Abstract
We employ the symmetry principle to obtain rate-independent hysteresis
operators associated with a nonlinear diffusion equation. The direct Lie
approach to this problem leads to an overdetermined system of equations
for symmetries that are extremely difficult to analyze. To mitigate this
problem, we employ the classification of low-dimensional Lie algebra
following the suggestion of Lahno \textit{et al.}
(Journal of Physics A: Mathematical and General
\textbf{32} (42) (1999): 7405). After excluding linear
models, we obtain a one-dimensional principal symmetry Lie algebra,
seven symmetry Lie algebras of dimension two, twenty-six symmetry Lie
algebras of dimension three, and two symmetry Lie algebras of dimension
four. The curves of invariant solutions for symmetry Lie algebras of
dimensions three and four are exhibited in three forms. Under
appropriate selection of parameters, all invariant solutions show
one-to-many relationships between inputs and outputs, except for one in
dimension three.