Geometrical optics in nonlinear media and integrable equations

It is shown that the geometrical optics limit of the Maxwell equations for certain nonlinear media with slow variation along one axis and particular dependence of dielectric constant on the frequency and fields gives rise to the dispersionless Veselov-Novikov equation for refractive index. It is demonstrated that the last one is amenable to the quasiclassical ∂̄-dressing method. A connection is noted between geometrical optics phenomena under consideration and quasiconformal mappings on the plane. PACS numbers: 02.30.Ik; 42.15.Dp Great variety of nonlinear phenomena from various fields of physics, applied physics, mathematics and applied mathematics can be modeled by nonlinear integrable equations [1]-[8]. A subclass of such equations , the so-called dispersionless integrable equations, has attracted a particular interest during the last decade [9]-[17]. In this letter, we will show that the propagation of electromagnetic waves of high frequency in certain nonlinear media is governed by the dispersionless Veselov-Novikov (dVN) equation. Namely, we will demonstrate that the Supported in part by the COFIN PRIN “SINTESI” 2002. e-mail [email protected] e-mail [email protected]