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Symmetry properties of the membrane shape equation. (English) Zbl 1382.35005
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 14th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 8–13, 2012. Sofia: Bulgarian Academy of Sciences. Geometry, Integrability and Quantization, 152-159 (2013).
Summary: Here we consider the Helfrich’s membrane shape model from a group-theoretical viewpoint. By making use of the conformal metric on the associated surface the model is represented by a system of four second order nonlinear partial differential equations. In order to construct the determining system for the symmetries of the metric we rely on the previously developed package LieSymm-PDE within Mathematica\(^\circledR\). In this way we have obtained the determining system consisting of 206 equations. Using the above mentioned programs we have solved the equations in a semi-automatic way. As a result we end up with an infinite dimensional symmetry Lie algebra of the Helfrich‘s model in conformal metric representation which we present here in explicit form.
For the entire collection see [Zbl 1267.00032].
MSC:
35-04 Software, source code, etc. for problems pertaining to partial differential equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
76B99 Incompressible inviscid fluids
Software:
LieSymm; Mathematica
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