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

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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.