Algebraic-geometric solutions of the Krichever-Novikov equation

Novikov, D. P.

doi:10.1007/bf02557203

Cited by 12 publications

(23 citation statements)

References 13 publications

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“…It is known that the Landau-Lifshitz equation and the Krichever-Novikov equation possess so 3 (C)valued ZCRs parametrized by points of this curve [28,3,20,21]. (For the Krichever-Novikov equation, the paper [20] presents a ZCR with values in the Lie algebra sl 2 (C) ∼ = so 3 (C).) Proposition 1 is proved in [8], using some results of [11,21].…”

Section: Introductionmentioning

confidence: 99%

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On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Igonin

Manno

2020

Journal of Geometry and Physics

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Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs.In [arXiv:1303.3575], for any (1+1)-dimensional scalar evolution equation E, we defined a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. In [arXiv:1804.04652] we showed that, using these algebras, one obtains necessary conditions for existence of a Bäcklund transformation between two given equations. The algebras F(E) are defined in [arXiv:1303.3575] in terms of generators and relations.In this preprint we show that, using the algebras F(E), one obtains some necessary conditions for integrability of the considered PDEs, where integrability is understood in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution equations of order 5. Also, we prove a result announced in [arXiv:1303.3575] on the structure of the algebras F(E) for certain classes of equations of orders 3, 5, 7, which include KdV, mKdV, Kaup-Kupershmidt, Sawada-Kotera type equations.In this approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation E. The algebras F(E) generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs.

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Section: Introductionmentioning

confidence: 99%

“…For any e 1 , e 2 , e 3 ∈ C, we have the Krichever-Novikov equation KN(e 1 , e 2 , e 3 ) given by (18). Consider also the algebraic curve (20) C…”

Section: Introductionmentioning

confidence: 99%

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Igonin

Manno

2020

Journal of Geometry and Physics

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“…For the Krichever-Novikov equation, in [11] we show that some infinite-dimensional Lie algebra of certain matrix-valued functions on an elliptic curve, which arises from the elliptic ZCR [15,21] of this equation, plays the main role in the description of F p (E, a).…”

Section: Introductionmentioning

confidence: 94%

“…which implies that formulas(21) determine an action of the group of G-valued gauge transformations (20) on the set of g-valued ZCRs of order ≤ p.…”

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confidence: 99%

Lie algebras responsible for zero-curvature representations of scalar evolution equations

Igonin

Manno

2019

Journal of Geometry and Physics

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Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs.For any (1+1)-dimensional scalar evolution equation E, we define a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. To achieve this, we find a normal form for ZCRs with respect to the action of the group of local gauge transformations.As we show in other publications, using these algebras, one obtains some necessary conditions for integrability of the considered PDEs (where integrability is understood in the sense of soliton theory) and necessary conditions for existence of a Bäcklund transformation between two given equations. Examples of proving non-integrability and applications to obtaining non-existence results for Bäcklund transformations are presented in other publications as well.In our approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation E. The algebras F(E) generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs.In this paper we describe general properties of F(E) and present generators and relations for these algebras. In other publications we study the structure of F(E) for equations of KdV, Krichever-Novikov, Kaup-Kupershmidt, Sawada-Kotera types. Among the obtained algebras, one finds infinite-dimensional Lie algebras of certain matrix-valued functions on rational and elliptic algebraic curves.2010 Mathematics Subject Classification. 37K30, 37K35.

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“…Theorem 4 can be directly verified. Note that the finite-gap solutions of the Krichever-Novikov equation (4.2) were studied in [22].…”

Section: The Krichever-novikov Hierarchymentioning

confidence: 99%

Self-adjoint commuting differential operators of rank 2 and their deformations given by soliton equations

Davletshina

2015

Math Notes

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Algebraic-geometric solutions of the Krichever-Novikov equation

Cited by 12 publications

References 13 publications

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Lie algebras responsible for zero-curvature representations of scalar evolution equations

Self-adjoint commuting differential operators of rank 2 and their deformations given by soliton equations

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