# Soliton and Similarity Solutions of Ν = 2, 4 Supersymmetric Equations

^{1}

^{2}

^{*}

## Abstract

**:**

**MSC**35Q51; 35Q53; 81Q60

## 1. Introduction

_{1},θ

_{2})which is assumed to be bosonic to get nontrivial extensions. The independent variables are given as a set of even (commuting) space x and time t variables and a set of odd (anticommuting) variables variables θ

_{1}, θ

_{2}. Since the odd variables satisfy , the dependent variable A admits the following finite Taylor expansion

_{1}and ξ

_{2}are fermionic complex valued functions. In this paper, we show that some of these extensions can be related to a linear partial differential equation (PDE) by assuming that A is a chiral superfield [9]. Proving the integrability of an equation by linearization has been largely studied in the classical case [10,11] and has found new developments in the N=1 formalism [12]. We propose a similar development in the N=2 formalism. In N=2 SUSY, we consider a pair of supercovariant derivatives defined as

_{1},D

_{2}} = 0. We consider also the complex supercovariant derivatives

_{±},D

_{±}}=0 and . In terms of the complex Grassmann variables , the derivatives Equation (3) admits the following representation

_{±}are defined as .

_{+}A=0. In terms of components, we get

_{2}= iξ

_{1}and v= -iu

_{x}.

_{α}), modified Korteweg–de Vries [6] (SmKdV) and Burgers [5] (SB) equations from a chiral superfield point of view. In this instance, the equations, in terms of the complex covariant derivatives Equation (3), reads, respectively, as

_{-2}and SmKdV together and construct classical N super soliton solutions [4,7,8,13] and an infinite set of similarity solutions [7]. In Section IV, we demonstrate the existence of special N super soliton solutions, called virtual solitons [5], for the SUSY extensions of the KdV equation with α=1,4 and the Burgers equation using a related linear partial differential equation. The last section is devoted to a N=4 extension of the KdV equation [6] in an attempt to construct a general N super virtual soliton solution.

## 2. General Approach and Chiral Solutions

_{+}A = 0, we get the chiral property {D

_{+},D

_{-}}A = D

_{+}D

_{-}A = A

_{x}and the Equations (7–9) reduce to

_{+}and θ

_{-}derivatives in Equations (10–12) indicates that the odd sectors of chiral solutions should be free from fermionic constraint. This property is in accordance with the integrability of these extensions due to arbitrary bosonization of the fermionic components [15] of the bosonic superfield A.

_{x}in Equation (10) and after one integration with respect to x, we get

_{x}in the nonlinear terms. This is standard in Hirota formalism. The choice α = -2 in Equation (13) gives, up to a slight change of variable, the SmKdV Equation (11). This means that the known [7] N super soliton solutions and similarity solutions of SKdV

_{-2}will lead to similar types of solutions for the SmKdV Equation (11).

_{1}=i and α=4 with . For α=-2, Equation (16) writes

_{-2}=i. It is discussed in the next Section.

_{B}log H

_{B}and in Equation (14) is assumed and leads to the linear Schrödinger Equation

## 3. SKdV_{-2} and SmKdV Equations

_{-2}=i and , where are bosonic chiral superfields for i=1,2. Equation (11) leads to the set of bilinear equations

_{1 }and τ

_{2}.

_{+}τ

_{i }= 0 for i=1,2. It will lead to new solutions of the SmKdV equation which are related to our recent contribution [7].

#### 3.1. N Super Soliton Solutions

_{1}is an even parameter. Ψ

_{1}is a N=2 chiral bosonic superfield defined as

_{1}and ξ

_{1}are, respectively, even and odd. The τ-functions Equation (23) together with Equation (24) solve the set of bilinear Equations (21,22) and give rise to a one super soliton solution. Since D

_{+}Ψ

_{1}=0, the resulting traveling wave solution is chiral.

_{1}and τ

_{2}:

_{i}'s are defined as in Equation (24). The functions τ

_{1}and τ

_{2}solves the bilinear Equations (21) and (22) and are such that D

_{+}τ

_{i}= 0 for i=1,2. The generalization to a N super soliton solution is direct using the τ-functions expressed above. The forms of the τ-functions given above are new representations of super soliton solutions and have never been introduced before.

**Figure 1.**The function Im(v) of the three soliton solution of the SmKdV equation where and t = -20,0,20

**Figure 2.**The density plots of the functions f

_{1}, f

_{2}and f

_{3}, respectively from left to right, of the three soliton solution of the SmKdV equation where .

_{i}=i in Equations (25) and (26) and t=-20,0,20. In Figure 2, we explore the behavior of the fermionic component ρ

_{-}of the superfield A for the same special values. To achieve this, we write ρ

_{-}as

_{1}, f

_{2}and f

_{3}.

#### 3.2. Similarity Solutions

_{-2}using a SUSY version of the Yablonskii–Vorob'ev polynomials [16,17,18]. We propose in this subsection to retrieve those solutions and find an infinite set of similarity solution for the SmKdV equation. To give us a hint into what change of variables we have to cast, we have used the symmetry reduction method associated to a dilatation invariance [2].

_{+}τ

_{i,n}= 0 for all integers n. Taking τ

_{2,n}= τ

_{1,n+1}, we have an infinite set of similarity solutions of the SmKdV Equation given by

_{1,n}defined as in Equation (29). To get similarity solutions A

_{n}of the SKdV

_{-2}, we use the above solution with . Plots of some similarity solutions are given in our recent contribution [7].

## 4. SKdV_{1}, SKdV_{4} and SB Equations and Virtual Solitons

_{1}, SKdV

_{4}and SB. Virtual solitons are soliton-like solutions which exhibit no phase shifts in nonlinear interactions. In terms of classical N soliton solutions [3,4,5,7,14,16,19], this is equivalent to say that the interaction coefficients A

_{ij}between soliton i and soliton j are zero, . They manifest as traveling wave solutions for negative time t«0 and decrease spontaneously at time t=0 to split into a N soliton profile which exhibit no phase shifts. It is often said that the traveling wave solution was charged with N-1 soliton, called virtual solitons [5].

_{α}must be a chiral superfield and solve the linear dispersive Equation (17) when α=1 and α=4. For the Burgers equation, the bosonic field H

_{B}had to be chiral and solves Equation (19).

_{i}are given as

_{i}) are such that ω(κ

_{i})= - κ

_{i}

^{3}for SKdV

_{α}and ω(κ

_{i})=- κ

_{i}

^{2}for SB. It looks like a typical KdV type soliton solution where all the interaction coefficients A

_{ij}are set to zero.

_{1}and SKdV

_{4}equations are completely similar due to the form of Ã which differs only by the constant value of β

_{α}. The expression of the original bosonic field is obtained from

_{α}for the SKdV

_{α}equation and β=β

_{B}for the SB equation. Thus, we can give the explicit forms of the superfield components u and ρ

_{-}. Indeed, we have

_{i}=κ

_{i}x+ ω(κ

_{i})t and the bosonic functions f

_{i}(x,t) are defined as

_{1}Equation for and α

_{i}= 1 in Equation (36) and t= 0,10,20. In Figure 4, we observe the behavior of the function v where v = - iu

_{x}, and α

_{i}=1 in Equation (36) and t=20,0,20. For the same special values, Figure 5 gives the density plots of the bosonic functions f

_{1}, f

_{2}and f

_{3}as given in Equation (40).

**Figure 3.**The function Im(u) of the three virtual soliton solution of the SKdV

_{1}equation where and t=0,10,20.

**Figure 4.**The function v of the three virtual soliton solution of the SKdV

_{1}equation where and t=20,0,20.

**Figure 5.**The density plots of the functions f

_{1}, f

_{2}and f

_{3}, respectively from left to right, of the three virtual soliton solution of the SKdV

_{1}equation where .

## 5. SUSY N=4 KdV Equation and Virtual Solitons

_{i},D

_{j}}=2δ

_{ij}∂

_{x}, where δ

_{ij}is the Kronecker delta, we have that the supercovariant derivatives Equation (42) satisfy the anticommutation rules

_{3}=θ

_{4}=0 and in Equation (41), we retrieve the SmKdV Equation (8).

_{1}is an even constant. This result can thus be generalized to give a N super virtual soliton solution of the SUSY N=4 KdV Equation (41) by taking

_{i}are defined as in Equation (50) for i=1,…,N.

## 6. Concluding Remarks and Future Outlook

_{α}(α=1,4). These special solutions are a direct generalization of the solutions obtained in a recent contribution [5] where N super virtual solitons have been found by setting to zero the fermionic contributions ξ

_{1}and ξ

_{2}in the bosonic superfield A given as in Equation (1). We retrieve those solutions by setting ς

_{i}= 0 in the exponent terms Equation (37). Thus the chirality property, exposed in this paper, has produced a nontrivial fermionic sector for a N super virtual soliton. Furthermore, to obtain such solutions we have related the SUSY equations to linear PDE's showing the true origin of those special solutions.

## Acknowledgments

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**MDPI and ACS Style**

Delisle, L.; Hussin, V.
Soliton and Similarity Solutions of* Ν = 2, 4 *Supersymmetric Equations. *Symmetry* **2012**, *4*, 441-451.
https://doi.org/10.3390/sym4030441

**AMA Style**

Delisle L, Hussin V.
Soliton and Similarity Solutions of* Ν = 2, 4 *Supersymmetric Equations. *Symmetry*. 2012; 4(3):441-451.
https://doi.org/10.3390/sym4030441

**Chicago/Turabian Style**

Delisle, Laurent, and Véronique Hussin.
2012. "Soliton and Similarity Solutions of* Ν = 2, 4 *Supersymmetric Equations" *Symmetry* 4, no. 3: 441-451.
https://doi.org/10.3390/sym4030441