Decay Mode Solutions for the Supersymmetric Cylindrical KdV Equation

1 Introduction

Supersymmetric integrable systems have been studied extensively during the past decade. Thus, a large number of well-known integrable equations have been extended into the supersymmetric context, such as KdV equation, KP hierarchy, and Boussinesq equation [1–3]. It has been shown that these supersymmetric integrable systems possess the Bäcklund transformation, the Hamiltonian formalism, Darboux transformation, bilinear form, and multi-soliton solutions [4–8]. The bilinear form of supersymmetric integrable was introduced by Carstea [8], it requires an extension of the Hirota bilinear operator [9, 10] to the supersymmetric systems. In recent years, Carstea, Liu, and Zhang have done a lot of work on the supersymmetric equations [8, 11–13]. However, to our knowledge, the supersymmetric equation for the variable coefficient KdV equation has been considered rarely so far.

The cylindrical KdV

was first studied by Maxon and Viecelli [14]. They solved (1) numerically and obtained that (1) has cylindrical soliton solutions which consists of a pulse moving inward rapidly at an ever increasing speed, leaving behind a flat wake that moves inward at sound speed. In 1980 [15], Nakamura showed analytical soliton solution by Hirota method that agrees with numerical and experiment results.

In this article, we consider the cylindrical KdV equation. We will show the supersymmetric cylindrical KdV equation and obtain the decay mode soliton solutions for it.

2 Supersymmetrical KdV Equation

The supersymmetric extension of a nonlinear evolution equation refers to a system coupled equations for a bosonic u(x, t) and a fermionic field η(t, x), which reduces to the initial equation in the limit where the fermionic field is zero (the bosonic limit). We extend the classical space (x, t) to a large space (superspace) (t, x, θ), where θ is a Grassmann variable and also to extend the pair of fields (u, η) to a large fermionic or bosonic superfield ϕ(t, x, θ). In order to have a nontrivial extension for the cylindrical KdV, we choose ϕ to be fermionic, having the expansion

We consider the space supersymmetric invariance x→x–ϵθ, θ→θ+ϵ (ϵ is an anticommuting parameter). Multiplying θ in (1), each term in the space supersymmetry is

where D = ∂∂θ + θ∂∂x is the superderivative and

Reduce to the initial equation in the limit where the fermionic field is zero (the bosonic limit). Cylindrical KdV can be extended to the supersymmetric

3 Bilinear Form and Decay Mode Solutions

In order to derive the bilinear form for the cylindrical KdV, we consider the transformation

where f(t, x, θ) is bosonic. Equation (6) can be transformed into the following superbilinear form

here

We are going to derive the supersoliton solutions through the classical perturbative method. Expanding f into power series of a small parameter ϵ as

Substituting (10) into (8) and equating coefficients

In order to obtain the solutions, we introduce the quantity a_ij defined by

where ρ_i, ρ_j are arbitrary constant parameters

and ω(z_k), k=i, j present Airy function Ai or Bi which are two linearly independent solutions of the ordinary differential equation

ω″(z) − zω(z) = 0.

To the definition of a_ij for the case i=j, we denote as follows

By the direct calculation, we can find

By the direct calculation, we have found

If taking

f(1) = a11 = ρ12(12t)−1/3[z1ω1(z1)ω(z1) − ω′(z1)ω′(z1)],

we can obtain the one-decay solution. To get further than one solution, we must generalise closed relation (20) and (21). For the arbitrary functions a₁, a₂, b₁, and b₂, we can directly verify the following

where

Here, prime presents differentiation with respect to x. The relation can be extended to

where

We can prove (24–26) by mathematical induction. We consider (n+1)×(n+1) in (24) product as F(a₁···a_n+1)· (b₁···b_n+1)=F(a₁···a_n)a_n+1·(b₁···b_n)b_n+1 and apply 2×2 formula given by (22) to the right-hand side of this. Using relation (24), we see that F(a₁···a_n+1)·(b₁···b_n+1) reduces to the form (24, 26) with n replaced by n+1.

Let

and rewrite the suffix

Then

where ∑l1l2 ⋯ ln and ∑s1s2 ⋯ sm take sum over all possible permutation of (l₁, l₂, …, l_n) and (s₁, s₂, …, s_m). And c(n, m) represents the same form as (26) in which replacement of (27) and (28) is made.

In (29), c(n, m) only the term

remain nonvanishing after taking sum over n! and m! permutations. This is seen as follows. Of total n! or m! permutations, there always exist equal number of even and odd permutations, thus we have the relation

In this way, all the terms in c(n, m) except (30) and ∑a″ilia′jljb′ksk vanish. As to the remaining two types of terms, the latter also vanishes as

Thus, in this way, all terms in c(n, m) except (30) vanish. By the (20) and (24), (29) can be written to

Especially

Several lowest orders of these are written explicitly as

From (35–43), we can obtain the one-, two-, and three-decay mode solutions as following