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

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

${u}_{t}-6u{u}_{x}+{u}_{xxx}=0$ (1)

we extend the classical spacetime $x,t$ to a super-spacetime $x,t,\theta $, where $\theta $ is a Grassmannian odd number ${\theta}^{2}=0$. Now we write the N = 1 supersymmetric KdV (sKdV) equation accompanied with a fermionic super-variable $\Phi =\Phi \left(x,t,\theta \right)$ under the compact form

${\Phi}_{t}-3{D}^{2}\left(\Phi D\Phi \right)+{D}^{6}\Phi =0$ (2)

Here the covariant super-derivative D is defined by $D={\partial}_{\theta}+\theta {\partial}_{x}$. Mathieu found that a unique extension

$\Phi =\chi +\epsilon {\chi}_{x}+{\epsilon}^{2}\chi \left(D\chi \right)$ (3)

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

${\chi}_{t}-3{D}^{2}\left(\chi D\chi \right)-3{\epsilon}^{2}\left(D\chi \right){D}^{2}\left(\chi D\chi \right)+{D}^{6}\chi =0$ (4)

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: $x\to x-\eta \theta $, $\theta \to \theta +\eta $ ( $\eta $ is an anticommuting parameter.). The component form of the above equation with the superfield $\chi =\xi +\theta u$ can be rewritten as

${u}_{t}+{u}_{xxx}-6u{u}_{x}+3\xi {\xi}_{xx}-{\epsilon}^{2}\left[6{u}^{2}{u}_{x}-3{\left(\xi {\xi}_{x}u\right)}_{x}\right]=0$ (5a)

${\xi}_{t}+{\xi}_{xxx}-3{\left(\xi u\right)}_{x}-3{\epsilon}^{2}u{\left(\xi u\right)}_{x}=0$ (5b)

Note that $\xi $ 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 $\epsilon \to 0$.

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 [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.

2. 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 $G\left(V\right)$. Mathematically, such uncomplicated method used for vanishing fermionic fields is based on direct sum of superspace $G\left(V\right)={\Lambda}_{0}\oplus {\Lambda}_{1}$, where ${\Lambda}_{0}$ and ${\Lambda}_{1}$ represents subspace containing even and odd elements, respectively [25]. Here and in the following we omit the exterior algebra sign $\wedge $ 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 ${\theta}_{1}$ and ${\theta}_{2}$ with ${\theta}_{1}^{2}={\theta}_{2}^{2}=0$, let the two component fields u and $\xi $ be expanded as

$u={u}_{0}+{u}_{1}{\theta}_{1}{\theta}_{2}$, $\xi ={v}_{1}{\theta}_{1}+{v}_{2}{\theta}_{2}$, (6)

here ${u}_{i}={u}_{i}\left(x,t\right)\left(i=0,1\right)$, ${v}_{i}={v}_{i}\left(x,t\right)\left(i=1,2\right)$ 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)

${u}_{0}{}_{t}+{u}_{0}{}_{xxx}-6{u}_{0}{u}_{0}{}_{x}f\left({u}_{0}\right)=0$ (7a)

${v}_{it}+{v}_{ixxx}-3{\left({v}_{i}{u}_{0}\right)}_{x}f\left({u}_{0}\right)=0$ (7b)

${u}_{1t}+{u}_{1xxx}-6{\left({u}_{0}{u}_{1}\right)}_{x}f\left({u}_{0}\right)-6{\epsilon}^{2}{u}_{0}{u}_{0x}{u}_{1}+{F}_{1}=0$ (7c)

where $f\left({u}_{0}\right)=1+{\epsilon}^{2}{u}_{0}$ and ${F}_{1}=3{\left[f\left({u}_{0}\right)\left({v}_{1}{v}_{2x}-{v}_{2}{v}_{1x}\right)\right]}_{x}$.

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

$\omega {u}_{0}{}_{X}+{k}^{3}{u}_{0}{}_{XXX}-6k{u}_{0}{u}_{0}{}_{X}f\left({u}_{0}\right)=0$ (8a)

$\omega {v}_{iX}+{k}^{3}{v}_{iXXX}-3k{\left({v}_{i}{u}_{0}\right)}_{X}f\left({u}_{0}\right)=0\text{\hspace{0.17em}}\left(i=1,2\right)$ (8b)

$\omega {u}_{1X}+{k}^{3}{u}_{1XXX}-6k{\left({u}_{0}{u}_{1}\right)}_{X}f\left({u}_{0}\right)-6{\epsilon}^{2}k{u}_{0}{u}_{0X}{u}_{1}+{F}_{1}\left(X\right)=0$. (8c)

Note that we denote ${F}_{1}\left(X\right)=3{k}^{2}{\left[f\left({u}_{0}\right)\left({v}_{1}{v}_{2X}-{v}_{2}{v}_{1X}\right)\right]}_{X}$ in here.

The traveling waves we discuss are only in the usual spacetime $x,t$ but not in the super-spacetime $x,t,\theta $, for example, $\chi \left(x,t,\theta \right)=\chi \left(X+\theta \zeta \right)$ with Grassmannian constant $\zeta $ 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 ${u}_{0X}$ from Equation (8a), and the result reads

${u}_{0X}=\frac{{a}_{0}}{{k}^{2}}\sqrt{k\lambda \left(z\right)}$, (9)

where, and are two arbitrary integral constants, here and below the new variable z stands for function.

To get the mapping relations of, and, we introduce the variable transformations as follows, and . 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

(10a)

(10b)

where linear operators read

(11a)

(11b)

and the nonhomogeneous term is

. (12)

while is just an arbitrary integral constant. On this basis, the mapping and deformation relations are obtained as

(13a)

. (13b)

where and are arbitrary constants, and with auxiliary function . Coefficients for used are defined as , and.

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

(14a)

(14b)

with the known solution or Equation (8a) of the usual Gardner equation which have been extensively studied in many literatures [18] [26]. For a special case which takes to eliminate nonhomogeneous terms, the above traveling wave solution becomes

(15)

It is interesting to see that the expression (9) is a trivial type of the symmetries or conservation quantity of standard Gardner equation ;,. In fact, for any given being a solution of the usual Gardner equation, a certain type of solutions of the bosonic-looking equation can be constructed as follows:, , , where represents any symmetry of the usual Gardner equation, and we have much freedom to choose such that it can construct solutions without restricting to the traveling wave type solutions and infinitely many symmetries of the superfield Gardner equations.

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

,(16)

where, modulus m and constants are related to other known constants through following relations:

(17)

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

(18)

where

(19)

and are all arbitrary integral constants, the Jacobian elliptic functions are denoted as and, the incomplete elliptic integrals are denoted as and. We also note that for the critical modulus m = 1, above solutions read:

(20)

where

(21)

3. Three-Fermionic-Parameter Bosonization

For the case of three fermionic parameters with, let the two component fields u and be expressed as

, ( 22)

here and are all usual bosonic functions with respect to spacetime variable x and t. The symbol is the third-order Levi-Civita tensor. Then from (5), we get PDEs in component form as

(23a)

(23b)

(23c)

(23d)

where the somewhat complex nonhomogeneous terms read

(24a)

(24b)

Introducing the traveling wave variable with constants k, and and variable transformations and, similar to the previous case, the above bosonization system becomes following ODEs

(25a)

(25b)

(25c)

where nonhomogeneous terms are

(26a)

(26b)

with arbitrary integral constants. Mapping and deformation relations with respect to via traveling wave variable are constructed as

(27a)

(27b)

(27c)

where and are some arbitrary integral

constants, and functioin with integral constant.

Therefore, we have obtained the three-fermionic-parameter traveling wave solutions of the superfield Gardner system. While one of the Grassmann numbers tends to zero, the solution turns back to above section. Similar to the two-fermionic-parameter case, we write a special type solution

(28a)

(28b)

where are some constants, are arbitrary symmetries of the usual Gardner equation, and is an arbitrary solution of the usual Gardner equation.

4. 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 can be expressed as

, (29)

Here and below we denote by the set of multi-indices which satisfies, and the upper bound of

summations. The elements and are bosonic smooth real or complex valued functions defined in the commutative algebra. Thus the super Gardner model (5) is transformed to a new pure bosonic-looking system with coupled nonlinear PDEs:

(30a)

(30b)

(30c)

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

, (31a)

(31b)

(31c)

and nonhomogeneous terms are

(32)

We have also used the shorthand notations

and

with

, ,

and for even case; for odd case. The parity function is defined in [25]. The symbol is the generalized Levi-Civita anti-symmetric tensor in m dimensions. Equation (30a) apparently suggests that it is the standard Gardner equation. Each solution in the case of fermionic parameters is a portion of solutions of the system consisting of n homologous Grassmannian variables. Introducing the traveling wave variable and variable transformations, mapping transformations are arranged as follows:

1) Equation of: (33a)

2) Equations of: (33b)

3) Equations of: (33c)

Here the nonhonogenerous terms are, each usual bosonic functions and are represented by and, respectively.

Finally, the mapping and deformation relations of and with N fermionic parameters can be formally rewritten as

(34a)

(34b)

in which auxiliary functions are

(35a)

(35b)

, and are some arbitrary integral constants.

5. Discussion and Summary

With relationships at hand, in case of preserving the square and cubic non-linear terms in Equations (5) or taking limitation, one can conveniently acquire the corresponding result of the integrable N = 1 sKdV regardless of some integral constants. Similarly to Section 4, the general traveling wave solutions of the N = 1 sKdV Equation (2) with N fermionic parameters are

(36a)

(36b)

where

.

Due to the case of taking limitation and the absence of non-linear quadratic term in Equation (4), traveling wave solutions of the usual integrable N = 1 smKdV Equation [10] [27]

(37)

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

(38)

where

(39a)

(39b)

with

(40a)

(40b)

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 [8] [25] [28] [29] [30]. For

example, in the case of two fermionic parameters, letting, , in bosonic field, the

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

(41a)

(41b)

where and are the usual super derivatives, S represents and, E, , , and are integral constants, is a solution of the standard sine-Gordon equation.

Acknowledgements

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.

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