A Supersymmetric AKNS Problem and Its Darboux‐Bäcklund Transformations and Discrete Systems

Xue, Lingling; Liu, Q. P.

doi:10.1111/sapm.12080

Cited by 19 publications

(25 citation statements)

References 34 publications

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“…In this paper, on one hand, we construct some novel, integrable, noncommutative (Grassmann) Boussinesq type systems, namely systems (28) and (39), together with a 3D-consistent limit (39), namely system (41). Moreover, we derive a novel Yang-Baxter map (13) together with its Grassmann extension (24). On the other hand, this paper answers an important question regarding the 3D consistency of systems, when they are extended to the Grassmann case.…”

Section: Discussionmentioning

confidence: 87%

On the 3D consistency of a Grassmann extended lattice Boussinesq system

Konstantinou-Rizos

2020

Nuclear Physics B

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In this paper, we formulate a "Grassmann extension" scheme for constructing noncommutative (Grassmann) extensions of Yang-Baxter maps together with their associated systems of P∆Es, based on the ideas presented in [11]. Using this scheme, we first construct a Grassmann extension of a Yang-Baxter map which constitutes a lift of a lattice Boussinesq system. The Grassmann-extended Yang-Baxter map can be squeezed down to a novel, integrable, Grassmann lattice Boussinesq system, and we derive its 3D-consistent limit. We show that some systems retain their 3D-consistency property in their Grassmann extension. The formulation of the ideas presented in [11] into a Grassmann extension scheme;2. The derivation of a new Boussinesq-type Yang-Baxter map together with its Grassmann extension;3. The construction of an integrable, noncommutative (Grassmann) extension of a discrete Boussinesq system and its 3D-consistent limit. The latter gives rise to the following important point.4. We show that, for some systems, the 3D-consistency property does not break in their noncommutative extension.The paper is organised as follows: The next section provides with preliminary knowledge, essential for the text to be self-contained. In particular, we fix the notation we use throughout the text, and we give the basic definitions of systems on quad-graphs and Yang-Baxter maps. Furthermore, we demonstrate the relation between the former and the latter and the relation between the 3D consistency property and the Yang-Baxter equation. We also explain what a Lax representation is for both equations on quad graphs and Yang-Baxter maps. Finally, we present the basic steps of a simple scheme for constructing Grassmann extensions of discrete integrable systems and their associated Yang-Baxter maps; the related ideas were presented in [11]. In section 3, we apply the aforementioned scheme to system (1). Specifically, we consider the associated Yang-Baxter lift of (1), for which we construct a Grassmann extension. Then, we show that the latter can be squeezed down to a novel integrable system of lattice equations which can be considered as the Grassmann extension of system (1). Finally, in section 4, we present a Boussinesq-type system associated via a conservation law to the one obtained in section 3, and we prove the integrability-in the sense of 3D-consistency-for a limit of this system. Finally, the last section deals with some concluding remarks and thoughts for future work. Preliminaries NotationHere, we explain the notation we shall be using throughout the text.

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Section: Discussionmentioning

confidence: 87%

On the 3D consistency of a Grassmann extended lattice Boussinesq system

Konstantinou-Rizos

2020

Nuclear Physics B

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“…Bäcklund transformations have been known to be an effective approach to construction of solutions for nonlinear systems, furthermore they may be applied to generate new integrable systems, both continuous and discrete [26,27,18]. It is remarked that the applications of Bäcklund transformations to integrable discretization of super or supersymmetric integrable systems were developed only recently [16,48,45,46,47,4,31].…”

Section: Introductionmentioning

confidence: 99%

Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and Applications

Mao¹,

Liu²,

Xue³

2021

JNMP

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In this paper, we construct a Darboux transformation and the related Bäcklund transformation for the supersymmetric Sawada-Kotera (SSK) equation.The associated nonlinear superposition formula is also worked out. We demonstrate that these are natural extensions of the similar results of the Sawada-Kotera equation and may be applied to produce the solutions of the SSK equation. Also, we present two semi-discrete systems and show that the continuum limit of one of them goes to the SKK equation.

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“…With a Bäcklund transformation in hand, one may either construct various solutions for the associated nonlinear system or produce new integrable systems of both continuous and discrete types [27,28,29]. It is noted that Bäcklund transformations for the supersymmetric integrable systems emerged already as early as the later seventies of last century [30], their applications to integrable discretizations of super integrable systems were developed only recently [31,32,33,34,35]. Very recently, the relationship between these results and extensions of Yang-Baxter map was explored [36,37].…”

Section: Introductionmentioning

confidence: 99%

Bäcklund–Darboux transformations and discretizations of N = 2 a = −2 supersymmetric KdV equation

Mao

Liu

2018

Physics Letters A

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The N = 2 a = −2 supersymmetric KdV equation is studied. A Darboux transformation and the corresponding Bäcklund transformation are constructed for this equation. Also, a nonlinear superposition formula is worked out for the associated Bäcklund transformation. The Bäcklund transformation and the related nonlinear superposition formula are used to construct integrable super semi-discrete and full discrete systems.The continuum limits of these discrete systems are also considered.

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A Supersymmetric AKNS Problem and Its Darboux‐Bäcklund Transformations and Discrete Systems

Cited by 19 publications

References 34 publications

On the 3D consistency of a Grassmann extended lattice Boussinesq system

On the 3D consistency of a Grassmann extended lattice Boussinesq system

Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and Applications

Bäcklund–Darboux transformations and discretizations of N = 2 a = −2 supersymmetric KdV equation

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