Un teorema di esistenza e unicità in ottica non lineare dispersiva

For systems of quasilinear hyperbolic partial differential equations in bicharacteristic canonical form L.Cesari proved existence and uniqueness theorems, and continuous dependence on the data, for boundary conditions represented by linear combinations of the unknowns on possibly distinct parallel hyperplanes. L.Cesari then applied the results to nonlinear optics, namely, the generation of a second harmonic with laser light through a crystal slab. In a previous paper P. Bassanini and M.C.Salvatori extended the existence and uniqueness theorems to the case in which the quasilinear hyperbolic system in bicharacteristic canonical form contains hereditary terms. In the present paper we conside the case of a monochromatic laser light crossing a crystal slab, say 0≤x≤a, assuming here that dispersion cannot be disregarded. Then, for plane waves and in the crystal slab 0≤x≤a, the Maxwell-type nonlinear integral-differential equations hold with boundary conditions at x=0 and x=a, namely, suitable linear combinations of E and H on the two faces of the slab, and where now the integral term contains the function of memory φ. We show that by a standard transformation, the problem can be reduced to a quasilinear hyperbolic system in bicharacteristic canonical form and with hereditary terms , and we show that the existence and uniqueness theorems apply. We also study in detail the function of memory φ on which the only assumption is that φ˙ is L-integrable.