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

TL;DR

This paper develops two supersymmetric extensions of the Fokas-Gel'fand formula for immersing 2D surfaces linked to supersymmetric integrable systems, providing geometric characterizations and applying them to supersymmetric sine-Gordon equations.

Contribution

It introduces novel supersymmetric versions of the Fokas-Gel'fand formula, expanding the geometric framework for supersymmetric integrable systems.

Findings

Constructed bosonic and fermionic supermatrix surfaces

Applied to supersymmetric sine-Gordon equation

Derived surfaces with constant Gaussian curvature

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.