Забродин Антон Владимирович

Multicomponent DKP hierarchy and its dispersionless limit

2025 в печати · ARTICLE · en

Using the free fermions technique and bosonization rules, we introduce the multicomponent DKP hierarchy as a generating bilinear integral equation for the tau-function. A number of bilinear equations of the Hirota–Miwa type are obtained as its corollaries. We also consider the dispersionless version of the hierarchy as a set of nonlinear differential equations for the dispersionless limit of logarithm of the tau-function (the F-function). We show that there is an elliptic curve built in the structure of the hierarchy, with the elliptic modulus being a dynamical variable. This curve can be uniformized by elliptic functions, and in the elliptic parametrization many dispersionless equations of the Hirota–Miwa type become equivalent to a single equation having a nice form.

Multi-component Toda lattice hierarchy

2025 · ARTICLE · en

We give a detailed account of the N -component Toda lattice hierarchy, which can be regarded as a generalization of the well-known Toda chain model and its non-abelian version. This hierarchy is an extension of the one intro- duced earlier by Ueno and Takasaki. Our version contains N discrete vari- ables rather than one. We start from the Lax formalism, deduce the bilinear relation for wave functions from it, and then, based on the latter, prove the existence of the tau-function. We also show how the multi-component Toda lattice hierarchy is embedded into the universal hierarchy, which is basically the multi-component Kadomtsev–Petviashvili hierarchy. Finally, we show how the bilinear integral equation for the tau-function can be obtained using the free fermion technique. An example of exact solutions (a multi-component analogue of one-soliton solutions) is given.

Elliptic Cauchy matrices

2024 · ARTICLE · en

Some identities that involve the elliptic version of the Cauchy matrices are presented and proved. They include the determinant formula, the formula for the inverse matrix, the matrix product identity and the factorization formula.

Tau-function of the multi-component CKP hierarchy

2024 · ARTICLE · en

We consider multi-component Kadomtsev-Petviashvili hierarchy of type C (the multi-component CKP hierarchy) originally defined with the help of matrix pseudo-differential operators via the Lax-Sato formalism. Starting from the bilinear relation for the wave functions, we prove existence of the tau-function for the multi-component CKP hierarchy and provide a formula which expresses the wave functions through the tau-function. We also find how this tau-function is related to the tau-function of the multi-component Kadomtsev-Petviashvili hierarchy. The tau-function of the multi-component CKP hierarchy satisfies an integral relation which, unlike the integral relation for the latter tau-function, is no longer bilinear but has a more complicated form.

Dispersionless limit of the B-Toda hierarchy

2024 · ARTICLE · en

We study the dispersionless limit of the recently introduced Toda lattice hierarchy with constraint of type B (the B-Toda hierarchy) and compare it with that of the DKP and C-Toda hierarchies. The dispersionless limits of the B-Toda and C-Toda hierarchies turn out to be the same.

Dispersionless version of the multicomponent KP hierarchy revisited

2024 · ARTICLE · en

We revisit dispersionless version of the multicomponent KP hierarchy considered previously by Takasaki and Takebe. In contrast to their study, we do not fix any distinguished component treating all of them on equal footing. We obtain nonlinear equations for dispersionless tau-function (the -function) and represent them using the trigonometric parametrization. In this trigonometric uniformization the equations considerably simplify and acquire a nice form.

Quasi-Periodic Solutions of the Universal Hierarchy

2024 · ARTICLE · en

We construct quasi-periodic solutions of the universal hierarchy which includes the multi-component KP and Toda hierarchies and show how they fit into the bilinear formalism. The tau-function is expressed in terms of the Riemann theta-function multiplied by exponential function of a quadratic form in the hierarchical times.

Elliptic Solutions of the Toda Lattice with Constraint of Type B and Deformed Ruijsenaars–Schneider System

2023 в печати · ARTICLE · en

Toda lattice with constraint of type B

2023 · ARTICLE · en

We introduce a new integrable hierarchy of nonlinear differential-difference equations which is a subhierarchy of the 2D Toda lattice defined by imposing a constraint to the Lax operators of the latter. The 2D Toda lattice with the constraint can be regarded as a discretization of the BKP hierarchy. We construct its algebraic-geometrical solutions in terms of Riemann and Prym theta-functions.

Об интегрируемости деформированной системы Руйсенарса–Шнайдера

2023 · ARTICLE · ru

Найдены интегралы движения для недавно введенной деформированной многочастичной системы Руйсенарса–Шнайдера, которая является динамической системой для полюсов эллиптических решений решетки Тоды со связью типа B. Наш метод основан на том факте, что уравнения движения этой системы совпадают с уравнениями движения для частиц Руйсенарса–Шнайдера, слипающихся в пары, в которых расстояние между частицами фиксировано и принимает специальное значение. Также для деформированной системы Руйсенарса–Шнайдера найдены преобразования Бэклунда и интегрируемая версия этой системы в дискретном времени. Показано, что эта последняя является динамической системой для полюсов эллиптических решений полностью дискретного уравнения Кадомцева–Петвиашвили типа B. Кроме того, предложен полевой аналог деформированной системы Руйсенарса–Шнайдера на пространственно-временной решетке.