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In this work, symmetry groups are used to determine symmetry reductions of a (2 + 1)‐dimensional Keller–Segel system depending on two arbitrary functions. We show that the point symmetries of the considered Keller–Segel system comprise an infinite‐dimensional Lie algebra which involves three arbitrary functions. By way of example, we have used these point symmetries to reduce straightaway the given system of second‐order partial differential equations to a system of second‐order ordinary differential equations. Moreover, we are allowed to substitute one of the dependent variables from one of the equations into the other, leading to an equivalent fourth‐order nonlinear ordinary differential equation. This equation is reduced through the use of solvable symmetry subalgebras, and some exact solutions are obtained for a particular case. [ABSTRACT FROM AUTHOR]

Nonlocally related partial differential equation (PDE) systems can play an important role in the analysis of a given PDE system. In this paper, a new systematic method for obtaining nonlocally related PDE systems is developed. In particular, it is shown that if a PDE system admits q point symmetries whose infinitesimal generators form a q-dimensional solvable Lie algebra, then, for each resulting q-dimensional solvable algebra chain, one can obtain systematically q nonlocally related PDE systems. Such nonlocally related systems are obtained for a general class of nonlinear reaction–diffusion equations admitting two- to four-dimensional solvable algebras. [ABSTRACT FROM AUTHOR]

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs. [ABSTRACT FROM AUTHOR]