Solution of the dispersionless Hirota equations

Abstract

The dispersionless differential Fay identity is shown to be equivalent to a kernel expansion providing a universal algebraic characterization and solution of the dispersionless Hirota equations. Some calculations based on D-bar data of the action are also indicated.

The $\bar{\partial}$-approach to the dispersionless KP hierarchy

“…The above discussion suggests to take equation (15), for appropriate functions W , as the quasi-classical version of the linear∂-problem (11). The function S is widely used in the discussions of the dispersionless limit of the integrable hierarchies [4]- [10].…”

“…Dispersionless hierarchies have been described and analysed by different methods. In particular, the quasiclassical versions of the inverse scattering transform and Riemann-Hilbert problem method have been applied to the study of some 1 + 1-dimensional integrable equations [23,20,11,12,13]. In contrast, similar study of the 2 + 1-dimensional integrable equations and hierarchies is missing.…”

On kernel formulas and dispersionless Hirota equations of the extended dispersionless BKP hierarchy

“…Let us briefly recall the method of kernel formula [5] for the derivation of dHirota equations of the dKP hierarchy. The dKP hierarchy is defined by the Lax equations…”

“…A few equations from such an expansion have been given in [5]. Here we just mention the simplest nontrivial equation 1 2 F 2 11 − 1 3 F 13 + 1 4 F 22 = 0 which, after noticing u = u 2 = F 11 , is just the (2 + 1)-dimensional dKP equation [13] …”

Classification of Integrable (2+1)-Dimensional Quasilinear Hierarchies

“…The function Φ 11 itself satisfies a complicated over-determined system of third order PDE's (see Sect. 2 where we re-derive this system based on the method of hydrodynamic reductions [7], [4], [6], [9]). Its general solution can be reduced to either of the four essentially different canonical forms…”