Abstract
In this paper, a supersymmetric extension of the minimal surface equation is formulated. Based on this formulation, a Lie superalgebra of infinitesimal symmetries of this equation is determined. A classification of the one-dimensional subalgebras is performed, which results in a list of 143 conjugacy classes with respect to action by the supergroup generated by the Lie superalgebra. The symmetry reduction method is used to obtain invariant solutions of the supersymmetric minimal surface equation. The classical minimal surface equation is also examined and its group-theoretical properties are compared with those of the supersymmetric version.
Highlights
In recent years, there has been a considerable amount of interest in supersymmetric (SUSY)models involving odd Grassmann variables and superalgebras
Supersymmetry was introduced in the theory of elementary particles and their interactions and forms an essential component of attempts to obtain a unification of all physical forces [1]
A number of supersymmetric extensions have been formulated for both classical and quantum mechanical systems. Such supersymmetric generalizations have been constructed for hydrodynamic-type systems as well as other nonlinear wave equations, e.g., the Schrödinger equation [8] and the sine/sinh-Gordon equation [9,10,11]
Summary
There has been a considerable amount of interest in supersymmetric (SUSY). We formulate a supersymmetric extension of the minimal surface equation and investigate its group-theoretical properties. The minimal surface and its related equations appear in many areas of physics and mathematics, such as fluid dynamics [15], continuum mechanics [16], nonlinear field theory [17,18], plasma physics [12,19], nonlinear optics [20] and the theory of fluid membranes [21,22]. We construct a supersymmetric extension of the minimal surface Equation (2). The space {( x, y)} of independent variables is extended to the superspace {( x, y, θ1 , θ2 )} while the bosonic surface function u( x, y) is replaced by the bosonic superfield Φ( x, y, θ1 , θ2 ) defined in terms of bosonic and fermionic-valued fields of x and y. We revisit and expand the group-theoretical analysis of the classical minimal surface equation and compare the obtained results to those found for the supersymmetric extension of the minimal surface equation
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