A theorem of existence and uniqueness in nonlinear optics.

We consider quasilinear hyperbolic systems of canonical form with boundary conditions for which L. Cesari [Ann. Scuola Norm. Sup. Pisa Cl. Sci. 1 (1974), 311-358 (1975)] proved an existence and uniqueness theorem requiring among other assumptions that the two k×k matrices involved have "dominant main diagonal'' in a suitable sense and a parameter a is assuned sufficiently small. Examples show that the conclusion may fail if the dominant main diagonal condition is not satisfied. When strong monochromatic laser radiation is focused on a thin crystal, then after emerging, the radiation shows a component of double frequency. A nonlinear partial differential system, as an initial model for this phenomenon with uniaxial crystals, was proposed by D. Graffi [Atti Mem. Accad. Naz. Sci. Lett. Arti 9 (1967), 172-196]. Boundary conditions were later supplied by L. Cesari who then reduced the problem to one studied in this paper for dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., 1974), Vol. I, pp. 251-261, Academic Press, New York, 1976] with two unknowns E, H and two independent variables x, t (a 2,2-problem), satisfying the dominant main diagonal condition. The first author [Z. Angew. Math. Phys. 27 (1976), no. 4, 409--422; MR0430544 (55 #3549)] then showed that, for a>0 of the order of magnitude used in experiments and quartz crystals, the remaining conditions of the existence and uniqueness theorem were also satisfied. In this paper we take into consideration a crystal slab made up of a piezoelectrical crystal sheet of a large class with the nonlinear optic axes and the linear optic axis. The Graffi-type nonlinear system is now a quasilinear system of equations in four unknowns and again two independent variables x,t (a 4,2-problem), with Cesari-type boundary conditions. There are many ways to reduce such a system to the canonical form studied in this paper. We discuss this reduction process, also in connection with our previous work [Riv. Mat. Univ. Parma 5 (1979), part 1, 55-76 (1980)], and show that the reduction can be done in such a way that the dominant main diagonal condition is satisfied, and, for a>0 and sufficiently small, the other conditions are also satisfied. Thus, for a strong monochromatic radiation of period T, the problem has a unique solution, also periodic of period T, whose second harmonic represents the phenomenon under discussion, the other harmonic being in general too small to observe. The analysis is made in terms of a rather general class of 4,2-problems, and shows that the systems can be reduced to hyperbolic problems (1,2) satisfying the dominant main diagonal condition, provided the characteristic roots of the underlying algebraic equation are real and distinct. If they coalesce, then the system may not even be hyperbolic in the sense of Lax, and if hyperbolic may not be reducible to a canonic form with dominant main diagonal.