Dispersionless integrable systems in 3D and Einstein-Weyl geometry

For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of these dispersionless PDEs can be seen from the geometry of their formal linearizations.Comment: In this version we add Preliminary Section and the Appendix, where we discuss the geometry of PDEs and the method of hydrodynamic reductions. Also we add Lax pairs for the 5 integrable equations of type I, and supply the ancillary files (Maple verifications of the calculations, Maple and Mathematica form of Integrability Conditions, together with their PDF versions) for completenes

On a class of integrable systems of Monge-Amp\`ere type

We investigate a class of multi-dimensional two-component systems of Monge-Amp\`ere type that can be viewed as generalisations of heavenly-type equations appearing in self-dual Ricci-flat geometry. Based on the Jordan-Kronecker theory of skew-symmetric matrix pencils, a classification of normal forms of such systems is obtained. All two-component systems of Monge-Amp\`ere type turn out to be integrable, and can be represented as the commutativity conditions of parameter-dependent vector fields. Geometrically, systems of Monge-Amp\`ere type are associated with linear sections of the Grassmannians. This leads to an invariant differential-geometric characterisation of the Monge-Amp\`ere property.Comment: arXiv admin note: text overlap with arXiv:1503.0227

Lie sphere geometry and integrable systems

書誌情報のみTwo basic Lie-invariant forms uniquely defining a generic (hyper)surface in Lie sphere geometry are introduced. Particularly interesting classes of surfaces associated with these invariants are considered. These are the diagonally cyclidic surfaces and the Lie-minimal surfaces, the latter being the extremals of the simplest Lie-invariant functional generalizing the Willmore functional in conformal geometry. Equations of motion of a special Lie sphere frame are derived, providing a convenient unified treatment of surfaces in Lie sphere geometry. In particular, for diagonallycyclidic surfaces this approach immediately implies the stationary modified Veselov-Novikov equation, while the case of Lie-minimal surfaces reduces in a certain limit to the integrable coupled Tzitzeica system. In the framework of the canonical correspondence between Hamiltonian systms of hydrodynamic type and hypersurfaces in Lie sphere geometry, it is pointed out that invariants of Lie-geometric hypersurfaces coincide with the reciprocal invariants of hydrodynamic type systems. Integrable evolutions of surfaces in Lie sphere geometry are introduced. This provides an interpretation of the simplest Lie-invariant functional as the first local conservation law of the (2+1)-dimensional modified Veselov-Novikov hierarchy. Parallels between Lie sphere geometry and projective differential geometry of surfaces are drawn in the conclusion

Integrable equations in 2 + 1 dimensions: deformations of dispersionless limits

We classify integrable third-order equations in 2 + 1 dimensions which generalize the examples of Kadomtsev–Petviashvili, Veselov–Novikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of integrable systems in 2 + 1 dimensions possess infinitely many multi-phase solutions coming from the so-called hydrodynamic reductions. In this paper, we adopt a novel perturbative approach to the classification problem. Based on the method of hydrodynamic reductions, we first classify integrable quasilinear systems which may (potentially) occur as dispersionless limits of soliton equations in 2 + 1 dimensions. To reconstruct dispersive deformations, we require that all hydrodynamic reductions of the dispersionless limit be inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third-order equations, some of which are apparently new

Projective-geometric aspects of homogeneous third-order Hamiltonian operators

none3E.V. Ferapontov;M.V. Pavlov;R.F. VitoloE. V., Ferapontov; M. V., Pavlov; Vitolo, Raffael