Supersymmetric versions of the Fokas–Gel’fand formula for immersion

Abstract

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.

On geometric aspects of the supersymmetric Fokas–Gel’fand immersion formula

“…In addition, we investigated the different ZCCs and their relation to the original SUSY integrable system. The use of these three types of LSPs provides a more complete geometric characterization of the supermanifolds under investigation than in the previous paper [15]. Using the super Killing form on the four g-valued tangent vectors D j F , j = 1, 2, 3, 4, we compute the 16 metric coefficients g ij (instead of the 4 coefficients given in [15]).…”

“…The present paper is a follow-up of the investigation performed in [15] concerning the SUSY extensions of the FGIF. In the paper [15], the SUSY FGIF only considers (bosonic and fermionic) deformations of the LSP involving covariant fermionic derivatives.…”

“…The present paper is a follow-up of the investigation performed in [15] concerning the SUSY extensions of the FGIF. In the paper [15], the SUSY FGIF only considers (bosonic and fermionic) deformations of the LSP involving covariant fermionic derivatives. Hence, the geometric characterization of the immersion function describes a two-dimensional supermanifold allowing only fermionic infinitesimal displacement in the first and second fundamental forms even though the immersion function depends on two fermionic and two bosonic independent variables.…”

“…The continuous deformations of manifolds are much better understood in the classical case than in their SUSY versions for which only few examples are known. Recently, generalizations of the FGIF were constructed to consider bosonic and fermionic deformations associated with SUSY integrable systems [15]. Since soliton supermanifolds associated with SUSY integrable models behave rather differently than the classical smooth manifolds, the rich character of the SUSY versions of the FGIF makes a rather special and interesting object of study.…”

On integrability aspects of the supersymmetric sine-Gordon equation

“…One should note that the solution of the LSP (25) with the fixed parameter λ 0 has been used in order to construct the new solution. The index j = 0 for the first (or higher) iteration transformation correspond to the trivial solution Φ = 0.…”