{"arxiv_id":"hep-th/9405096","incoming":{"papers":24,"sentences":47,"groups":[{"kind":"method_or_tool","label":"Uses its method or tool","papers":3,"sentences":3,"quotes":[{"arxiv_id":"1312.7184","title":"Generalized Ablowitz-Ladik hierarchy in topological string theory","url":"/paper/1312.7184","section":"Introduction","sentence":"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] .","intent_p":0.97,"contests_p":null},{"arxiv_id":"2309.12176","title":"KP integrability through the $x-y$ swap relation","url":"/paper/2309.12176","section":"Semi-classical limit equation and implications","sentence":"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] ).","intent_p":1.0,"contests_p":null},{"arxiv_id":"2301.06486","title":"Integrability structures in string theory","url":"/paper/2301.06486","section":"Lax triples and volume preserving flows","sentence":"The idea is to generalize the method of [ref] developed to study area preserving diffeomorphic KP equation.","intent_p":0.82,"contests_p":null}]},{"kind":"comparison","label":"Compares with it","papers":1,"sentences":1,"quotes":[{"arxiv_id":"1003.5767","title":"Non-degenerate solutions of universal Whitham hierarchy","url":"/paper/1003.5767","section":"Riemann-Hilbert problem and non-degenerate solutions","sentence":"This is the same as Propositions 4 and 5 of [ref] , but let us prove it here in our language as in [ref] .","intent_p":0.93,"contests_p":null}]},{"kind":"supports_this_paper","label":"Cites it as support","papers":1,"sentences":1,"quotes":[{"arxiv_id":"2506.07208","title":"Revisiting B\\\"acklund-Darboux transformations for KP and BKP integrable hierarchies","url":"/paper/2506.07208","section":"The KP hierarchy and its discretization","sentence":"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] .","intent_p":0.97,"contests_p":null}]},{"kind":"background","label":"Cites it as background","papers":11,"sentences":12,"quotes":[{"arxiv_id":"1410.5060","title":"Orbifold melting crystal models and reductions of Toda hierarchy","url":"/paper/1410.5060","section":"Introduction","sentence":"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] .","intent_p":0.81,"contests_p":null},{"arxiv_id":"1206.1151","title":"Old and New Reductions of Dispersionless Toda Hierarchy","url":"/paper/1206.1151","section":"Introduction","sentence":"In this paper, we consider the dispersionless Toda hierarchy [ref] .","intent_p":0.96,"contests_p":null},{"arxiv_id":"2608.28942","title":"Action-angle variables and phase space formulation of Hermitian matrix models","url":"/paper/2608.28942","section":"Orthogonal polynomials and the Jacobi operator","sentence":"The recurrence formulation and its relation to Toda-type integrable structures are standard in the literature [ref] .","intent_p":0.99,"contests_p":null},{"arxiv_id":"1201.4392","title":"c-map as c=1 string","url":"/paper/1201.4392","section":"Dispersionless limit","sentence":"In this limit one recovers what is known as dispersionless Toda hierarchy [ref] .","intent_p":0.84,"contests_p":null},{"arxiv_id":"2309.12176","title":"KP integrability through the $x-y$ swap relation","url":"/paper/2309.12176","section":"KP integrability","sentence":"There is a version of it that takes into account topological expansions of the corresponding $τ$ functions, which is sometimes called the $ħ$-KP integrability, see [ref] .","intent_p":0.94,"contests_p":null}]}],"unlabelled_sentences":30},"outgoing":{"papers":0,"sentences":0,"groups":[],"unlabelled_sentences":0},"claims":[],"thresholds":{"intent_p":0.8,"supports_p":0.95,"contests_p":0.7},"method":"Citing sentences come from each paper's LaTeX source; every \\cite marks the sentence and the works it cites. Pith's classifier typed each sentence. A label is shown at intent probability 0.8 or more (0.95 for support), a contest at 0.7 or more. Measured accuracy against two blind model families is at https://pith.science/accuracy. The quoted sentences are the citing authors' own.","held_up":null,"_links":{"html":"https://pith.science/paper/hep-th/9405096"}}