On integrability aspects of the supersymmetric sine-Gordon equation

Abstract

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.

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

“…where φ = φ(x + , x − , θ + , θ − ) is a bosonic G-valued function. Its associated FLSP (2.2) is defined by the potential matrices [26][27][28]]…”

Is the $\mathbb{Z}_2\times \mathbb{Z}_2$-graded sine-Gordon equation integrable?

“…Other approaches to the integrability of supersymmetric sine-Gordon exist, such as constructing Lax pairs via superconformally affine Toda theories (see [13,16]) and the construction of solitons (see for example [11,19]). For a study of the different aspects of the integrability of the supersymmetric sine-Gordon see Bertrand [5]. It would be interesting to see how these other approaches generalise to this "higher graded" setting.…”

Soliton solutions of coupled complex modified Korteweg-de Vries system through Binary Darboux transformation

“…A century later in 1970, V. B Matveev extended his idea to important different partial differential equations (PDEs).Now a days Darboux transformation is very powerful tool to get soliton solutions and analyses it. A few authors studied different integrable models like nonlinear Schrodinger equation (NLS), Sine-Gordon equation and Korteweg-de Vries (KdV) equation and obtain soliton solutions by using Darboux transformation [27,30,17,4,5,2].…”