相关论文: Haantjes Manifolds and Veselov Systems
The paper aims to point out a novel geometric characterisation of the WDVV equations of 2D topological field theory.
A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or $\omega \mathscr{H}$ manifolds), as a natural…
We consider hydrodynamic chains in $(1+1)$ dimensions which are Hamiltonian with respect to the Kupershmidt-Manin Poisson bracket. These systems can be derived from single $(2+1)$ equations, here called hydrodynamic Vlasov equations, under…
The idea of a $\bigvee$-system was introduced by Veselov in the study of rational solutions of the WDVV equations of associativity. These are algebraic/geometric conditions on the set of covectors that appear in rational solutions to the…
We consider trigonometric solutions of WDVV equations and derive geometric conditions when a collection of vectors with multiplicities determines such a solution. We incorporate these conditions into the notion of trigonometric Veselov…
We introduce the notion of Haantjes algebra: It consists of an assignment of a family of operator fields on a differentiable manifold, each of them with vanishing Haantjes torsion. They are also required to satisfy suitable compatibility…
A special class of solutions to the generalised WDVV equations related to a finite set of covectors is investigated. Some geometric conditions on such a set which guarantee that the corresponding function satisfies WDVV equations are found…
The Witten-Dijkgraaf-Verlinde-Verlinde(WDVV) equations appeared in the study of two-dimensional topological field theoies in the early 1990s. An extension of the WDVV equations, called the open WDVV equations, was introduced by A.Horev and…
This paper has two purposes. The first is to introduce the definition of Haantjes manifolds with symmetry. The second is to explain why these manifolds appear in the theory of integrable systems of hydrodynamic type and in topological field…
In the context of the theory of symplectic-Haantjes manifolds, we construct the Haantjes structures of generalized St\"ackel systems and, as a particular case, of the quasi-bi-Hamiltonian systems. As an application, we recover the Haantjes…
A special class of solutions to the generalised WDVV equations related to a finite set of covectors is considered. We describe the geometric conditions ($\vee$-conditions) on such a set which are necessary and sufficient for the…
A theory of partial separability for classical Hamiltonian systems is proposed in the context of Haantjes geometry. As a general result, we show that the knowledge of a non-semisimple symplectic-Haantjes manifold for a given Hamiltonian…
An exact formula for the solutions to the WDVV equation in terms of horizontal sections of the corresponding flat connection is found.
The notion of Jacobi-Haantjes manifold, consisting of a Jacobi manifold endowed with an algebra of extended Haantjes operator fields, is proposed as a natural geometric framework which allows us to define the notion of integrability of both…
We consider the bi-Hamiltonian representation of the two-component coupled KdV equations discovered by Drinfel'd and Sokolov and rediscovered by Sakovich and Foursov. Connection of this equation with the supersymmetric…
Various links connecting well-known hydrodynamic chains and corresponding 2+1 nonlinear equations are described.
We present a complete proof that solutions of the WDVV equations in Seiberg-Witten theory may be constructed from root systems. A generalization to weight systems is proposed.
This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order…
The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general…
This manuscript embarks on an in-depth exploration of the modified Korteweg-de Vries (mKdV) equation, with a particular emphasis on unraveling the intricate structure of its infinite symmetries and their physical interpretations. Central to…