Integrable Hierarchies and Dispersionless Limit

Kanehisa Takasaki (Kyoto University); Takashi Takebe (University of Tokyo)

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Integrable Hierarchies and Dispersionless Limit

Kanehisa Takasaki (Kyoto University), Takashi Takebe (University of Tokyo)

High Energy Physics - Theory · Submitted 16 May 1994

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The authors' main claim An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quatization of the similar construction of solutions of dispersionless hierarchies.

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arXiv IDhep-th/9405096 (version 3)

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CitationKanehisa Takasaki (Kyoto University), Takashi Takebe (University of Tokyo). "Integrable Hierarchies and Dispersionless Limit." arXiv:hep-th/9405096 (1994). https://pith.science/paper/ZRQHT6AU

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.74 · mentions code or data availability 0.02.

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Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied. Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is dressing operators of the ordinary KP and Toda hierarchy. An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quatization of the similar construction of solutions of dispersionless hierarchies. These results as well as those obtained in previous papers are presented with proofs and necessary technical details.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied.	contribution 0.86	0.70	0.12	asserted 0.98
s1	Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is dressing operators of the ordinary KP and Toda hierarchy.	contribution 0.77	0.85	0.22	asserted 1.00
s2	An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quatization of the similar construction of solutions of dispersionless hierarchies.	contribution 0.76	0.86	0.23	asserted 1.00
s3	These results as well as those obtained in previous papers are presented with proofs and necessary technical details.	contribution 0.75	0.58	0.07	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from proof or derivation 1.00.

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Original abstract

Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied. Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is dressing operators of the ordinary KP and Toda hierarchy. An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quatization of the similar construction of solutions of dispersionless hierarchies. These results as well as those obtained in previous papers are presented with proofs and necessary technical details.

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References

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Action-angle variables and phase space formulation of Hermitian matrix models
hep-th 2026-08 accept novelty 7.0 of 10

The orthogonal polynomial recursion in a Hermitian matrix model defines an action-angle pair (I, θ) that maps canonically to eigenvalue and momentum variables, with the polynomial zero density determining the momentum...
Dispersionless Hirota equations of two-component BKP hierarchy
nlin.SI 2006-04 conditional novelty 7.0 of 10

The dispersionless limit of the two-component BKP hierarchy is shown to be governed by a system of Hirota-type equations equivalent to Hamilton-Jacobi equations, from which dispersionless Lax equations follow.
Revisiting B\"acklund-Darboux transformations for KP and BKP integrable hierarchies
nlin.SI 2025-06 unverdicted novelty 4.0 of 10

Revisits Bäcklund-Darboux transformations for KP, BKP and related hierarchies in bilinear tau-function and fermionic operator frameworks, extending naturally to fully discrete cases.

How the literature uses it

24 arXiv papers cite this in 47 sentences

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Papers citing it · 30 sentences without a confident label

Uses its method or tool 3 papers

We deformed the modified melting crystal model by two sets of external potentials with coupling constants $= (t_1,t_2,…)$ and $= (t̅_1,t̅_2,…)$, and employed the method [ref] developed for the genuine melting crystal model [ref] to show that the deformed partition function is essentially a tau function $τ(s,,)$ of the 2D Toda hierarchy [ref] .

Generalized Ablowitz-Ladik hierarchy in topological string theory · Introduction · 0.97
At the first step of the proof we are going to solve this equation in the space of formal symmetric power expansions in $x_1$ and $x_2$ (as it is done in [ref] ).

KP integrability through the $x-y$ swap relation · Semi-classical limit equation and implications · 1.00
The idea is to generalize the method of [ref] developed to study area preserving diffeomorphic KP equation.

Integrability structures in string theory · Lax triples and volume preserving flows · 0.82

Compares with it 1 paper

This is the same as Propositions 4 and 5 of [ref] , but let us prove it here in our language as in [ref] .

Non-degenerate solutions of universal Whitham hierarchy · Riemann-Hilbert problem and non-degenerate solutions · 0.93

Cites it as support 1 paper

The fact that equation ([ref]), or, equation ([ref]), is equivalent to the whole hierarchy (i.e., to the generating equation ([ref])), was proved in [ref] .

Revisiting B\"acklund-Darboux transformations for KP and BKP integrable hierarchies · The KP hierarchy and its discretization · 0.97

Cites it as background 11 papers

In the course of refining this observation, Brini and his collaborators [ref] reformulated this integrable hierarchy as a special “rational reduction” of the 2D Toda hierarchy [ref] .

Orbifold melting crystal models and reductions of Toda hierarchy · Introduction · 0.81
In this paper, we consider the dispersionless Toda hierarchy [ref] .

Old and New Reductions of Dispersionless Toda Hierarchy · Introduction · 0.96
The recurrence formulation and its relation to Toda-type integrable structures are standard in the literature [ref] .

Action-angle variables and phase space formulation of Hermitian matrix models · Orthogonal polynomials and the Jacobi operator · 0.99

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