Quasilinear systems of Jordan block type and the mKP hierarchy

Xue, Lingling; Ferapontov, E. V.

doi:10.1088/1751-8121/ab859a

Cited by 8 publications

(9 citation statements)

References 18 publications

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“…• each diagonal block has upper-triangular Toeplitz form as in Xue and Ferapontov (2020), with the unique eigenvalue; here is a 3 × 3 Toeplitz matrix with eigenvalue v:…”

Section: Discussionmentioning

confidence: 99%

“…Remark Integrable quasilinear systems of a single Jordan block type (of arbitrary size) have appeared in the literature as degenerations of hydrodynamic systems associated with multi-dimensional hypergeometric functions (Kodama and Konopelchenko 2016), in the context of parabolic regularisation of the Riemann equation (Konopelchenko and Ortenzi 2018) and as reductions in hydrodynamic chains and linearly degenerate dispersionless PDEs in 3D (Pavlov 2018). A connection of such systems with the modified KP hierarchy was established in Xue and Ferapontov (2020).…”

Section: Introductionmentioning

confidence: 99%

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Kinetic Equation for Soliton Gas: Integrable Reductions

Ferapontov

Павлов

2022

J Nonlinear Sci

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Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several $$2\times 2$$ 2 × 2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev’s theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.

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“…• each diagonal block has upper-triangular Toeplitz form as in Xue and Ferapontov (2020), with the unique eigenvalue; here is a 3 × 3 Toeplitz matrix with eigenvalue v:…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Kinetic Equation for Soliton Gas: Integrable Reductions

Ferapontov

Павлов

2022

J Nonlinear Sci

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“…The generalized hodograph transform has been developed for HTS which are semi-hamiltonian, diagonalizable and with real distinct eigenvalues; so, strictly speaking, we cannot say that our systems are integrable. However, the fact that there are coinciding eigenvalues is not a strong restriction to the applicability of the generalized hodograph transform, as recent results show [38].…”

Section: Remark 13mentioning

confidence: 97%

Homogeneous Hamiltonian operators and the theory of coverings

Vergallo¹,

Vitolo²

2020

Preprint

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A new method (by Kersten, Krasil'shchik and Verbovetsky), based on the theory of differential coverings, allows to relate a system of PDEs with a differential operator in such a way that the operator maps symmetries/conserved quantities into symmetries/conserved quantities of the system of PDEs. When applied to a quasilinear first-order system of PDEs and a Dubrovin-Novikov homogeneous Hamiltonian operator the method yields conditions on the operator and the system that have interesting differential and projective geometric interpretations.

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“…In the latter case, the hydrodynamic-type systems reduce to the form ut=(−12v3w2)x,1emvt=(u+32v2w)x, wt=vx and uy=(14v4w3)x,1emvy=(−12v3w2)x, wy=ux. Their velocity matrices have a single common characteristic root only; such degenerate cases are very interesting. For instance, hydrodynamic-type systems with a unique characteristic root were recently investigated [11–13]. Let us consider the first of the above three-component hydrodynamic-type system (6.1): right left right left right left right left right left right left3pt0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em ut=−3v22w2vx+v…”

Section: An Example In Three Components: Witten–dijkgraaf–verlinde–verlinde Equationmentioning

confidence: 99%

“…Their velocity matrices have a single common characteristic root only; such degenerate cases are very interesting. For instance, hydrodynamic-type systems with a unique characteristic root were recently investigated [11][12][13]. Let us consider the first of the above three-component hydrodynamic-type system (6.1):…”

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confidence: 99%

Classification of bi-Hamiltonian pairs extended by isometries

2021

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The aim of this article is to classify pairs of the first-order Hamiltonian operators of Dubrovin–Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such a bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of two dependent variables, and a significant new example with three dependent variables that is an extension of a hydrodynamic-type system obtained from a particular solution of the Witten–Dijkgraaf–Verlinde–Verlinde equations.

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Quasilinear systems of Jordan block type and the mKP hierarchy

Abstract: We demonstrate that commuting quasilinear systems of Jordan block type are parametrised by solutions of the modified KP hierarchy. Systems of this form naturally occur as hydrodynamic reductions of multi-dimensional linearly degenerate dispersionless integrable PDEs. MSC: 35Q51, 37K10.

Cited by 8 publications

References 18 publications

Kinetic Equation for Soliton Gas: Integrable Reductions

Kinetic Equation for Soliton Gas: Integrable Reductions

Homogeneous Hamiltonian operators and the theory of coverings

Classification of bi-Hamiltonian pairs extended by isometries

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