On Darboux transformation of the supersymmetric sine-Gordon equation

Abstract. The objective of this paper is to formulate two distinct supersymmetric (SUSY) extensions of the Gauss-Weingarten and Gauss-Codazzi (GC) equations for conformally parametrized surfaces immersed in a Grassmann superspace, one in terms of a bosonic superfield and the other in terms of a fermionic superfield. We perform this analysis using a superspace-superfield formalism together with a SUSY version of a moving frame on a surface. In constrast with the classical case, where we have three GC equations, we obtain six such equations in the bosonic SUSY case and four such equations in the fermionic SUSY case. In the fermionic case the GC equations resemble the form of the classical GC equations. We determine the Lie symmetry algebra of the classical GC equations to be infinite-dimensional and perform a subalgebra classification of the one-dimensional subalgebras of its largest finite-dimensional subalgebra. We then compute superalgebras of Lie point symmetries of the bosonic and fermionic SUSY GC equations respectively, and classify the onedimensional subalgebras of each superalgebra into conjugacy classes. We then use the symmetry reduction method to find invariants, orbits and reduced systems for two one-dimensional subalgebras for the classical case, two one-dimensional subalgebras for the bosonic SUSY case and two one-dimensional subalgebras for the fermionic SUSY case. We find explicit solutions of these reduced SUSY systems, which correspond to different surfaces immersed in a Grassmann superspace. Within this framework for the SUSY versions of the GC equations, a geometrical interpretation of the results is discussed.

Section: Susy Extensions Of the Gw And Gc Equationsmentioning

confidence: 99%

Section: Invariant Solutions Of the Susy Gc Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Supersymmetric versions of the equations of conformally parametrized surfaces

Bertrand¹,

Grundland²,

Hariton³

2015

“…Some of these integrability properties have been investigated for the case of the SUSY sine-Gordon equation, see e.g. [2,[15][16][17][19][20][21][22][23][24]. The approach proposed in this paper goes deeper into certain integrability properties of the SUSY sine-Gordon equation than [17,24], especially the Darboux transformation of the SUSY sine-Gordon equation, which will later enable us to obtain new results for the bosonic SUSY version of the Sym-Tafel formula for the immersion of surfaces in Lie superalgebras.…”

Section: Introductionmentioning

confidence: 95%

On integrability aspects of the supersymmetric sine-Gordon equation

Bertrand

2017

Abstract. In this paper we study certain integrability properties of the supersymmetric sine-Gordon equation. We construct Lax pairs with their zerocurvature representations which are equivalent to the supersymmetric sine-Gordon equation. From the fermionic linear spectral problem, we derive coupled sets of super Riccati equations and the auto-Bäcklund transformation of the supersymmetric sine-Gordon equation.In addition, a detailed description of the associated Darboux transformation is presented and non-trivial super multisoliton solutions are constructed. These integrability properties allow us to provide new explicit geometric characterizations of the bosonic supersymmetric version of the Sym-Tafel formula for the immersion of surfaces in a Lie superalgebra. These characterizations are expressed only in terms of the independent bosonic and fermionic variables.

“…Let us consider the fermionic sl(2|1, G)-valued gauge 28) which is associated with the deformation supermatrices…”

Section: A Fermionic Gauge Deformationmentioning

confidence: 99%

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

Bertrand¹

2017

In this paper, we develop a new geometric characterization for the supersymmetric versions of the Fokas-Gel'fand formula for the immersion of soliton supermanifolds with two bosonic and two fermionic independent variables into Lie superalgebras. In order to do so, from a linear spectral problem of a supersymmetric integrable system using the covariant fermionic derivative, we provide a technique to obtain two additional linear spectral problems for that integrable system, one using the bosonic variable derivatives and the other using the fermionic variable derivatives. This allows us to investigate, through the first and second fundamental forms, the geometry of the (1 + 1|2)-supermanifolds immersed in Lie superalgebras. Whenever possible, the mean and Gaussian curvatures of the supermanifolds are calculated. These theoretical considerations are applied to the supersymmetric sine-Gordon equation.