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We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg–de Vries and modified KdV equations. We give new representations of the

The study of _{1},_{2})which is assumed to be bosonic to get nontrivial extensions. The independent variables are given as a set of even (commuting) space _{1}, _{2}. Since the odd variables satisfy

where _{1} and _{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

which satisfy the anticommutation relations _{1},_{2}} = 0. We consider also the complex supercovariant derivatives

which satisfy {_{±},_{±}}=0 and

and the superfield

The fermionic complex valued functions _{±} are defined as

Chiral superfields are superfields of type Equation (5) satisfying _{+}A

or equivalently _{2} = _{1} and _{x}

In the subsequent sections, we produce solutions of _{α}

where [

In this paper, we start by presenting a general reduction procedure of these equations using chiral superfields (_{-2} and SmKdV together and construct classical

Here, we propose a general approach for the construction of chiral solutions of SUSY extensions. This approach avoids treating SUSY extensions in terms of components of the bosonic field _{+}_{+},_{-}_{+}D_{-}A = A_{x}

Note that these equations may be evidently treated as classical [

The absence of the Grassmannian variables _{+} and _{-} 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 [

From the classical case, we know that the methods of resolution of all these equations are similar. The same could be said for the SUSY case. Indeed, if we assume the introduction of a potential _{x}

where the constant of integration is set to zero. The same is done on Equation (12) and leads to

We thus observe that the Equations (11,13,14) are now on an equal footing, _{x}_{-2} will lead to similar types of solutions for the SmKdV Equation (11).

Now setting

in Equation (13), we obtain

The above equation reduces to the linear dispersive equation [

for the special and only values _{1}=

which does not linearize but can be bilinearized taking _{-2}=

A similar change of variable as in Equation (15) but with _{B} log _{B} and

It is well known [

Equation (11) can be bilinearized using the Hirota derivative defined as

Indeed, we take _{-2}=

This set is analogous to the corresponding bilinear equations in the classical mKdV equation [_{1 }and _{2}.

In order to get chiral solutions, we have to solve the set of bilinear equations with the additional chiral property _{+}_{i }= 0 for

The one soliton solution is easily retrieved. Indeed, we cast

where _{1} is an even parameter. Ψ_{1} is a

and never appears on this form in other approaches of _{1} and _{1} are, respectively, even and odd. The _{+}Ψ_{1}=0, the resulting traveling wave solution is chiral.

Since we exhibit the three super soliton solution of the SmKdV equation in _{1} and _{2}:

Where _{i}_{1} and _{2} solves the bilinear Equations (21) and (22) and are such that _{+}_{i}

The function Im(

The density plots of the functions _{1}_{2}_{3}

In

as a function of _{i}=_{-} of the superfield _{-} as

and trace out the bosonic functions _{1}_{2}_{3}

In a recent paper [_{-2} using a SUSY version of the Yablonskii–Vorob'ev polynomials [

Let us define the following

where

with

we have that the pair of bilinear Equations (21) and (22) are such that [

From the choice of the variable _{+}_{i,n}_{2,n} = τ_{1,n+1}

for all integers _{1,n} defined as in Equation (29). To get similarity solutions _{n} of the SKdV_{-2}, we use the above solution with

In this section, we exhibit _{1}, SKdV_{4} and SB. Virtual solitons are soliton-like solutions which exhibit no phase shifts in nonlinear interactions. In terms of classical _{ij}

Using the change of variable Equation (15) for the unknown bosonic field _{α}_{B}

It is easy to show that they admit the following solution

where the bosonic superfields Ψ_{i}

The frequencies _{i}_{i}_{i}^{3} for SKdV_{α}_{i}_{i}^{2} for SB. It looks like a typical KdV type soliton solution where all the interaction coefficients _{ij}

We see that the virtual soliton solutions of the SKdV_{1} and SKdV_{4} equations are completely similar due to the form of _{α}

where _{α}_{α}_{B}_{-}. Indeed, we have

where _{i}_{i}x_{i})_{i}(x,t)

In _{1} Equation for _{i}_{x}_{i}_{1}, _{2} and _{3} as given in Equation (40).

The function Im(_{1} equation where

The function _{1} equation where

The density plots of the functions _{1}, _{2} and _{3}, respectively from left to right, of the three virtual soliton solution of the SKdV_{1} equation where

The SUSY

where Г is a bosonic superfield and the complex supercovariant derivatives are defined as

where _{i}_{j}_{ij}_{x}_{ij}

where _{3}=_{4}=0

To construct virtual solitons of

A bosonic superfield Ξ satisfying the chiral conditions Equation (44) has the following general form

where

Equation (46) is, up to a slight change of variable, similar to Equation (13) for the integrable cases

The above equation can be linearized into the linear dispersive Equation (17) by the change of variable

Thus to obtain solutions of Equation (41), the superfield

A solution to this system is

where

with _{1} is an even constant. This result can thus be generalized to give a

where the superfields _{i}

It is interesting to note that by setting

In this paper, we have studied special solutions of supersymmetric extensions of the Burgers, KdV and mKdV equations in a unified way and using a chirality of the superfield

We have recovered interacting super soliton solutions (often called KdV type solitons) and an infinite set of rational similarity solutions. To produce such rational solutions, we have used an SUSY extension of the Yablonskii–Vorob'ev polynomials. We have introduce a new representation of the

We have shown the existence of non-interacting super soliton solutions, called virtual solitons, for the Burgers and SKdV_{α} (_{1} and _{2} in the bosonic superfield _{i}

A

L. Delisle acknowledges the support of a FQRNT doctoral research scholarship. V. Hussin acknowledges the support of research grants from NSERC of Canada.