, Volume 25, Issue 2, pp 323-329
Date: 06 Nov 2012

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

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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.