Rational differential systems, loop equations, and application to the \(q\)th reductions of KP. (English) Zbl 1343.37061
The authors associate a matrix kernel and correlators to any invertible matrix solution \(\Psi(x)\) of a linear differential system. They show that the correlators satisfy a set of linear equations and also a set of quadratic equations, which they refer to collectively as “loop equations”. It is shown that when the correlators have an expansion of topological type, it can be computed by the topological recursion of [the third author and N. Orantin, Commun. Number Theory Phys. 1, No. 2, 347–452 (2007; Zbl 1161.14026)] (see also [V. Bouchard and the third author, J. High Energy Phys. 2013, No. 2, Article ID 143, 34 p. (2013; Zbl 1342.81513)] for the case of arbitrary ramifications). Also, in the presence of isomonodromic times \(\vec t\), they define the isomonodromic Tau function \({\mathcal T}(\vec t)\) and give the expansion for \(\ln {\mathcal T}(\vec t)\). Finally, they apply their theory to the linear system associated to the \(q\)th reduction of KP and illustrate it more specifically with examples of the \((p,q)\) models.
Reviewer: Andrew Pickering (Madrid)
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
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