Cauchy problem for integrable discrete equations on quad-graphs

Адлер, Всеволод Эдуардович; Веселов, А. П.

doi:10.1007/s10440-005-4557-8

Cited by 40 publications

(84 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Using such Bäcklund transformation and starting with a simple initial solution, one can obtain, in principle, a multi-soliton solution. See [6,8] for the construction of some examples of solutions.…”

Section: The Abs Equations As Bäcklund Transformations Of the Ydkn Eqmentioning

confidence: 99%

On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

Levi

Petrera

Scimiterna

et al. 2008

SIGMA

View full text Add to dashboard Cite

Abstract. We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Bäcklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.

show abstract

Section: The Abs Equations As Bäcklund Transformations Of the Ydkn Eqmentioning

confidence: 99%

On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

Levi

Petrera

Scimiterna

et al. 2008

SIGMA

View full text Add to dashboard Cite

show abstract

“…This interpretation follows naturally if one starts from the two dimensional grid situation studied in [12] and generalizes to higher dimension inspired by the multi-dimensional consistency [13,9] and the Cauchy problem on quad-graphs [14].…”

Section: Explicit Auto-transformations Of Integrable Chains As 2-fielmentioning

confidence: 99%

Yang–Baxter maps and multi-field integrable lattice equations

Papageorgiou¹,

Tongas²

2007

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

A variety of Yang-Baxter maps are obtained from integrable multi-field equations on quadgraphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice equations introduced by Adler and Yamilov and which are related to the nonlinear superposition formulae for the Bäcklund transformations of the nonlinear Schrödinger system and specific ferromagnetic models.

show abstract

“…Remark 7.5. The transition from the canonical initial-value-problem on the hypercube domain for quad-equations, to a general planar quad-graph domain embedded within it, has been explained by Adler & Veselov [37]. A simple example of this, illustrating the two-dimensional discrete dynamics contained in the system of proposition 7.3, is a description of the regular N 2 sub-domain, which proceeds as follows.…”

Section: The Cremona Systemmentioning

confidence: 99%

On the lattice-geometry and birational group of the six-point multi-ratio equation

Atkinson

2015

Proc. R. Soc. A.

View full text Add to dashboard Cite

AustraliaThe inherent self-consistency properties of the sixpoint multi-ratio equation allow it to be considered on a domain associated with a T-shaped Coxeter-Dynkin diagram. This extends the Kadomtsev-Petviashvili lattice, which has A N symmetry, and incorporates also Korteweg-de Vries-type dynamics on a sub-domain with D N symmetry, and Painlevé dynamics on a subdomain withẼ 8 symmetry. More generally, it can be seen as a distinguished representation of Coble's Cremona group associated with invariants of point sets in projective space.

show abstract

Cauchy problem for integrable discrete equations on quad-graphs

Cited by 40 publications

References 22 publications

On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

Yang–Baxter maps and multi-field integrable lattice equations

On the lattice-geometry and birational group of the six-point multi-ratio equation

Contact Info

Product

Resources

About