Bäcklund–Darboux transformations for the coupled KP hierarchy

Leur, Johan van de

doi:10.1088/0305-4470/37/15/008

Cited by 9 publications

(3 citation statements)

References 20 publications

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“…4, which, up to the moduli e and g , have appeared in Ref. 27 (see also references therein) as the first few members in the bilinear DKP hierarchy.…”

Section: Closed‐form Nonlinear System and Associated Linear Problemmentioning

confidence: 99%

On a coupled Kadomtsev–Petviashvili system associated with an elliptic curve

Nijhoff

2022

Stud Appl Math

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The coupled Kadomtsev-Petviashvili system associated with an elliptic curve, proposed by Date, Jimbo, and Miwa [J. Phys. Soc. Jpn., 52:766-771, 1983], is reinvestigated within the direct linearization framework, which provides us with more insights into the integrability of this elliptic model from the perspective of a general linear integral equation. As a result, we successfully construct for the elliptic coupled Kadomtsev-Petviashvili system not only a Lax pair composed of differential operators in 2 × 2 matrix form but also multisoliton solutions with phases parametrized by points on the elliptic curve. Dimensional reductions based on the direct linearization, to the elliptic coupled Kortewegde Vries and Boussinesq systems, are also discussed. In addition, a novel class of solutions is obtained for the 𝐷 ∞ -type Kadomtsev-Petviashvili equation with nonzero constant background as a by-product.

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“…4, which, up to the moduli e and g , have appeared in Ref. 27 (see also references therein) as the first few members in the bilinear DKP hierarchy.…”

Section: Closed‐form Nonlinear System and Associated Linear Problemmentioning

confidence: 99%

On a coupled Kadomtsev–Petviashvili system associated with an elliptic curve

Nijhoff

2022

Stud Appl Math

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“…Concerning the cKP equations (1.4)-(1.6), some research has been conducted [1,3,10,[12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

“…The main difficulty lies in that one does not know what exact form the Bäcklund transformation takes for the coupled KP equation although the Bäcklund transformation for the KP is available. Very recently, van de Leur constructed an elementary Bäcklund-Darboux transformation for the coupled KP hierarchy using the geometry of the infinite isotropic Grassmannian related to the coupled KP hierarchy [12]. The remarkable result of van de Leur has shed light on solving Hirota's problem.…”

Section: Introductionmentioning

confidence: 99%

Commutativity of Pfaffianization and Bäcklund transformations: the KP equation

Hu¹,

Zhao²

2005

Inverse Problems

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The Pfaffianization procedure is applied to the modified KP equation. As a result, we derive a coupled modified KP system, which is shown to constitute a bilinear Bäcklund transformation for the coupled KP (cKP) equation introduced by Hirota and Ohta. This means that commutativity of Pfaffianization and Bäcklund transformation is valid for the KP equation. As a by-product, we explicitly derive a coupled system from Bäcklund transformation formulae of the cKP equation, which solves an open problem proposed by Hirota. In addition to starting from the Bäcklund transformation, a Lax pair for the coupled KP equation is worked out. Finally, two kinds of Pfaffian solutions to the coupled modified KP equation are derived.

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