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Functional relation for commuting quantum transfer matrices of quantum integrable models is identified with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. The standard objects of quantum integrable models are identified with elements of classical nonlinear integrable difference equation. In particular, elliptic solutions of Hirota's equation give complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to Bethe ansatz are studied. The nested Bethe ansatz equations for A k−1 -type models appear as discrete time equations of motions for zeros of classical τ -functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation and a new determinant formula for eigenvalues of the quantum transfer matrices are obtained.

We establish the equivalence of 2D contour dynamics to the dispersionless limit of the integrable Toda hierarchy constrained by a string equation. Remarkably, the same hierarchy underlies 2D quantum gravity. PACS number(s):02.30. Em, 02.30,.Dk, 68.70.+w, 1. Laplacian growth. Contour dynamics takes place in many physical processes, where an interface moves between two immiscible phases. The key example of interface dynamics to illustrate the main result of this work is the Laplacian growth (LG) [1]. This process is dissipative, unstable, ubiquitous (applications range from oil/gas recovery to tumor growth), and universal: a steady self-similar pattern appears governed by scaling laws, most of which still yet to be derived [1,2].In this paper we show that an arbitrary interface dynamics has an integrable structure which is the same as the one that underlies models of 2D quantum gravity. This structure links the interface dynamics, and especially LG, with other branches of theoretical physics, where scaling laws are also expected [17].2. To be specific, we will speak about Hele-Shaw flow [1]: a viscous fluid (oil) and a non-viscous fluid (water) are confined in a narrow gap between two parallel plates. The interior water domain, D + , is surrounded by an exterior oil domain, D − , occupying the rest of the plane. Water is supplied from the origin and pushes the oil/water interface, C(t). Both liquids are incompressible, so oil is extracted at infinity at the same rate q as water is supplied.-the normal velocity of the interface is V n = −∂ n p (the D'Arcy law); the pressure p is kept constant (p = 0) inside the water domain D + (t) and on the interface (surface tension and viscosity of the water are neglected); and pressure is a harmonic function, ∇ 2 p = 0, inside the oil domain D − (t), while p → −(q/2π) log x 2 + y 2 at infinity. This (idealized)LG problem has an important property: The harmonic moments of the oil domain C k = D−(t) z −k dx dy (k = 1, 2, . . ., and z = x + iy) 1 do not change in time, while the area of water domain, grows linearly in time [3]. The proof:because V n = −∂ n p and p = 0 along the C(t) . By virtue of the Gauss' theorem, it equalsThis property may be used as the definition of the idealized LG problem:-To find the form of the domain whose area increases while all harmonic moments remain fixed.This problem is known to be ill-defined [1]. For almost all sets of harmonic moments, the boundary develops cusp-like singularities in finite time (area) [1]. Once a singularity occurs, the idealized LG model is no longer valid. Surface tension, omitted above, stabilizes the growth and simultaneously ruins the conservation of harmonic moments. Simulations and experiments show that different mechanisms of regularization of singularities (surface tension, lattice etc.) exhibit the same self-similar pattern [1,2]. This suggests a fixed point (or points) in the space of harmonic moments, which correspond to observed stable patterns. To identify the fixed points and their scaling properties is the cha...