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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.

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 [

The integrability of sKdV equation can be established in the way to supersymmetrize the unique Gardner transformation. For the well known KdV equation

u t − 6 u u x + u x x x = 0 (1)

we extend the classical spacetime x , t to a super-spacetime x , t , θ , where θ is a Grassmannian odd number θ 2 = 0 . Now we write the N = 1 supersymmetric KdV (sKdV) equation accompanied with a fermionic super-variable Φ = Φ ( x , t , θ ) under the compact form

Φ t − 3 D 2 ( Φ D Φ ) + D 6 Φ = 0 (2)

Here the covariant super-derivative D is defined by D = ∂ θ + θ ∂ x . Mathieu found that a unique extension

Φ = χ + ε χ x + ε 2 χ ( D χ ) (3)

of establishing the integrability maps a solution of the superfield Gardner equation

χ t − 3 D 2 ( χ D χ ) − 3 ε 2 ( D χ ) D 2 ( χ D χ ) + D 6 χ = 0 (4)

into a solution of the sKdV equation [

This map was also used to recover an infinite number of conservation laws for the sKdV equation, and construct interesting BT [

u t + u x x x − 6 u u x + 3 ξ ξ x x − ε 2 [ 6 u 2 u x − 3 ( ξ ξ x u ) x ] = 0 (5a)

ξ t + ξ x x x − 3 ( ξ u ) x − 3 ε 2 u ( ξ u ) x = 0 (5b)

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 ε → 0 .

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 [

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.

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 G ( V ) = Λ 0 ⊕ Λ 1 , where Λ 0 and Λ 1 represents subspace containing even and odd elements, respectively [

u = u 0 + u 1 θ 1 θ 2 , ξ = v 1 θ 1 + v 2 θ 2 , (6)

here u i = u i ( x , t ) ( i = 0 , 1 ) , v i = v i ( x , t ) ( i = 1 , 2 ) are all usual bosonic functions with respect to spacetime variable x and t, thus we get nonlinear Partial Differential Equations (PDEs) in the component form by using (5)

u 0 t + u 0 x x x − 6 u 0 u 0 x f ( u 0 ) = 0 (7a)

v i t + v i x x x − 3 ( v i u 0 ) x f ( u 0 ) = 0 (7b)

u 1 t + u 1 x x x − 6 ( u 0 u 1 ) x f ( u 0 ) − 6 ε 2 u 0 u 0 x u 1 + F 1 = 0 (7c)

where f ( u 0 ) = 1 + ε 2 u 0 and F 1 = 3 [ f ( u 0 ) ( v 1 v 2 x − v 2 v 1 x ) ] x .

Next we introduce the traveling wave variable X = k x + ω t + x 0 along with the constants of wavenumber k, angular frequency ω and phase x 0 , therefore, above equations would be changed to a system consisting of ordinary differential equations(ODEs):

ω u 0 X + k 3 u 0 X X X − 6 k u 0 u 0 X f ( u 0 ) = 0 (8a)

ω v i X + k 3 v i X X X − 3 k ( v i u 0 ) X f ( u 0 ) = 0 ( i = 1 , 2 ) (8b)

ω u 1 X + k 3 u 1 X X X − 6 k ( u 0 u 1 ) X f ( u 0 ) − 6 ε 2 k u 0 u 0 X u 1 + F 1 ( X ) = 0 . (8c)

Note that we denote F 1 ( X ) = 3 k 2 [ f ( u 0 ) ( v 1 v 2 X − v 2 v 1 X ) ] X in here.

The traveling waves we discuss are only in the usual spacetime x , t but not in the super-spacetime x , t , θ , for example, χ ( x , t , θ ) = χ ( X + θ ζ ) with Grassmannian constant ζ is different from those in the usual spacetime. In addition to a directly integrable ODE in u 0 , the solution of the residual system which are related to third-order linear (non)homogeneous ODEs in u 1 , v 1 and v 2 , 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 u 0 X from Equation (8a), and the result reads

u 0 X = a 0 k 2 k λ ( z ) ,

where

To get the mapping relations of

where linear operators read

and the nonhomogeneous term

while

where

Thus, we have constructed the general two-fermionic-parameter traveling wave solutions of the supersymmetric version of Gardner system

with the known solution

It is interesting to see that the expression (9) is a trivial type of the symmetries or conservation quantity of standard Gardner equation

It is clear that the solution (9) can be expressed by the form of the Jacobian elliptic sine functions, i.e.,

where

Therefore, we derive special type solutions of the superfield Gardner equation:

where

where

For the case of three fermionic parameters

, ( 22)

here

where the somewhat complex nonhomogeneous terms read

Introducing the traveling wave variable

where nonhomogeneous terms are

with arbitrary integral constants

where

constants, and functioin

Therefore, we have obtained the three-fermionic-parameter traveling wave solutions of the superfield Gardner system. While one of the Grassmann numbers

where

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

, (29)

Here and below we denote by

summations. The elements

Operators related to Gâteaux derivative of Equation (4) or Equations (5) with an operator decomposition read

, (31a)

and nonhomogeneous terms are

We have also used the shorthand notations

and

with

and

1) Equation of

2) Equations of

3) Equations of

Here the nonhonogenerous terms are

Finally, the mapping and deformation relations of

in which auxiliary functions are

With relationships at hand, in case of preserving the square and cubic non-linear terms in Equations (5) or taking limitation

where

Due to the case of taking limitation

with N-fermionic-parametric Bosonization procedure can be derived as

where

with

and

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 applied or formulated in the N = 1 supersymmetric sine-Gordon equation, especially in the N = 2 version of KdV (SKdVa) equations [

example, in the case of two fermionic parameters, letting

simple traveling wave solutions of the N = 1 supersymmetric Sine-Gordon equation

where

We thank the Editor and the referee for their comments. This research is funded by the National Natural Science Foundation of China (Grant nos. 31570364) and the National Key Research and Development Program of China (Grant No. 2018YFE0103700). This support is greatly appreciated.

The authors declare no conflicts of interest regarding the publication of this paper.

Wang, S.T., Yu, H.G., Dai, C.J. and Zhao, M. (2020) Bosonization Approach and Novel Traveling Wave Solutions of the Superfield Gardner Equation. Journal of Applied Mathematics and Physics, 8, 443-455. https://doi.org/10.4236/jamp.2020.83034