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arXiv · 1610.01029

Supersymmetric versions of the Fokas-Gel'fand formula for immersion

S. Bertrand·A. M. Grundland
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Abstract

In this paper, we construct and investigate two supersymmetric versions of the Fokas-Gel'fand formula for the immersion of 2D surfaces associated with a supersymmetric integrable system. The first version involves an infinitesimal deformation of the zero-curvature condition and the linear spectral problem associated with this system. This deformation leads the surfaces to be represented in terms of a bosonic supermatrix immersed in a Lie superalgebra. The second supersymmetric version is obtained by using a fermionic parameter deformation to construct surfaces expressed in terms of a fermionic supermatrix immersed in a Lie superalgebra. For both extensions, we provide a geometrical characterization of deformed surfaces using the super Killing form as an inner product and a super moving frame formalism. The theoretical results are applied to the supersymmetric sine-Gordon equation in order to construct super soliton surfaces associated with five different symmetries. We find integrated forms of these surfaces which represent constant Gaussian curvature surfaces and nonlinear Weingarten-type surfaces.

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S. Bertrand, A. M. Grundland. 2016-10-04. Supersymmetric versions of the Fokas-Gel'fand formula for immersion. https://doi.org/10.1088/1751-8113%2F49%2F30%2F305201

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