envisioned. Recent results show that third-order NLO phenomena of considerable amplitude (including, for example, nonlinear refractive index) can be expected both in an x-ray laser plasma and, most interestingly, in solid-state materials. The theory of these effects will have to include quantum transitions of (initially) bounded electrons, both between atomic shelves and into a free-electron continuum. This poses new mathematical challenges that are (initially) related to the fast relaxation of hard-driven electrons in the continuum.

Lastly, one of the most fascinating recently discovered phenomena in the nonlinear interaction of light with atoms and ions is that high-order harmonic generation (HHG) spectra deviate drastically from perturbation theory predictions. The physics of this phenomenon has two major components: phase-matching conditions and the nonlinear response of individual atoms. It has recently became clear that the major features of HHG, and, in particular, its plateau, result mainly from general properties of nonlinear atomic response. Yet, there exists no simple model or theory that explains even those major features. A number of multiparameter theoretical models of the single-atom response to intensive optical fields in HHG have been suggested; most of them reproduce qualitatively the gross picture of the process. The most successful theoretical approach thus far has been direct numerical simulation using Hartry-Slater approximation of the atomic Schrödinger equations for many-electron atoms. It requires, however, a tremendous amount of calculation, provides little insight into the physics of the process, and hardly allows for general conclusions. A great theoretical challenge is to develop a mathematical and physical model that describes the major features of the phenomenon and identifies the most substantial factors resulting in HHG.



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