Related papers: Dispersionless Multi-Dimensional Integrable System…
Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaces with the underlying governing equation being revealed as the general heavenly equation. The formalism developed here may be used to link…
Einstein-Weyl geometry is a triple (D,g,w), where D is a symmetric connection, [g] is a conformal structure and w is a covector such that: (i) connection D preserves the conformal class [g], that is, Dg=wg; (ii) trace-free part of the…
We consider dispersionless Lax systems and present a new systematic method of deriving new integrable systems from a given one. We provide examples that include: the dispersionless Hirota equation, the general heavenly equation and the web…
We consider a $D$-dimensional cosmological model with a dilaton field and two $(D-d-1)$-form field strengths which have nonvanishing fluxes in extra dimensions. Exact solutions for the model with a certain set of couplings are obtained by…
From a specific series of exchange conditions for a one-parameter Hamiltonian vector field, we establish an integrable hierarchy using Lax pairs derived from the dispersionless partial differential equation. An exterior differential form of…
We develop a Hamiltonian formulation of Bianchi type-I cosmological model in conformal gravity, i.e. the theory described by a Lagrangian which involves the quadratic curvature invariant constructed from the Weyl tensor, in four dimensions.…
We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS$_{3}$ algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions. The analysis…
The group of automorphisms of the geometry of an integrable system is considered. The geometrical structure used to obtain it is provided by a normal form representation of integrable systems that do not depend on any additional geometrical…
A new method for the construction of conformally invariant equations in an arbitrary four dimensional (pseudo-) Riemannian space is presented. This method uses the Weyl geometry as a tool and exploits the natural conformal invariance we can…
The formalism of quantization deformation is reviewed and the Weyl-Moyal like deformation is applied to systematic construction of the field and lattice integrable soliton systems from Poisson algebras of dispersionless systems.
We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential $m$-form $\Omega_m$ with the coefficients satisfying the Pl\"ucker relations, which is gauge-invariantly closed and its…
We consider a Lie algebra generalizing the Virasoro algebra to the case of two space variables. We study its coadjoint representation and calculate the corresponding Euler equations. In particular, we obtain a bi-Hamiltonian system that…
There are investigated the Lie algebraic structure and integrability properties of a very interesting class of nonlinear dynamical systems called the heavenly equations, which were initiated by Pleba\'nski and later analyzed in a series of…
We consider a radiating shear-free spherically symmetric metric in higher dimensions. Several new solutions to the Einstein's equations are found systematically using the method of Lie analysis of differential equations. Using the five Lie…
We present a simple, systematic and practical method to construct conformally invariant equations in arbitrary Riemann spaces. This method that we call "Weyl-to-Riemann" is based on two features of Weyl geometry. i) A Weyl space is defined…
It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual…
A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra f. Under certain conditions f-invariant systems of differential…
We demonstrate that the dispersionless $\bar\partial$-dressing method developed before for general heavenly equation is applicable to the $4+4$ and $2N+2N$ - dimensional symmetric heavenly type equations. We introduce generating relation…
A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie…
We examine the variational and conformal structures of higher order theories of gravity which are derived from a metric-connection Lagrangian that is an arbitrary function of the curvature invariants. We show that the constrained first…