Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour

Nilson, Tomas; Schiebold, Cornelia

doi:10.1080/14029251.2020.1683978

Cited by 4 publications

(2 citation statements)

References 31 publications

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“…The solutions of A ∞ -and A (1) 1 -types coincide with those obtained from the operator approach and the so-called generalised Cauchy matrix approach, see e.g. [24,29,30] and also [19,34,35]. Here we extended these results to nonlinear integrable systems associated with other Lie algebras.…”

Section: Discussionsupporting

confidence: 66%

Linear integral equations and two-dimensional Toda systems

Yin,

2021

Preprint

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A general framework is presented for the two-dimensional Toda equations associated with the infinite-dimensional Lie algebras A ∞ , B ∞ and C ∞ , as well as the Kac-Moody algebras A (1) r , A (2) 2r , C (1) r and D (2) r+1 for arbitrary integers r ∈ Z + , from the aspect of a set of linear integral equations in a certain form. Our scheme not only provides a unified perspective to understand the underlying integrability structure, but also induces a general solution potentially leading to the universal solution space, for each class of the two-dimensional Toda system. As particular applications of this framework to the two-dimensional Toda lattices, we rediscover the Lax pairs and the adjoint Lax pairs and simultaneously construct the generalised Cauchy matrix solutions.

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Section: Discussionsupporting

confidence: 66%

Linear integral equations and two-dimensional Toda systems

Yin,

2021

Preprint

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“…The solutions of

A_{\infty}

‐ and

A_{1}^{false(1 false)}

‐types coincide with those obtained from the operator approach and the so‐called generalized Cauchy matrix approach, see, for example, Refs. 30–32 and also Refs. 26–28.…”

Section: Discussionmentioning

confidence: 90%

Linear integral equations and two‐dimensional Toda systems

Yin

2021

Stud Appl Math

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The direct linearization framework is presented for the two‐dimensional (2D) Toda equations associated with the infinite‐dimensional Lie algebras A∞, B∞, and C∞, as well as the Kac–Moody algebras Arfalse(1false), A2rfalse(2false), Crfalse(1false), and Dr+1false(2false) for arbitrary integers r∈double-struckZ+, from the aspect of a set of linear integral equations in a certain form. Such a scheme not only provides a unified perspective to understand the underlying integrability structure, but also induces the direct linearizing type solution potentially leading to the universal solution space, for each class of the 2D Toda system. As particular applications of this framework to the 2D Toda lattices, we rediscover the Lax pairs and the adjoint Lax pairs and simultaneously construct the generalized Cauchy matrix solutions.

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From the operator KP equation to scalar and matrix-valued solutions

Porten,

Schiebold

2024

Contemporary Mathematics

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In this paper, which has a partially introductory character, the point of departure is very general solution formulas to the operator-valued Kadomtsev–Petviashvili equations (KPI and KPII). Their generality relies on the presence of parameters, which are allowed to be linear mappings between Banach spaces. While this freedom gives access to very complicated solutions, like countable nonlinear superpositions of solitons, the applications in the present article are restricted to matrix parameters. The necessary techniques to ‘project’ to scalar and matrix-valued solutions are explained in detail. The applications part is focused on the KPII. For scalar solutions, the testing ground is the celebrated classification of web structures of interacting line-solitons by Biondini, Chakravarty, Kodama, et al. We establish an explicit link between the different approaches and add some details about interactions of few line-solitons and solutions with singularities. Starting from more complicated algebraic data, we construct solutions beyond web structures, for which we prove asymptotic movement along logarithmic curves in time slices. The article is concluded by a section on solutions to the d × d {\mathsf {d}}\times {\mathsf {d}} matrix KPII. We realize N N -solitons in the sense of Gilson, Nimmo, and Sooman (and originally Goncharenko) and discuss the question about nonscalar Miles structures.

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Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour

Cited by 4 publications

References 31 publications

Linear integral equations and two-dimensional Toda systems

Linear integral equations and two-dimensional Toda systems

Linear integral equations and two‐dimensional Toda systems

From the operator KP equation to scalar and matrix-valued solutions

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