Darboux transformations of the supersymmetric constrained B and C type KP hierarchies

doi:10.1016/j.geomphys.2021.104216

Journal of Geometry and Physics

Volume 165, July 2021, 104216

https://doi.org/10.1016/j.geomphys.2021.104216 Get rights and content

Abstract

Based on Darboux transformations for the supersymmetric constrained KP(ScKP) hierarchy, we construct the supersymmetric constrained B type KP(ScBKP) and supersymmetric constrained C type KP(ScCKP) hierarchies of Manin–Radul and Jacobian types, and derive Darboux transformations on them. On the basis of the seed solutions, the new solutions can be generated by these Darboux transformations.

Introduction

The Kadomtsev–Petviashvili (KP) hierarchy is an interesting target of integrable system. By considering the reduction of the Lax operator, we know that the two sub-hierarchies as C type KP(CKP) hierarchy [5], B type KP(BKP) hierarchy [6] and their generalization [16] have very good integrable properties. In order to maintain the invariance of Lax equations, Darboux transformations of the BKP hierarchy and the CKP hierarchy cannot be applied to either $T_{a} = ϕ \partial ϕ^{- 1}$ or $T_{b} = ψ^{- 1} \partial^{- 1} ψ$ alone, but must be used by combining $T_{a}$ and $T_{b}$ together. Here $ϕ$ and $ψ$ represent eigenfunctions and adjoint eigenfunctions respectively. In particular, the functions generated by $T_{a}$ and $T_{b}$ in one brick depend on each other, by a restriction on the Lax operator [18]. In addition, the so-called “constrained” KP (cKP) hierarchy [2], [4], [7] is a very interesting sub-hierarchy developed from the perspective of the symmetry constraint. In some sense, the cKP hierarchy can be regarded as the promotion of the $n$ th reduction of the KP hierarchy. Two Darboux transformation operators $T_{a}$ and $T_{b}$ can also be used to construct solutions of the constraint KP hierarchy. In [8], gauge transformations of the constrained BKP and constrained CKP hierarchies are constructed.

In mathematical physics, various extensions of KP hierarchy and supersymmetric extensions are extremely important [20], [22], especially in the theory of Lie algebras. In [9], the theory of super Lie algebras was studied through the super Boson–fermion correspondence. The first supersymmetric KP(SKP) hierarchy was proposed by Manin and Radul [15]. Another related supersymmetric KP hierarchy was proposed by Mulase and Rabin [17], [20]. As we all know, the Darboux transformation is an effective method to generate soliton solutions of integrable systems. Therefore, we consider using Darboux transforms to find soliton solutions of supersymmetric integrable systems. Aratyn et al. pointed out that the natural candidate of Darboux transformation cannot maintain fermionic flows [3]. In [14], Darboux transformations for the supersymmetric KP hierarchy of Manin–Radul and Jacobian types were constructed. In [3], [19], Darboux transformations of the ScKP hierarchy were constructed. In [10], [12], [13], [21], Darboux transformations of the SB(C) KP hierarchies were constructed by using Darboux transformation operators $T_{D} = Φ D Φ^{- 1}$ and $T_{I} = Ψ^{- 1} D^{- 1} Ψ$ . Here $Φ$ and $Ψ$ of $T_{D}$ and $T_{I}$ represent super eigenfunctions and adjoint super eigenfunctions respectively, and $D$ represents super-derivation.

Therefore, it is natural to use the above-mentioned Darboux transformation operators $T_{D}$ and $T_{I}$ to construct Darboux transformations for the supersymmetric cCKP and cBKP hierarchies. Through previous studies, we come to the conclusion: the fermionic flows of Manin-Radul supersymmetric constrained KP(MR-ScKP) [3], [13] hierarchy is retained through non-trivial modification; the Darboux transformation of Manin-Radul supersymmetric constrained B and C type KP(MR-ScB(C)KP) hierarchies only retain the bosonic flows. The purpose of this article is to provide proper Darboux transformations for the ScB(C)KP hierarchies. In order to achieve this goal, we must recall that following Ref [14], similarly to supersymmetric KP hierarchy. There are four ScB(C)KP hierarchies, namely ScB(C)KP $_{k}, k = 0, 1$ . In fact, we consider from ScB(C)KP $_{k}$ to ScB(C)KP $_{k + 1}$ (mod $2)$ by reversing the fermionic flows. We will prove that the natural candidate of the elementary Darboux transformation is a mapping ScB(C)KP $_{k}$ $\to$ ScB(C)KP $_{k + 1}$ (mod $2)$ . New solutions of ScB(C)KP $_{k}$ are given, when we composed with a reversion of fermionic flows.

The arrangement of this article is as follows. In Section 2 we will review calculation rules and some necessary facts about the ScKP hierarchy. In Section 3, we think about using the Darboux transformation operator $T_{I}$ to do Darboux transformation on the supersymmetric cKP hierarchy, and give the Wronskian expression of super- $τ$ function. In Sections 4 The supersymmetric constrained BKP hierarchy, 6 The supersymmetric constrained CKP hierarchy, we briefly describe the theoretical knowledge of the Manin–Radul and Jacobian ScB(C)KP hierarchies. In Sections 5 Darboux transformations of the supersymmetric constrained BKP hierarchy, 7 Darboux transformations of the supersymmetric constrained CKP hierarchy, we construct Darboux transforms of the supersymmetric constrained B(C)KP hierarchies respectively.

Section snippets

The supersymmetric constrained KP hierarchy

We first look back to some basic knowledge about supersymmetric KP hierarchy [3], [15]. Suppose that $Ϝ$ is an algebra of super quasi-differential operators, which takes the spatial variable $x$ , the Grassmann variable $θ$ and a super-derivation $D = \partial_{θ} + θ \partial$ . The algebra $Ϝ$ satisfies the following algorithm $D^{n} \cdot f = \sum_{n = 0}^{\infty} [\begin{matrix} n \\ n - i \end{matrix}] {(- 1)}^{| f | (n - i)} f^{[i]} D^{n - i},$ $[\begin{matrix} n \\ n - i \end{matrix}] = \{\begin{matrix} 0 i < 0 o r (n, i) = (0, 1) (m o d 2), \\ (\begin{matrix} [\frac{n}{2}] \\ [\frac{n - i}{2}] \end{matrix}) i \geq 0, (n, i) \neq (0, 1) (m o d 2) . \end{matrix}$

The value $| f |$ here represents the super degree of the operator $f$ , indicating that the operator $f$ is

Darboux transformations of the supersymmetric constrained KP hierarchy

In this section, we will construct the Darboux transformation of the ScKP hierarchy, from which we find the relationship between the new solutions and the old solutions.

Now we consider the Darboux transformation of the ScKP hierarchy from the following Lax operator with new solutions $\tilde{f}, \tilde{Φ}, \tilde{Ψ}$ : $\tilde{L} = D + \tilde{f} + \tilde{Φ} D^{- 1} \tilde{Ψ} .$ In order to maintain the invariance of Lax equations of the ScKP hierarchy, the new super-pseudo-differential operator should satisfy the following form: $\frac{\partial}{\partial t_{l}} \tilde{L} = [{\tilde{L}}_{+}^{2 l}, \tilde{L}}, D_{n} \tilde{L} = [{\tilde{L}}_{+}^{2 n - 1}, \tilde{L}}$

The supersymmetric constrained BKP hierarchy

In this section, based on the above research on the ScKP hierarchy, we will construct the ScBKP hierarchy, and give the form of the Lax operator for the ScBKP hierarchy. The B type condition of the ScBKP hierarchy is defined as $L^{*} = - D L D^{- 1} .$ The Lax operator of the ScBKP hierarchy can has a form: $L = \partial + q D^{- 1} r - e D^{- 1} f, r = e^{[1]} f = q^{[1]},$ where $D (f) = f^{[1]}, D (q) = q^{[1]}$ , $q$ and $e$ are bosonic. Lax equations corresponding to the above are $\partial_{n} L = [L_{+}^{\frac{4 n - 2}{2}}, L}, D_{n} L = [L_{+}^{\frac{4 n - 1}{2}}, L},$ where $\partial_{n} = \frac{\partial}{\partial t_{4 n - 2}}, D_{n} = \{\begin{matrix} \frac{\partial}{\partial ℓ_{4 n - 1}} - \sum ℓ_{4 m - 1} \frac{\partial}{\partial t_{4 n + 4 m - 2}} \end{matrix}$

Darboux transformations of the supersymmetric constrained BKP hierarchy

In this section, we will construct Darboux transformations of the ScBKP hierarchy. It is easy to know that elementary Darboux transformations do not meet the appropriate conditions. Now we consider Darboux transformation of the ScBKP hierarchy from the following new Lax operator: $\tilde{L} = \partial + \tilde{q} D^{- 1} \tilde{r} - \tilde{e} D^{- 1} \tilde{f} = T_{1} L T_{1}^{- 1},$ where $T_{1}$ plays the role of the Darboux transformation operator.

Lemma 5.1

Introduce the odd normal operator: $T_{D} = Φ D Φ^{- 1}$ [14] and $T_{I} = Ψ^{- 1} D^{- 1} Ψ$ , where $Φ$ is the invertible wave function of linear system

The supersymmetric constrained CKP hierarchy

In this section, on the basis of the above research on the ScBKP hierarchy, we will construct the ScCKP hierarchy. The Lax operator of the ScCKP hierarchy should satisfy the C type condition: $L^{*} = - L .$ The pseudo-differential operator of the ScCKP hierarchy has the following form $L = \partial + q D^{- 1} r + e D^{- 1} f, r = e f = q,$ where $q$ and $e$ are not both bosonic or fermionic. The Lax equation corresponding to Eq. (6.2) is as follows: $\partial_{n} L = [L_{+}^{\frac{4 n - 2}{2}}, L}, D_{n} L = [L_{+}^{\frac{4 n - 3}{2}}, L},$ where $\partial_{n} = \frac{\partial}{\partial t_{4 n - 2}}, D_{n} = \{\begin{matrix} \frac{\partial}{\partial ℓ_{4 n - 3}} - \sum ℓ_{4 m - 3} \frac{\partial}{\partial t_{4 n + 4 m - 6}} \end{matrix}$

Darboux transformations of the supersymmetric constrained CKP hierarchy

In this section, we will construct Darboux transformations of the ScCKP hierarchy, which is similar to Darboux transformations of the ScBKP hierarchy. We need to complete the two steps of the elementary Darboux transformation, so as to retain the odd flows. Now we consider the Darboux transformation of the ScCKP hierarchy from the following newly constructed Lax operator: $\tilde{L} = \partial + \tilde{q} D^{- 1} \tilde{r} + \tilde{e} D^{- 1} \tilde{f} = T_{1} L T_{1}^{- 1},$ where $T_{1}$ is a Darboux transformation operator.

Now we consider the following lemmas.

Lemma 7.1

The operator:

Acknowledgements

Chuanzhong Li is supported by the National Natural Science Foundation of China under Grant No. 12071237 and K.C. Wong Magna Fund in Ningbo University in China .

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Darboux transformations of the supersymmetric constrained B and C type KP hierarchies

Abstract

Introduction

Section snippets

The supersymmetric constrained KP hierarchy

Darboux transformations of the supersymmetric constrained KP hierarchy

The supersymmetric constrained BKP hierarchy

Darboux transformations of the supersymmetric constrained BKP hierarchy

The supersymmetric constrained CKP hierarchy

Darboux transformations of the supersymmetric constrained CKP hierarchy

Acknowledgements

Phys. Lett. A

J. Geom. Phys.

Nuclear Phys. B

Phys. Lett. B

Nuclear Phys. B

Method of squared eigenfunction potentials in integrable hierarchies of KP type

Comm. Math. Phys.

Supersymmetric Kadomtsev–Petviashvili hierarchy: Ghost symmetric structure, reductions, and Darboux-Bäcklund solutions

J. Math. Phys.

Solving the constrained KP hierarchy by gauge transformations

J. Math. Phys.

Transformation groups for solition equation. 3. Operator approach to the Kadomtsev-Petviahvili equation

J. Phys. Soc. Japan

Transformation groups for solition equations

Soliton Equations and Hamiltonian Systems