Lie symmetry classification of a coupled nonlinear cross-diffusion system in radial geometry

TL;DR

This paper performs Lie symmetry analysis on a coupled nonlinear cross-diffusion system with radial geometry, classifying its symmetries and revealing conditions for additional invariances.

Contribution

It provides a comprehensive classification of symmetries for the system, highlighting the impact of nonlinear coupling and geometry on invariance properties.

Findings

System admits time translation and parabolic scaling symmetries.

Additional symmetries depend on structural assumptions of constitutive functions.

Larger symmetry algebras occur only in degenerate or linearizable cases.

Abstract

In this work, Lie symmetry analysis is performed on a coupled nonlinear cross-diffusion system with varying cross-section geometry. The system describes two interacting quantities whose material properties, namely the capacity functions and the diffusion coefficients, depend nonlinearly on the dependent variables. The classical Lie invariance criterion produces a set of sixteen determining equations for infinitesimal symmetry generators. The determining equations are solved by first establishing the universal geometric structure of the admitted generators and then classifying the constitutive functions according to their invariance properties in the state space. It is shown that the system always admits time translation and parabolic scaling as kernel symmetries, with an additional spatial translation admitted only in the Cartesian case. Further symmetries, such as translations, scalings, and rotations in the dependent-variable plane, are obtained by making precise structural assumptions about the constitutive functions. The analysis shows that the strong nonlinear coupling in the governing equations prohibits any new point symmetries from arising in the general case, and that larger symmetry algebras are only attainable in degenerate or linearizable special cases. The symmetries obtained in this work are geometrically consistent with parabolic and radial structure of governing equations.