New solvable Matrix Integrals

We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one can reduce matrix integral to the integral over eigenvalues, which in turn is certain tau function. Soliton theory. KP hierarchy of integrable equations, which is the most popular example, consists of semi-infinite set of nonlinear partial differential equations ∂tmu = Km[u], m = 1, 2, . . . (0.0.1) which are commuting flows: [∂tk , ∂tm ] u = 0 The first nontrivial one is Kadomtsev-Petviashvili equation ∂t3u = 1 4 ∂ t1u+ 3 4 ∂ t1 ∂ 2 t2 u+ 3 4 ∂t1u 2 (0.0.2) which originally served in plasma physics [4], now plays a very important role both, in physics and in mathematics. Another very important equation is the equation of two-dimensional Toda lattice (TL) [11] ∂t1∂t1φn = e φn−1−φn − enn+1 (0.0.3) This equation gives rise to TL hierarchy which contains derivatives with respect to higher times t1, t2, . . . and t ∗ 1, t ∗ 2, . . .. The key point of soliton theory is the notion of tau function, introduced by Sato (for KP tau-function see [5]). Tau function is a sort of potential which gives rise both to TL hierarchy and KP hierarchy. It depends on two semi-infinite sets of higher times t1, t2, . . . and t ∗ 1, t ∗ 2, . . ., and discrete variable n: τ = τ(n, t, t). More explicitly we have [5],[11]: u = 2∂ t1 log τ(n, t, t ), φn(t, t ) = − log τ(n + 1, t, t) τ(n, t, t) (0.0.4) In soliton theory so-called Hirota-Miwa variables x,y, which are related to higher times as mtm = ∑