Searcharxiv⌕ Search

arXiv subjects

Youjin Zhang

Publications and source records attributed to Youjin Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Finite Group Reduction of the DR/DZ Hierarchies

We show that the finite group reduction of the Dubrovin-Zhang/Double Ramification hierarchy associated to a semisimple CohFT preserves its bihamiltonian structure and its tau structure. By applying this result to the orbifold Gromov-Witten theory with the target $\mathbb{P}^1_{2,2,2,2}$, we see that there are two different and interesting integrable hierarchies that can be associated to the Frobenius manifold structure on the Hurwitz space $M_{1;1}$. One of them is equivalent to the genus $1$ topological recursion.

math.AG↗
arXiv2025

On a class of integrable deformations of the integrable hierarchy of topological type associated to a semisimple Frobenius manifold

Given a semisimple Frobenius manifold, we construct a class of integrable deformations of its hierarchy of topological type. We show that these integrable deformations have polynomial tau-structures, and conjecture that for the one-dimensional Frobenius manifold they give a universal object for integrable deformations of the Riemann--Hopf hierarchy having a tau-structure.

math-ph↗
arXiv2024

Legendre transformations of a class of generalized Frobenius manifolds and the associated integrable hierarchies

For two generalized Frobenius manifolds related by a Legendre-type transformation, we show that the associated integrable hierarchies of hydrodynamic type, which are called the Legendre-extended Principal Hierarchies, are related by a certain linear reciprocal transformation; we also show, under the semisimplicity condition, that the topological deformations of these Legendre-extended Principal Hierarchies are related by the same linear reciprocal transformation.

math-ph↗
arXiv2024

On Tautological Flows of Partial Difference Equations

We propose a new analyzing method, which is called the tautological flow method, to analyze the integrability of partial difference equations (P$\Delta$Es) based on that of partial differential equations (PDEs). By using this method, we prove that the discrete $q$-KdV equation is a discrete symmetry of the $q$-deformed KdV hierarchy and its bihamiltonian structure, and we also demonstrate how to directly search for continuous symmetries and bihamiltonian structures of P$\Delta$Es by using the approximated tautological flows and their quasi-triviality transformation.

nlin.SI↗
arXiv2024

Generalized Frobenius Manifolds with Non-flat Unity and Integrable Hierarchies

For any generalized Frobenius manifold with non-flat unity, we construct a bihamiltonian integrable hierarchy of hydrodynamic type which is an analogue of the Principal Hierarchy of a Frobenius manifold. We show that such an integrable hierarchy, which we also call the Principal Hierarchy, possesses Virasoro symmetries and a tau structure, and the Virasoro symmetries can be lifted to symmetries of the tau-cover of the integrable hierarchy. We derive the loop equation from the condition of linearization of actions of the Virasoro symmetries on the tau function, and construct the topological deformation of the Principal Hierarchy of a semisimple generalized Frobenius manifold with non-flat unity. We also give two examples of generalized Frobenius manifolds with non-flat unity and show that they are closely related to the well-known integrable hierarchies: the Volterra hierarchy, the q-deformed KdV hierarchy and the Ablowitz-Ladik hierarchy.

math-ph↗
arXiv2024

Solutions of the loop equations of a class of generalized Frobenius manifolds

We prove the existence and uniqueness of solution of the loop equation associated with a semisimple generalized Frobenius manifold with non-flat unity, and show, for a particular example of one dimensional generalized Frobenius manifold, that the deformation of the Principal Hierarchy induced by the solution of the loop equation is the extended q-deformed KdV hierarchy.

math-ph↗
arXiv2023

Variational Bihamiltonian Cohomologies and Integrable Hierarchies III: Linear Reciprocal Transformations

For an integrable hierarchy which possesses a bihamiltonian structure with semisimple hydrodynamic limit, we prove that the linear reciprocal transformation with respect to any of its symmetry transforms it to another bihamiltonian integrable hierarchy. Moreover, we show that the central invariants of the bihamiltonian structure are preserved under such a linear reciprocal transformation.

nlin.SI↗
arXiv2022

Reduction of the 2D Toda Hierarchy and Linear Hodge Integrals

We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the intermediate long wave hierarchy, and the remaining flows coincide with a certain limit of the flows of the fractional Volterra hierarchy which controls the special cubic Hodge integrals.

nlin.SI↗
arXiv2021

The Virasoro-like Algebra of a Frobenius Manifold

For an arbitrary calibrated Frobenius manifold, we construct an infinite dimensional Lie algebra, called the Virasoro-like algebra, which is a deformation of the Virasoro algebra of the Frobenius manifold. By using the Virasoro-like algebra we give a family of quadratic PDEs that are satisfied by the genus-zero free energy of the Frobenius manifold. We also derive, under the semisimplicity assumption, the Virasoro constraints for the corresponding abstract Hodge partition function.

math-ph↗
arXiv2021

Linearization of Virasoro symmetries associated with semisimple Frobenius manifolds

For any semisimple Frobenius manifold, we prove that a tau-symmetric bihamiltonian deformation of its Principal Hierarchy admits an infinite family of linearizable Virasoro symmetries if and only if all the central invariants of the corresponding deformation of the bihamiltonian structure are equal to $\frac{1}{24}$. As an important application of this result, we prove that the Dubrovin-Zhang hierarchy associated with the semisimple Frobenius manifold possesses a bihamiltonian structure which can be represented in terms of differential polynomials.

math-ph↗
arXiv2021

Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations

This series of papers is devoted to the study of deformations of Virasoro symmetries of the principal hierarchies associated to semisimple Frobenius manifolds. The main tool we use is a generalization of the bihamiltonian cohomology called the variational bihamiltonian cohomology. In the present paper, we give its definitions and compute the associated cohomology groups that will be used in our study of deformations of Virasoro symmetries. To illustrate its application, we classify the conformal bihamiltonian structures with semisimple hydrodynamic limits.

math.DG↗
arXiv2021

Super tau-covers of bihamiltonian integrable hierarchies

We consider a certain super extension, called the super tau-cover, of a bihamiltonian integrable hierarchy which contains the Hamiltonian structures including both the local and non-local ones as odd flows. In particular, we construct the super tau-cover of the principal hierarchy associated with an arbitrary Frobenius manifold, and the super tau-cover of the Korteweg-de Vries (KdV) hierarchy. We also show that the Virasoro symmetries of these bihamiltonan integrable hierarchies can be extended to symmetries of the associated super tau-covers.

math.DG↗
arXiv2021

Tri-Hamiltonian Structure of the Ablowitz-Ladik Hierarchy

We construct a local tri-Hamiltonian structure of the Ablowitz-Ladik hierarchy, and compute the central invariants of the associated bihamiltonian structures. We show that the central invariants of one of the bihamiltonian structures are equal to 1/24, and the dispersionless limit of this bihamiltonian structure coincides with the one that is defined on the jet space of the Frobenius manifold associated with the Gromov-Witten invariants of local CP1. This result provides support for the validity of Brini's conjecture on the relation of these Gromov-Witten invariants with the Ablowitz-Ladik hierarchy.

math-ph↗