Related papers: A note on the extended dToda hierarchy
Integrable dispersionless Kadomtsev-Petviashvili (KP) hierarchy of B type is considered. Addition formula for the $\tau$-function and conformally invariant equations for the dispersionless BKP (dBKP) hierarchy are derived. Symmetry…
The dispersionless limit of the scalar nonlocal dbar-problem is derived. It is given by a special class of nonlinear first-order equations. A quasi-classical version of the dbar-dressing method is presented. It is shown that the algebraic…
A review of selected topics in Hirota's bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most important types of soliton equations.…
The dDS (dispersionless Davey-Stewartson) hierarchy is constructed by two eigenfunctions of a special vector field. This hierarchy consists the infinite symmetries of the dDS system. Further, this paper explores the tau function, the…
We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy…
Geometric interpretation of the Hirota equation is presented as equation describing the Laplace sequence of two-dimensional quadrilateral lattices. Different forms of the equation are given together with their geometric interpretation: (i)…
We extend a recent result of [13] for the KdV hierarchy to the Toda lattice hierarchy. Namely, for an arbitrary solution to the Toda lattice hierarchy, we define a pair of wave functions, and use them to give explicit formulae for the…
The dispersionless KP hierarchy is considered from the point of view of the twistor formalism. A set of explicit additional symmetries is characterized and its action on the solutions of the twistor equations is studied. A method for…
Using the lax pair, nonlocal symmetry of the coupled Hirota equation is obtained. By introducing an appropriate auxiliary dependent variable the nonlocal symmetry is successfully localized to a Lie point symmetry. For the closed…
We solve the discrete Hirota equations (Kirillov-Reshetikhin Q-systems) for $A_r$, and their analogue for $D_r$, for the cases where the second variable ranges over either a finite set or over all integers. Until now only special solutions…
We exploit the gauge equivalence between the Hirota equation and the extended continuous Heisenberg equation to investigate how nonlocality properties of one system are inherited by the other. We provide closed generic expressions for…
It has been shown that the dispersionless KP hierarchy (or the Benney hierarchy) is reduced to the chordal L\"owner equation. We show that the radial L\"owner equation also gives reduction of a dispersionless type integrable system. The…
In this paper, we firstly give the definition of dipersionless bigraded Toda hierarchy (dBTH) and introduce some Sato theory on dBTH. Then we define Orlov-Schulman's $\M_L$, $\M_R$ operator and give the additional Block symmetry of dBTH.…
We exploit the correspondence between the three-dimensional Lorentzian Einstein-Weyl geometries of the hyper-CR type, and the Veronese webs to show that the former structures are locally given in terms of solutions to the dispersionless…
The inverse scattering transform for the defocusing-defocusing coupled Hirota equations with non-zero boundary conditions at infinity is thoroughly discussed. We delve into the analytical properties of the Jost eigenfunctions and scrutinize…
Nonlinear dispersionless equations arise as the dispersionless limit of well know integrable hierarchies of equations or by construction, such as the system of hydrodynamic type. Some of these equations are integrable in the Hamiltonian…
Applying the inverse scattering transform to study a focusing two-component Hirota equation with nonzero boundary conditions at infinity. Through the spectral problem and the adjoint spectral problem, the analyticity properties and symmetry…
We consider a (2+1)-dimensional Toda-like chain which can be viewed as a two-dimensional generalization of the Wu-Geng model and which is closely related to the two-dimensional Volterra, two-dimensional Toda and relativistic Toda lattices.…
We generalize the determinant representation of the KP $\tau$ functions to the case of the 2D Toda $\tau$ functions. The generating functions for the weighted Hurwitz numbers are a parametric family of 2D Toda $\tau$ functions; for which we…
In this paper we generalize the Sato theory to the extended bigraded Toda hierarchy (EBTH). We revise the definition of the Lax equations,give the Sato equations, wave operators, Hirota bilinear identities (HBI) and show the existence of…