Cauchy matrix approach to the SU(2) self‐dual Yang–Mills equation

Li, Shangshuai; Qu, Changzheng; Yi, Xiangxuan; Zhang, Da‐jun

doi:10.1111/sapm.12488

Cited by 20 publications

(13 citation statements)

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“…In reductions of the BKP and CKP hierarchies, the reduced measures admit antisymmetry and symmetry property, respectively [24]. It is worth noting that the Sylvester equation was also developed to study some interesting higher-dimensional physical models, such as a Yajima-Oikawa system, a Mel'nikov model and a self-dual Yang-Mills equation (see [25][26][27]). In [27], based on the Sylvester Equation (3), Li et al investigated the master function S (i,j) = s T K j (I + M) −1 K i r and revealed that this function has the symmetric property S (i,j) T = −σ 2 S (j,i) σ 2 , where σ 2 is a Pauli matrix.…”

Section: Introductionmentioning

confidence: 99%

The Sylvester Equation and Kadomtsev–Petviashvili System

Feng

Zhao

2022

Symmetry

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In this paper, we seek connections between the Sylvester equation and Kadomtsev–Petviashvili system. By introducing Sylvester equation LM are bold, please chekc if bold neceaasry, if not, please remove all bold of equation −MK = rsT together with an evolution equation set of r and s, master function S(i,j)=sTKjC(I + MC)−1Lir is used to construct the Kadomtsev–Petviashvili system, including the Kadomtsev–Petviashvili equation, modified Kadomtsev–Petviashvili equation and Schwarzian Kadomtsev–Petviashvili equation. The matrix M provides τ-function by τ = |I + MC|. With the help of some recurrence relations, the reductions to the Korteweg–de Vries and Boussinesq systems are discussed.

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Section: Introductionmentioning

confidence: 99%

The Sylvester Equation and Kadomtsev–Petviashvili System

Feng

Zhao

2022

Symmetry

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“…Note that the original Yang-Mills equation [17] is not integrable in general but the self-dual gauge field equations are integrable, e.g., in the sense of Painlevé property [5,12]. One can also refer to an early review [10] or a recent paper [6] and the references therein, or a recent theses [4], for more details.…”

Section: Introductionmentioning

confidence: 99%

“…In the recent paper [6] we developed an approach to construct solutions to the SU(2) SDYM equation by means of the Cauchy matrix method. It is based on a framework of the Cauchy matrix approach to the Ablowitz-Kaup-Newell-Segur (AKNS) equations.…”

Section: Introductionmentioning

confidence: 99%

“…It is based on a framework of the Cauchy matrix approach to the Ablowitz-Kaup-Newell-Segur (AKNS) equations. Through introducing an independent variable x 0 , we were able to establish a set of noncommutative recursive and evolution relations for master functions {S (i,j) } (see equations ( 22) and (23) in [6]), which enable us to construct solutions to the SU(2) SDYM equation. It is possible to extend the approach developed in [6] to the SU(N ) SDYM equation for N being even, however, not for arbitrary N .…”

Section: Introductionmentioning

confidence: 99%

“…Our plan is the following. We will begin by presenting a set of derivative and difference relations (see Lemma 1), which are the same as the set in [6] but the master functions S (i,j) are allowed to be N × N matrices. Then we will look for S (i,j) that can match these relations.…”

Section: Introductionmentioning

confidence: 99%

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Solutions to the SU($\mathcal{N}$) self-dual Yang-Mills equation

Li¹,

Qu²,

Zhang³

2022

Preprint

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In this paper we aim to derive solutions for the SU(N ) self-dual Yang-Mills (SDYM) equation with arbitrary N . A set of noncommutative relations are introduced to construct a matrix equation that can be reduced to the SDYM equation. It is shown that these relations can be generated from two different Sylvester equations, which correspond to the two Cauchy matrix schemes for the (matrix) Kadomtsev-Petviashvili hierarchy and the (matrix) Ablowitz-Kaup-Newell-Segur hierarchy, respectively. In each Cauchy matrix scheme we investigate the possible reductions that can lead to the SU(N ) SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one.

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Solutions of Local and Nonlocal Discrete Complex Modified Korteweg–De Vries Equations and Continuum Limits

Hu,

Shen,

Zhao

2024

Math Methods in App Sciences

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Cauchy matrix approach for the discrete Ablowitz–Kaup–Newell–Segur equations is reconsidered, where two “proper” discrete Ablowitz–Kaup–Newell–Segur equations and two “unproper” discrete Ablowitz–Kaup–Newell–Segur equations are derived. The “proper” equations admit local reduction, while the “unproper” equations admit nonlocal reduction. By imposing the local and nonlocal complex reductions on the obtained discrete Ablowitz–Kaup–Newell–Segur equations, two local and nonlocal discrete complex modified Korteweg–de Vries equations are constructed. For the obtained local and nonlocal discrete complex modified Korteweg–de Vries equations, soliton solutions and Jordan‐block solutions are presented by solving the determining equation set. The dynamical behaviors of 1‐soliton solution are analyzed and illustrated. Continuum limits of the resulting local and nonlocal discrete complex modified Korteweg–de Vries equations are discussed.

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Cauchy matrix approach to the SU(2) self‐dual Yang–Mills equation

Cited by 20 publications

References 47 publications

The Sylvester Equation and Kadomtsev–Petviashvili System

The Sylvester Equation and Kadomtsev–Petviashvili System

Solutions to the SU($\mathcal{N}$) self-dual Yang-Mills equation

Solutions of Local and Nonlocal Discrete Complex Modified Korteweg–De Vries Equations and Continuum Limits

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