Effect of background geometry on symmetries of the $(1+2)$-dimensional heat equation and reductions of the TDGL model

Abstract

The $(1+2)$-dimensional heat equation on the plane ($\mathbb R ^{2}$) and sphere ($\mathbb S ^{2}$) is considered respectively. For each surface a class of functions is presented for which considered equation has nontrivial symmetries. We consider whether the background metric ($\mathbb S ^{2}$) or the nonlinearity have the dominant role in the infinitesimal generators of the considered equation. Then the time dependent Ginzburg–Landau equation (TDGL model) is considered on $\mathbb S ^{2}$. Lie point symmetry generators are calculated and optimal systems of its subalgebras up to conjugacy classes are obtained. Similarity reductions for each class are performed.