite possible that equations in the class $\mathcal C$ are related by point transformations explicitly depending on its arbitrary elements. In the language of equivalence groups of classes of differential equations, this means that the class $\mathcal C$ admits a nontrivial generalized equivalence group $\bar {G}^\sim$. Although such groups were introduced over twenty years ago, there were no nontrivial examples provided until recently. In the talk we fill the niche by presenting several such examples.
In general, an equivalence group of a class of differential equations generates a subgroupoid of the equivalence groupoid of this class. The situation with the generalized equivalence group $\bar{G}^\sim$ is more subtle in the sense that there may exist its proper subgroups generating the same subgroupoid of the equivalence groupoid of the class $\mathcal C$ as the group $\bar{G}^\sim$ does. Minimal subgroups of the group $\bar{G}^\sim$ with this property are called effective generalized equivalence groups. In principle, an effective generalized equivalence group may coincide with the group $\bar{G}^\sim$ or be its proper subgroup, with in the latter case it even being nonunique.
For several classes of $(1+1)$-dimensional evolution equations we construct nontrivial effective generalized equivalence groups and show that they may or may not contain the corresponding usual equivalence groups.

The talk is based on joint papers [1,2] with Alex Bihlo, Vyacheslav Boyko and Roman O. Popovych.

[1] Opanasenko S., Bihlo A. and Popovych R.O., Group analysis of general Burgers-Korteweg-de Vries equations, J. Math. Phys. 58 (2017), 081511, 37 pp., arXiv:1703.06932.
[2] Opanasenko S., Boyko V. and Popovych R.O., Enhanced group classification of nonlinear diffusion-reaction equations with gradient-dependent diffusion, 2018, 21 pp., arXiv:1804.08776.