Additional symmetries of the extended bigraded Toda hierarchy

Bakalov, Bojko; Wheeless, William

doi:10.1088/1751-8113/49/5/055201

Cited by 9 publications

(13 citation statements)

References 29 publications

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“…Another interesting question is whether one can generate a W -algebra from the vertex operators Γ + and Γ − , as was done for the KP hierarchy in [1,2,14]. One can construct a Virasoro algebra based on [8,16], but it would be interesting to try to construct a more general W -algebra of symmetries by modifying the vertex operators Γ + and Γ − so that they depend explicitly on x (see [3,7,32]).…”

Section: Discussionmentioning

confidence: 99%

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Darboux transformations and Fay identities for the extended bigraded Toda hierarchy

Bakalov¹,

Yadavalli²

2020

J. Phys. A: Math. Theor.

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The extended bigraded Toda hierarchy (EBTH) is an integrable system satisfied by the Gromov-Witten total descendant potential of CP 1 with two orbifold points. We write a bilinear equation for the tau-function of the EBTH and derive Fay identities from it. We show that the action of Darboux transformations on the tau-function is given by vertex operators. As a consequence, we obtain generalized Fay identities.

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Section: Discussionmentioning

confidence: 99%

“…This section is a quick review of the EBTH following [8]. We first discuss the spaces of difference and differential-difference operators.…”

Section: Review Of the Extended Bigraded Toda Hierarchymentioning

confidence: 99%

Darboux transformations and Fay identities for the extended bigraded Toda hierarchy

Bakalov¹,

Yadavalli²

2020

J. Phys. A: Math. Theor.

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“…In [1] and [19], Bakalov and Wheeless introduced additional symmetries of the EBTH by constructing two operators M and M (extensions of those in [13]). These additional symmetries act on L via L p (M − M) and commute with the flows of the EBTH but not with one another.…”

Section: Introductionmentioning

confidence: 99%

The Lax operator fixed under the additional symmetries of the extended Toda hierarchy

Cox¹,

Sisson²

2022

J. Phys. A: Math. Theor.

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Additional symmetries of the extended bigraded Toda hierarchy (EBTH) were introduced by Bakalov and Wheeless. We describe properties of a Lax operator fixed under these additional symmetries and determine the unique such Lax operator in the special case of the extended Toda hierarchy (ETH). We further present differential equations for the wave functions and their general solutions that hint at a possible connection to the bispectral problem. Based on these solutions, we highlight a correspondence between the wave functions. Finally, we find the form of a tau function for our Lax operator and compute a second solution to the ETH by applying a Darboux transformation presented by Li and Song.

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“…The finite term recurrence relations generate solutions for the bi-graded Toda lattice hierarchy [15,55,56,9]. It is worth understanding their nature.…”

Section: Introductionmentioning

confidence: 99%

Vector Orthogonal Polynomials with Bochner’s Property

Horozov

2017

Constr Approx

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Abstract. Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre and Bessel have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a classical theorem by Bochner they are the only systems with this property. Similarly, the polynomials of Charlier, Meixner, Kravchuk and Hahn are both orthogonal and are eigenfunctions of a suitable difference operator of second order. We recall that according to the famous theorem of Favard-Shohat, the condition of orthogonality is equivalent to the 3-term recurrence relation. Vector orthogonal polynomials (VOP) satisfy finite-term recurrence relation with more terms, according to a theorem by J. Van Iseghem and this characterizes them.Motivated by Bochner's theorem we are looking for VOP that are also eigenfunctions of a differential (difference) operator. We call these simultaneous conditions Bochner's property.The goal of this paper is to introduce methods for construction of VOP which have Bochner's property. The methods are purely algebraic and are based on automorphisms of non-commutative algebras. They also use ideas from the so called bispectral problem. Applications of the abstract methods include broad generalizations of the classical orthogonal polynomials, both continuous and discrete. Other results connect different families of VOP, including the classical ones, by linear transforms of purely algebraic origin, despite of the fact that, when interpreted analytically, they are integral transformations.To Boris Shapiro on occasion of his 60-th birthday 2010 Mathematics Subject Classification. 34L20 (Primary); 30C15, 33E05 (Secondary).

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Additional symmetries of the extended bigraded Toda hierarchy

Cited by 9 publications

References 29 publications

Darboux transformations and Fay identities for the extended bigraded Toda hierarchy

Darboux transformations and Fay identities for the extended bigraded Toda hierarchy

The Lax operator fixed under the additional symmetries of the extended Toda hierarchy

Vector Orthogonal Polynomials with Bochner’s Property

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