A general scheme is proposed for introduction of lattice and qdifference variables to integrable hierarchies in frame of∂-dressing method . Using this scheme, lattice and q-difference Darboux-ZakharovManakov systems of equations are derived. Darboux, Bäcklund and Combescure transformations and exact solutions for these systems are studied.
Dunajski generalization of the second heavenly equation is studied. A dressing scheme applicable to Dunajski equation is developed, an example of constructing solutions in terms of implicit functions is considered. Dunajski equation hierarchy is described, its Lax-Sato form is presented. Dunajsky equation hierarchy is characterized by conservation of three-dimensional volume form, in which a spectral variable is taken into account.
Dunajski equationA preliminary sketch of the results presented here was given in the preprint [1].In this work we study an integrable model proposed by Dunajski [2]. This model is a representative of the class of integrable sytems arising in the context of complex relativity [3]- [7]. It is closely connected to the Plebański second heavenly equation [3] and in some sense generalizes it. The starting point is a (2,2) signature metric in canonical Plebański formVacuum Einstein equations and conformal anti-self-duality (ASD) condition for this metric lead to the selebrated Plebański second heavenly equation [3], Θ wx + Θ zy + Θ xx Θ yy − Θ 2 xy = 0.
The∂-dressing scheme based on local nonlinear vector∂-problem is developed. It is applicable to multidimensional nonlinear equations for vector fields, and, after Hamiltonian reduction, to heavenly equation. Hamiltonian reduction is described explicitely in terms of thē ∂-data. An analogue of Hirota bilinear identity for heavenly equation hierarchy is introduced, τ -function for the hierarchy is defined. Addition formulae (generating equations) for the τ -function are found. It is demonstrated that τ -function for heavenly equation hierarchy is given by the action for∂-problem evaluated on the solution of this problem.
A note on reductions of the dispersionless Toda hierarchy An analytic-bilinear approach for the construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allows us to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. A generalized hierarchy incorporates basic hierarchy, modified hierarchy, singularity manifold equation hierarchy, and corresponding linear problems. Different levels of generalized hierarchy are connected via invariants of Combescure symmetry transformation. The resolution of functional equations also leads to the function and addition formulas to it.
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