Seminar talk, 16 September 2026: Difference between revisions
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| title = Finite dimensional reductions of integrable differential-difference equations | | title = Finite dimensional reductions of integrable differential-difference equations | ||
| abstract = Integrable partial differential equations, such as the Korteweg-de Vries (KdV) equation, admit infinite hierarchies of commuting higher symmetries. Their symmetry reductions give rise to integrable finite-dimensional dynamical systems that are solvable in terms of Abelian functions. This observation underlies the finite-gap integration method for the KdV equation, introduced by S.P. Novikov and subsequently extended to a wide class of integrable systems. In this paper, we introduce a new and more general class of reductions for integrable differential-difference equations, leading to integrable finite-dimensional systems in both commutative and noncommutative settings. The reduction constraints define integrable maps that enable solutions of the reduced systems to be extended to solutions of the corresponding differential-difference equations. The construction is illustrated using the Volterra and Toda hierarchies. | | abstract = Integrable partial differential equations, such as the Korteweg-de Vries (KdV) equation, admit infinite hierarchies of commuting higher symmetries. Their symmetry reductions give rise to integrable finite-dimensional dynamical systems that are solvable in terms of Abelian functions. This observation underlies the finite-gap integration method for the KdV equation, introduced by S.P. Novikov and subsequently extended to a wide class of integrable systems. In this paper, we introduce a new and more general class of reductions for integrable differential-difference equations, leading to integrable finite-dimensional systems in both commutative and noncommutative settings. The reduction constraints define integrable maps that enable solutions of the reduced systems to be extended to solutions of the corresponding differential-difference equations. The construction is illustrated using the Volterra and Toda hierarchies. | ||
| video = | | video = https://video.gdeq.org/GDEq-zoom-seminar-20260916-Alexander_Mikhailov.mp4 | ||
| slides = | | slides = | ||
| references = {{arXiv|2609.13769}} | | references = {{arXiv|2609.13769}} | ||
| 79YY-MM-DD = 7973-90-83 | | 79YY-MM-DD = 7973-90-83 | ||
}} | }} | ||
Latest revision as of 22:35, 16 September 2026
Speaker: Alexander Mikhailov
Title: Finite dimensional reductions of integrable differential-difference equations
Abstract:
Integrable partial differential equations, such as the Korteweg-de Vries (KdV) equation, admit infinite hierarchies of commuting higher symmetries. Their symmetry reductions give rise to integrable finite-dimensional dynamical systems that are solvable in terms of Abelian functions. This observation underlies the finite-gap integration method for the KdV equation, introduced by S.P. Novikov and subsequently extended to a wide class of integrable systems. In this paper, we introduce a new and more general class of reductions for integrable differential-difference equations, leading to integrable finite-dimensional systems in both commutative and noncommutative settings. The reduction constraints define integrable maps that enable solutions of the reduced systems to be extended to solutions of the corresponding differential-difference equations. The construction is illustrated using the Volterra and Toda hierarchies.
Video
References:
arXiv:2609.13769