Bosonization Approach and Novel Traveling Wave Solutions of the Superfield Gardner Equation

In this paper, the bosonization of the superfield Gardner equation in the case of multifermionic parameters is presented and novel traveling wave solutions are extracted from the coupled bosonic equations by using the mapping and deformation relations. In the case of two-fermionic-parameter bosonization procedure, we provide a special solution in the form of Jacobian elliptic functions. Meanwhile, we discuss and formally derive traveling wave solutions of N fermionic parameters bosonization procedure. This technique can also be applied to treat the N = 1 supersymmetry KdV and mKdV systems which are obtained in two limiting cases.


Introduction
The supersymmetry (SUSY), applied to treat fermions and bosons in a unified way in elementary particle physics since the concept first arose in 1971 by Ramond, Golfand and Likhtman, has been researched extensively during past four decades [1]- [6]. The starting point of SUSY is the supersymmetric versions of well known KdV equation first by Kupershmidt in 1984 (a simple fermionic but not supersymmetric extension) and later found independently of the work of Manin-Radual on super KP hierarchy [7] [8] [9] [10]. It was pointed out afterwards that the latter is indeed a truly N = 1 (N refers to the number of super-symmetries, for N = 1 standard, for N > 1 extended) sKdV equation which is invariant under supersymmetric transformation [8]. Since then, various properties have been established for its supersymmetric versions, such as Lax representation, bi-Hamiltonian structures, Backlund transformation (BT), Painleve analysis, N-soliton solutions, (non) local conserved quantities, etc. [9]- [15].
The 3 0 Here the covariant super-derivative D is defined by into a solution of the sKdV equation [8] [9] [15]. The Gardner equation is also called the extended KdV equation with the variable-sign cubic non-linear term or the combined KdV and mKdV (KdV-mKdV) equation. It is widely used in various branches of physics, such as plasma physics, fluid physics, nonlinear phenomena and quantum field theory, etc., and it also describes a variety of wave phenomena in plasma and solid state [16] [17] [18] [19] [20].
This map was also used to recover an infinite number of conservation laws for the sKdV equation, and construct interesting BT [6]. It is easy to show that such super equation is invariant under the supersymmetry transformation: Note that ξ and u are new setted fermionic and bosonic functions, the usual Gardner equation is recovered by setting the fermionic variable to be absent and the sKdV equation is the limiting case where 0 ε → .
Nonlinear partial differential equations play an important role in nonlinear physics, even nonlinear science. Various effective methods have been proposed to derive explicit or formal solutions. Recently, a simple but powerful bosonization approach, which main idea is to consider fields in a Grassmannian algebra and rewrite a system in a basis of this algebra to arrive at a system of ordinary (commutative) evolution equations, can effectively simplify such systems containing anti-communicating fermionic fields [7] [21] [22]. In [23], B. Ren et al. used this approach in the N = 1 supersymmetric Burgers (SB) system, and the exact solutions of the usual pure bosonic systems are obtained with the mapping and deformation method and Lie point symmetries theory. In [24], the Lie point symmetries of the supersymmetric KdV-a system are considered and similarity reductions of it are conducted. Several types of similarity reduction solutions of the coupled bosonic equations are also simply obtained. The motivation and purpose of this paper is to show this procedure and outcome of the method by taking the superfield Gardner equation and to acquire novel traveling wave solutions of this equation.
The rest of this paper is organized as follows. Section 2 and 3 are brief reminders of fairly basic illustrations of the bosonization approach of the superfield Gardner equation with two and three fermionic parameters. In Section 4 we present the N fermionic parameters bosonization case. In Section 5 we give the N = 1 supersymmetric KdV and mKdV equations using parallel procedure in two particular cases, and we will also give a short summary.

Two-Fermionic-Parameter Bosonization
To get direct comprehension and fixed notations of the bosonizaion approach for superfield Gardner equation with multi-fermionic parameters, we first concentrate on a linear space ( ) G V . Mathematically, such uncomplicated method used for vanishing fermionic fields is based on direct sum of superspace where 0 Λ and 1 Λ represents subspace containing even and odd elements, respectively [25]. Here and in the following we omit the exterior algebra sign ∧ and denote it briefly by ordinary multiplication. Moreover, relevant solutions may involve rich (super) symmetries physically. An exact example is the cases of two and three fermionic parameters bosonization, thereby we can directly derive the usual Gardner equation and coupled equations appear below. For the case of two fermionic parameters 1 θ and 2 θ with 2 Next we introduce the traveling wave variable 0 X kx t x ω = + + along with the constants of wavenumber k, angular frequency ω and phase 0 x , therefore, above equations would be changed to a system consisting of ordinary differential equations(ODEs): The traveling waves we discuss are only in the usual spacetime , x t but not in the super-spacetime , , x t θ , for example, with Grassmannian constant ζ is different from those in the usual spacetime. In addition to a directly integrable ODE in 0 u , the solution of the residual system which are related to third-order linear (non)homogeneous ODEs in 1 u , 1 v and 2 v , can be obtained through the variable transformation from ordinary coordinates space to phase space on the base of periodic wave solutions of usual Gardner equation. We first solve out 0 X u from Equation (8a), and the result reads where 2 0 1 a = , 1 C and 2 C are two arbitrary integral constants, here and below the new variable z stands for function 0 u .
To get the mapping relations of 1 u , 1 v and 2 v , we introduce the variable transformations as follows Applying the transformation via Equation (8a), the linear ODEs (8b)-(8c) are reduced to mapping and deformation relations between the traveling wave solutions of the classical Gardner equation and its supersymmetric equation by exploiting the known solutions of classical Gardner equation where linear operators read and the nonhomogeneous term Coefficients for used are defined as Thus, we have constructed the general two-fermionic-parameter traveling wave solutions of the supersymmetric version of Gardner system with the known solution 0 u or Equation (8a) It is interesting to see that the expression (9) is a trivial type of the symmetries or conservation quantity of standard Gardner equation Therefore, we derive special type solutions of the superfield Gardner equation:  We also note that for the critical modulus m = 1, above solutions read: where ( ) ( )

Three-Fermionic-Parameter Bosonization
For the case of three fermionic parameters where the somewhat complex nonhomogeneous terms read where nonhomogeneous terms are with arbitrary integral constants

N-Fermionic-Parameter Bosonization
Motivated by above sections, we repeat same procedure to get traveling wave solutions of the superfield Gardner equation via bosonizaion approach with N fermionic parameters here. The component fields u and ξ can be expressed as

Discussion and Summary
With relationships at hand, in case of preserving the square and cubic non-linear terms in Equations (5) with N-fermionic-parametric Bosonization procedure can be derived as  In summary, the bosonization approach with multi-fermionic parameters to deal with supersymmetric systems is developed in the super Gardner equation with the role of traveling wave solution. The procedure and technique are also available for N = 1 sKdV and smKdV equations derived from two particular cases. We expect this procedure exhibited in our paper could be successfully ap-