On the models of nonlocal nonlinear optics

We show that under certain assumptions a general model of nonlocal nonlinear response in 1+1-dimension is equivalent to the model considered by Krolikowski and Bang for a Kerr-type medium. We derive the limit of weak nonlocality in high frequency regime and discuss the integrable cases.Comment: 6 page

Deformations of plane algebraic curves and integrable systems of hydrodynamic type

We describe our recent work on deformations of hyperelliptic curves by means of integrable hierarchy of hydrodynamic type (nlin.SI/0205012). We also discuss a further extension to the case of non-hyperelliptic curves.Comment: 7 pages, Submitted for the WSPC Proceedings of Gallipoli workshop July 26 - Aug. 6, 200

Symmetry constraints for real dispersionless Veselov-Novikov equation

Symmetry constraints for dispersionless integrable equations are discussed. It is shown that under symmetry constraints the dispersionless Veselov-Novikov equation is reduced to the 1+1-dimensional hydrodynamic type systems.Comment: 14 pages, no figures, to appear on Fund.Prikl.Mat. (russian version), Jour.Math.Sciences (english version

Ellipticity Conditions for the Lax Operator of the KP Equations

The Lax pseudo-differential operator plays a key role in studying the general set of KP equations, although it is normally treated in a formal way, without worrying about a complete characterization of its mathematical properties. The aim of the present paper is therefore to investigate the ellipticity condition. For this purpose, after a careful evaluation of the kernel with the associated symbol, the majorization ensuring ellipticity is studied in detail. This leads to non-trivial restrictions on the admissible set of potentials in the Lax operator. When their time evolution is also considered, the ellipticity conditions turn out to involve derivatives of the logarithm of the tau-function.Comment: 21 pages, plain Te

Integrable equations in nonlinear geometrical optics

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole-Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov-Novikov equation for the refractive index. It is demonstrated that the Veselov-Novikov hierarchy is amenable to the quasiclassical DBAR-dressing method. Under more specific requirements for the media, one gets the dispersionless Kadomtsev-Petviashvili equation. Geometrical optics interpretation of some solutions of the above equations is discussed.Comment: 33 pages, 7 figure

Singular sector of the Kadomtsev–Petviashvili hierarchy, δ̅ operators of nonzero index, and associated integrable systems

Integrable hierarchies associated with the singular sector of the Kadomtsev– Petviashvili (KP) hierarchy, or equivalently, with δ̅ operators of nonzero index are studied. They arise as the restriction of the standard KP hierarchy to submanifolds of finite codimension in the space of independent variables. For higher δ̅ index these hierarchies represent themselves as families of multidimensional equations with multidimensional constraints. The δ̅ -dressing method is used to construct these hierarchies. Hidden Korteweg–de Vries, Boussinesq, and hidden Gelfand–Dikii hierarchies are considered, too