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KdV hierarchies and quantum Novikov's equationsстатья
- Авторы: Buchstaber V.M., Mikhailov A.V.
- Журнал: Open Communications in Nonlinear Mathematical Physics
- Номер: 1
- Год издания: 2024
- Издательство: Episciences
- Местоположение издательства: France
- Первая страница: 1
- Последняя страница: 36
- DOI: 10.46298/ocnmp.12684
- Аннотация: This paper begins with a review of the well-known KdV hierarchy, the 𝑁-th Novikov equation, and its finite hierarchy in the classical commutative case. This finite hierarchy consists of 𝑁 compatible integrable polynomial dynamical systems in ℂ^{2𝑁}. We discuss a non-commutative version of the 𝑁-th Novikov hierarchy defined on the finitely generated free associative algebra 𝔅_𝑁 with 2𝑁 generators. Using the method of quantisation ideals in 𝔅_𝑁, for 𝑁=1,2,3,4, we obtain two-sided homogeneous ideals 𝔔_𝑁⊂𝔅_𝑁 (quantisation ideals) that are invariant with respect to the 𝑁-th Novikov equation and such that the quotient algebra ℭ_𝑁=𝔅_𝑁/𝔔+𝑁 has a well-defined Poincare-Birkhoff-Witt basis. This allows us to define the quantum 𝑁-th Novikov equation and its hierarchy on ℭ_𝑁. We derive 𝑁 commuting quantum first integrals (Hamiltonians) and represent the equations of the hierarchy in the Heisenberg form. Essential for our research is the concept of cyclic Frobenius algebras, which we introduced in our recent paper. In terms of the quadratic form that defines the structure of a cyclic Frobenius algebra, we explicitly express the first integrals of the 𝑁-th Novikov hierarchy in the commutative, free, and quantum cases.
- Добавил в систему: Бухштабер Виктор Матвеевич