Editorial of Papers Published in 2020–2021 in the Mathematics and Symmetry/Asymmetry Section

This editorial is a short review of papers accepted in the Mathematics and Symmetry/Asymmetry section in 2020–2021 about the symmetry methods.

We show these papers in order of publication. In [1], the authors apply a conditional Lie–Bäcklund symmetry method to investigate the functionally generalized separation of variables for quasi-linear diffusion equations with a source. The paper [2] is devoted to studying a generalised $(2+1)$ equation of the Zakharov–Kuznetsov (ZK) $(m,n,k)$ equation involving three arbitrary functions. Lie symmetries and line soliton solutions are derived. Moreover, the authors study the low-order conservation laws. Finally, when possible, the corresponding Hamiltonian formulation is shown. In [3], a geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. The authors stress that their results are applicable to the PDE systems of interest in applied mathematics and mathematical physics. In [4], the author recalls a 1892 paper of L. Bianchi where it was noticed that quite simple transformations of the formulas that describe the Bäcklund transformation of the sine-Gordon equation lead to what is called a nonlocal conservation law. Using the techniques of differential coverings, the author proves that this observation is of a quite general nature. The procedures to construct such conservation laws are described, and a number of illustrative examples are shown. Ref. [5] is devoted to review computations of joint invariants on a linear symplectic space, discuss variations for an extension of group and space and relate this to other equivalence problems and approaches, most importantly to differential invariants. In [6], the authors consider the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions of symplectic and contact linear spaces. This problem is solved by computing differential invariants via the Lie–Tresse theorem. In the paper [7], the authors consider a model from optimal investment theory. It is possible to show that the governing equation possesses an extensive contact symmetry and, through this, it is linearizable. Several exact solutions are provided, including a solution to a particular terminal value problem. The paper [8] considers the local symmetries of algebraic and ordinary differential equations involving a small parameter $ϵ$ in comparison to the symmetry structure of their unperturbed versions (small parameter equal to zero). Exact symmetries of the unperturbed equations, and exact and approximate symmetries (in the Baikov–Gazizov–Ibragimov framework) of the perturbed models are investigated. The main goal of the paper is to address the question of stability of symmetries, when some given equation is perturbed by the addition of small $O\left(ϵ\right)$ terms. Finally, Ref. [9] is devoted to describe the program ReLie, written in the Computer Algebra System Reduce, an open source program for all platforms since 2008. ReLie is able to perform almost automatically the needed computations for Lie symmetry analysis of differential equations. Its source code is freely available too. The use of the program is illustrated by means of some examples.