Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy

Chvartatskyi, Oleksandr

doi:10.3842/sigma.2015.028

Cited by 3 publications

(2 citation statements)

References 81 publications

(100 reference statements)

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“…Its scalar version has been studied in [1,4,7,[23][24][25][26][27]. The "noncommutative" generalization (2.12) already appeared in [28]. In our framework, the modification (2.10) is important.…”

Section: Introductionmentioning

confidence: 99%

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

2016

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We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey-Stewartson, two-dimensional Toda lattice and discrete KP equation. We also recover a (2+1)-dimensional version of the Yajima-Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.

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Section: Introductionmentioning

confidence: 99%

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

2016

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“…The soliton equations with self-consistent sources have many physical applications (see [7,14,20,21,22,27,28] and the references therein). For example, the speed of the solitons can be changed by the sources (see, e.g., [20,28]).…”

Section: Introductionmentioning

confidence: 99%

Bilinear identities for an extended B-type Kadomtsev–Petviashvili hierarchy

et al. 2016

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In this paper, we construct the bilinear identities for the wave functions of an extended B-type Kadomtsev-Petviashvili (BKP) hierarchy, which contains two types of (2+1)-dimensional Sawada-Kotera equation with a self-consistent source (2d-SKwS-I and 2d-SKwS-II). By introducing an auxiliary variable corresponding to the extended flow for the BKP hierarchy, we find the taufunction and the bilinear identities for this extended BKP hierarchy. The bilinear identities can generate all the Hirota's bilinear equations for the zerocurvature forms of this extended BKP hierarchy. As examples, the Hirota's bilinear equations for the two types of 2d-SKwS (both 2d-SKwS-I and 2d-SKwS-II) will be given explicitly.

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Inverse scattering transform in two spatial dimensions for the N-wave interaction problem with a dispersive term

Ismailov

2021

Journal of Inverse and Ill-Posed Problems

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A dispersive N-wave interaction problem ( N = 2 ⁢ n {N=2n} ), involving n velocities in two spatial and one temporal dimensions, is introduced. Explicit solutions of the problem are provided by using the inverse scattering method. The model we propose is a generalization of both the N-wave interaction problem and the ( 2 + 1 ) {(2+1)} matrix Davey–Stewartson equation. The latter examines the Benney-type model of interactions between short and long waves. Referring to the two-dimensional Manakov system, an associated Gelfand–Levitan–Marchenko-type, or so-called inversion-like, equation is constructed. It is shown that the presence of the degenerate kernel reads explicit soliton-like solutions of the dispersive N-wave interaction problem. We also present a discussion on the uniqueness of the solution of the Cauchy problem on an arbitrary time interval for small initial data.

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Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy

Cited by 3 publications

References 81 publications

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

Bilinear identities for an extended B-type Kadomtsev–Petviashvili hierarchy

Inverse scattering transform in two spatial dimensions for the N-wave interaction problem with a dispersive term

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