Multilinear Spectral Decomposition for Nonlinear Maxwell Equations

We study the frequency form, popular in physics, of Maxwell equations for nonlinear spatially inhomogeneous dielectric media. Using wavepackets based on eigenmodes of the underlying linear dielectric medium we develop a mathematically consistent interpretation of the frequency form of nonlinear Maxwell operators. In particular, we construct the operator of nonlinear polarization based on frequency dependent susceptibilities. To this end we use a multilinear variant of the spectral decomposition of selfadjoint operators. Introduction This work is motivated by the frequency form (popular in physics; see [1], [2], [3], [7], [8], [10], [11], [12], [22], [23], [26] and references therein) of nonlinear Maxwell equations, including representation of the nonlinear polarization in terms of frequency dependent susceptibilities. Another motivation for this study is our recent work [5] showing the fundamental importance of the spectral theory and frequency composition of waves in understanding of their nonlinear interactions. Consider first, as an illustration, the case of a homogeneous dielectric medium. In this case, the eigenmodes of the underlying linear medium are just plane waves. The corresponding time harmonic solutions of the linear Maxwell equation are of the form E = E (r, t)= ei(r·k−ω(k)t)v, and a similar expression holds for the magnetic field. Here r and t are the position vector and the time, respectively, v is a constant 3-component vector amplitude of the electric field E, k is the wavevector, and ω (k) is the corresponding eigenfrequency. 2000 Mathematics Subject Classification. Primary 78A60, 78M35, 78A40.