Non-commutative NLS-type hierarchies: Dressing &amp; solutions

Doikou, Anastasia; Findlay, Iain; Sklaveniti, Spyridoula

doi:10.1016/j.nuclphysb.2019.02.019

Cited by 7 publications

(25 citation statements)

References 33 publications

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“…Let us briefly discuss the continuous case using the Toda type Darboux transformation (2.9). Recall the U -operator (see also [10]) of the continuous Lax pair (U, V )…”

Section: • Simple Solutionsmentioning

confidence: 99%

“…If V 0 = 0, and alsoV 0 =û 0B , thenû 0 satisfies the linear equations (we will consider below the example of the NLSlike equations. For details on the V operators for the hierarchy we refer the interested reader in [10]): From the Darboux relations (2.62), and assuming A = aBB, we obtain:…”

Section: )mentioning

confidence: 99%

“…x n →û(x), y n → u(x), a n → a(x), d n → d(x) (2.84) whereû, u are the matrix AKNS fields (following the notation of [10]). The two soliton solutions are given by (2.83) in terms of one soliton solutionsû (i) (x), u (i) (x) (a (i) (x), d (i) (x)) reported in [10], subject to the "dictionary" (2.84).…”

Section: Bianchi's Permutability and The Soliton Latticementioning

confidence: 99%

“…and also K − ij = F ij + N l≥i K il F lj , j ≤ i. From the discrete GLM equation two independent sets of algebraic equations are obtained, in analogy to the continuous case [10]:…”

Section: Symmetric Schemementioning

confidence: 99%

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Discretizations of the generalized AKNS scheme

Doikou

Sklaveniti

2020

J. Phys. A: Math. Theor.

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We consider space discretizations of the matrix Zakharov-Shabat AKNS scheme, in particular the discrete matrix non-linear Scrhrödinger (DNLS) model, and the matrix generalization of the Ablowitz-Ladik (AL) model, which is the more widely acknowledged discretization. We focus on the derivation of solutions via local Darboux transforms for both discretizations, and we derive novel solutions via generic solutions of the associated discrete linear equations. The continuum analogue is also discussed. In this frame we also derive a discretization of the Burgers equation via the analogue of the Cole-Hopf transform. Using the basic Darboux transforms for each scheme we identify both matrix DNLS-like and AL hierarchies, i.e. we extract the associated Lax pairs, via the dressing process. We also discuss the global Darboux transform, which is the discrete analogue of the integral transform, through the discrete Gelfand-Levitan-Marchenko (GLM) equation. The derivation of the discrete matrix GLM equation and associated solutions are also presented together with explicit linearizations. Particular emphasis is given in the discretization schemes, i.e. forward/backward in the discrete matrix DNLS scheme versus symmetric in the discrete matrix AL model.

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“…Let us briefly discuss the continuous case using the Toda type Darboux transformation (2.9). Recall the U -operator (see also [10]) of the continuous Lax pair (U, V )…”

Section: • Simple Solutionsmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Bianchi's Permutability and The Soliton Latticementioning

confidence: 99%

“…and also K − ij = F ij + N l≥i K il F lj , j ≤ i. From the discrete GLM equation two independent sets of algebraic equations are obtained, in analogy to the continuous case [10]:…”

Section: Symmetric Schemementioning

confidence: 99%

See 2 more Smart Citations

Discretizations of the generalized AKNS scheme

Doikou

Sklaveniti

2020

J. Phys. A: Math. Theor.

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“…To achieve this let us also take into consideration the x-part of the auxiliary linear problem (see also e.g. [24] on similar arguments regarding the space-like matrix NLS model). Indeed, the linear equation associated to the x n flow reads in general as…”

Section: Matrix Riccati Equations and Conserved Quantitiesmentioning

confidence: 99%

Time-like boundary conditions in the NLS model

2019

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We focus on the non-linear Schrödinger model and we extend the notion of space-time dualities in the presence of integrable time-like boundary conditions. We identify the associated time-like "conserved" quantities and Lax pairs as well as the corresponding boundary conditions. In particular, we derive the generating function of the space components of the Lax pairs in the case of time-like boundaries defined by solutions of the reflection equation. Analytical conditions on the boundary Lax pair lead to the time like-boundary conditions. The time-like dressing is also performed for the first time, as an effective means to produce the space components of the Lax pair of the associated hierarchy. This is particularly relevant in the absence of a classical r-matrix, or when considering complicated underlying algebraic structures. The associated time Riccati equations and hence the time-like conserved quantities are also derived. We use as the main paradigm for this purpose the matrix NLS-type hierarchy.

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