nLab Hirota equation

Hirota equations are certain bilinear equations related to integrable models/hierarchies. The dependent variables can, in most cases, be identified with $\tau$-functions, as it was first observed by Miwa. One often speaks of Hirota’s direct method in solving integrable equations. We say that an equation is put in Hirota bilinear form if it is written as $P(D)(f\cdot f) = 0$ where $D$ is the Hirota derivative (or difference) operator. One defines it by

$$D_x^n (f\cdot g) = (\partial_{x_1}-\partial_{x_2})^n (f(x_1)g(x_2))|_{x_1=x_2=x}$$

The difference version of Hirota operator is obtained by exponentiating. The Hirota equations enable easier finding of multisoliton? solutions. Recent $Y$-systems may also be viewed as a version of generalized Hirota equations.

One can consider partition functions as made up from characters; $GL(N)$/characters are by Weyl formula given in terms of (ratios of) certain determinants. Flag minor determinants satisfy bilinear Pluecker relations. If the tau functions are interpreted in terms of infinite Grassmanians (fermionic approach) then the Hirota relations sometimes boil down to Pluecker relations.

Literature

• R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27, 1192 (1971).
• R. Hirota, Discrete analogue of a generalized Toda equation, J. Phys. Soc. Japan 50 (1981) 3785-3791.
• T. Miwa, On Hirota’s difference equations, Proc. Japan Acad. 58 (1982) 9-12 euclid
• S. Saito, String theories and Hirota’s bilinear diference equation, Pys. Rev. Lett. 59 (1987) 1798–1801.
• A. Zabrodin, Bethe ansatz and Hirota equation in integrable models, arxiv/1211.4428
• Graeme Segal, George Wilson, Loop groups and equations of KdV type, Inst. Hautes Etudes Sci. Publ. Math. 61 (1985) 5–65.

Relation to Y-systems

• A. Kuniba, T. Nakanishi, J. Suzuki, T-systems and Y-systems in integrable systems, J. Phys. A 44 (2011) 103001.
• S. Leurent, Hirota equation and Q-functions, slides