On non-autonomous differential-difference AKP, BKP and CKP equations

Fu, Wei; Nijhoff, F.W.

doi:10.1098/rspa.2020.0717

Cited by 11 publications

(14 citation statements)

References 39 publications

(63 reference statements)

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“…Multiplying this equation by (1 + 𝛀𝑪 𝑛 ) −1 from the right, we end up with (34b) in virtue of (24). Equations (34a) and (34c) are proven similarly.…”

Section: Closed-form Nonlinear Equationsmentioning

confidence: 90%

Linear integral equations and two‐dimensional Toda systems

Yin

2021

Stud Appl Math

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The direct linearization framework is presented for the two‐dimensional (2D) Toda equations associated with the infinite‐dimensional Lie algebras A∞, B∞, and C∞, as well as the Kac–Moody algebras Arfalse(1false), A2rfalse(2false), Crfalse(1false), and Dr+1false(2false) for arbitrary integers r∈double-struckZ+, from the aspect of a set of linear integral equations in a certain form. Such a scheme not only provides a unified perspective to understand the underlying integrability structure, but also induces the direct linearizing type solution potentially leading to the universal solution space, for each class of the 2D Toda system. As particular applications of this framework to the 2D Toda lattices, we rediscover the Lax pairs and the adjoint Lax pairs and simultaneously construct the generalized Cauchy matrix solutions.

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“…Multiplying this equation by (1 + 𝛀𝑪 𝑛 ) −1 from the right, we end up with (34b) in virtue of (24). Equations (34a) and (34c) are proven similarly.…”

Section: Closed-form Nonlinear Equationsmentioning

confidence: 90%

Linear integral equations and two‐dimensional Toda systems

Yin

2021

Stud Appl Math

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“…Therefore by Lemma 7 and spectral representation of q in (17), one can at last prove ( 31) is correct for m. Now let us discuss the expression of g 1 (t, z) by free Bosons, which is given in the proposition below.…”

Section: The Darboux Transformation Of the Ckp Hierarchy Via Free Bosonsmentioning

confidence: 93%

CKP hierarchy and free Bosons

Yang,

Geng,

Cheng

2021

Preprint

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In this paper, free Bosons are used to study some integrable properties of the CKP hierarchy, from the aspects of tau functions. Firstly, the modified CKP hierarchy is constructed by using free Bosons and the corresponding Lax structure is given. Then the constrained CKP hierarchy is found to be related with the modified CKP hierarchy, and the corresponding solutions are derived by using free Bosons. Next by using the relations between the Darboux transformations and the squared eigenfunction symmetries, we express the Darboux transformations of the CKP hierarchy in terms of free Bosons, by which one can better understand the essential of the CKP Darboux transformations. In particular, the additional symmetries of the CKP hierarchy can be viewed as the infinitesimal generator of the CKP Darboux transformations. Based upon these results, we finally obtain the actions of the CKP additional symmetries on the CKP tau functions constructed by free Bosons.

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“…The hyperdetermantal relations (2.7.2) were introduced as integrable systems of recurrence relations on lattices by Kashaev [22], who showed that the star triangle relations satisfied by Boltzmann weights for the Ising model imply these for a suitably defined τ -function on the Z 3 integer lattice. They were studied subsequently by Schief and others [30,2,20,21], as discrete analogs of the CKP hierarchy.…”

Section: 3)mentioning

confidence: 99%

“…and substituting eqs. (3.6.42a) -(3.6.42d) into (3.6.43) gives the form of the discrete CKP relations studied in [20,21].…”

Section: Hyperdeterminantal Relations In Infinite Dimensionsmentioning

confidence: 99%

Lagrangian Grassmannians, CKP hierarchy and hyperdeterminantal relations

Arthamonov¹,

Harnad²,

Hurtubise³

2022

Preprint

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This work concerns the relation between the geometry of Lagrangian Grassmannians and the CKP integrable hierarchy. For a complex vector space V of dimension N , and H N := V ⊕ V * the associated symplectic space, with canonical symplectic structure ω N , the exterior space Λ(H N ) is decomposed into a direct sum of irreducible Sp(H N , ω N ) submodules, and a basis adapted to this decomposition is constructed. The Lagrangian map) is defined by restricting the Plücker map to the Lagrangian Grassmannian Gr L V (H N , ω N ) of maximal isotropic subspaces and composing it with projection to the subspace of symmetric elements of Λ N (H N ) under dualization. In terms of the affine coordinate matrix on the big cell, this reduces to the principal minors map, whose image is cut out by the 2 × 2 × 2 quartic hyperdeterminantal relations. To apply this to the CKP hierarchy, the Lagrangian Grassmannian framework is extended to infinite dimensions, with H N replaced by a polarized Hilbert space H = H + ⊕ H − , with symplectic form ω. The fermionic Fock space F = Λ ∞/2 H is decomposed into a direct sum of irreducible Sp(H, ω) representations and the infinite dimensional Lagrangian map is defined. The linear constraints defining reduction to the CKP hierarchy are expressed as a fermionic null condition and the infinite analogue of the hyperdeterminantal relations is deduced. A multiparametric family of such relations is shown to be satisfied by the evaluation of the τ -function at translates of a point in the space of odd flow variables along the cubic lattices generated by power sums in the parameters.

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On non-autonomous differential-difference AKP, BKP and CKP equations

Cited by 11 publications

References 39 publications

Linear integral equations and two‐dimensional Toda systems

Linear integral equations and two‐dimensional Toda systems

CKP hierarchy and free Bosons

Lagrangian Grassmannians, CKP hierarchy and hyperdeterminantal relations

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