On Darboux transformation of the supersymmetric sine-Gordon equation

Darboux transformation is constructed for superfields of the super sine-Gordon equation and the superfields of the associated linear problem. The Darboux transformation is shown to be related to the super Bäcklund transformation and is further used to obtain N super soliton solutions. PACS: 02.30.Ik PACS: 12.60.Jv mohsin [email protected] On study leave from PRD (PINSTECH) Islamabad, Pakistan [email protected] usman [email protected] There has been an increasing interest in the study of supersymmetric integrable systems for the last few decades [1]-[20]. Among the many techniques used to study integrability and to obtain the multisoliton solutions for a given integrable model, Darboux transformation has been widely used and it has established itself as an economic, convenient and efficient way of generating solutions [21]-[23]. The Darboux transformation has been employed on some supersymmetric integrable models in recent years [16]-[15]. In these investigations multisoliton solutions have been constructed and the ideas are generalized to incorporate the Crum transformation, the Wronskian superdeterminant and Pfaffian type solutions. The super soliton solutions of the super KdV equation and super sine-Gordon equation have been investigated and it has been shown that the solitons of the KdV and sine-Gordon solitons appear as the body of the super solitons [12]-[16]. The purpose of this work is to provide a thorough investigation of Darboux transformation for super sine-Gordon equation in a systematic way and to obtain the explicit super multisoliton solutions by a Crum type transformation. Following [17], we write a linear problem in superspace whose compatibility condition is the super sine-Gordon equation. The linear problem then leads to a Lax formalism in superspace. We explicitly write the Darboux transformation for the fermionic and bosonic superfields of the linear system and for the scalar superfield of the super sine-Gordon equation. The approach adopted here is different from that adopted in [15]. In ref [15] the authors have not explicitly constructed the Darboux transformation and N -soliton solutions for the super sine-Gordon equation. We have extended the results of [15] to give N super soliton solutions of the super sineGordon equation in terms of known solutions of the linear problem and express it as a series of products of super determinants and fermionic superfields. The Darboux transformation is then shown to be related to the super Bäcklund transformation of the super sine-Gordon equation [5]. We follow the general procedure of writing manifestly supersymmetric sine-Gordon equation. The equation is defined in two dimensional super-Minkowski space with bosonic light-cone coordinates x±6 and fermionic coordinates θ±, which are Majorana spinors. The Originally the Darboux transformation was first introduced by Darboux back in 1882, in the study of pseudo-spherical surfaces and later used to generate solutions of the Sturm-Liouville differential equation. The Darboux transformation has now widely been used to generate multisoliton solutions of integrable as well as super integrable evolution equations. Our space-time conventions are such that the orthonormal and light-cone coordinates are related by x = 1 2 (t ± x) and ∂± = 12 (∂t ± ∂x). 1 superspace lagrangian density of N = 1 super sine-Gordon theory is given by L(Φ) = i 2 D+ΦD−Φ+ cosΦ, (1) where Φ is a real scalar superfield and D± are covariant superspace derivatives defined as D± = ∂ ∂θ± − iθ∂±, D ± = −i∂±, {D+, D−} = 0, where { , } is an anti-commutator. The superfield evolution equation follows from the lagrangian and is given by D+D−Φ = i sin Φ. (2) The equation (2) is invariant under N = 1 supersymmetry transformations. Let us first recall that the super sine-Gordon equation appears as the compatibility condition of the following linear system of equations [17] D±Ψ = A±Ψ, (3) where the superfield Ψ is expressed in terms of bosonic and fermionic superfield components as Ψ = 