Wave Progatation in Weakly Nonlinear Bi-Anisotropic and Bi-Isotropic Media

Electrodynamics, Topsy-Turvy Special Relativity, and Generalized Minkowski Constitutive Relations for Linear and Nonlinear Systems

“…Here we observe that at least symbolically, the differential form (23) is equivalent to its integral counterpart (28). It follows that the generalization of (28), as expressed in the Volterra functionals series (39), (40), should now be written as…”

“…Attempts of defining nonlinear constitutive parameters from first principles usually lead to Volterra's series of functionals [17,18]. For additional literature references and related work see [19][20][21][22][23][24][25][26][27][28]. A postulated model [29], applied to numerical simulation of rays in nonlinear media, was used in conjunction with experimental data [30] given in the literature, and close agreement of the experimental and simulation results was found.…”

“…and substitute (28) in (1), thus working again in the spatiotemporal domain. Two problems arise in this context: Firstly, solving (1) with (28) involves an integro-differential set of equations, which might be more complicated than computing a solution in the spectral domain, and secondly, one must be careful with the limits of integration in (28) so that causality in the relativistic sense is preserved.…”

“…Such circumstances exists for free space (vacuum), but might also be considered in other special cases as an adequate approximation. Here ε(K) in (27) becomes a constant, which is equivalent to ε(X 1 ) = εδ(X 1 − X) in (28), and the four-dimensional impulse function causes the integral to collapse, resulting in D(X) = εE(X) . But this special case refers once more to a homogeneous medium!…”

“…A postulated model [29], applied to numerical simulation of rays in nonlinear media, was used in conjunction with experimental data [30] given in the literature, and close agreement of the experimental and simulation results was found. Our prototype linear model for constitutive relations is given by (23), or its spectral equivalent in one of the forms (24) or (27), whichever is more convenient at a given time, or the spatiotemporal convolution integral equivalent (28). The corresponding nonlinear model (which is usually postulated, i.e., belongs to category (a) above) should satisfy a "correspondence principle" and reduce to the linear case in the limit of vanishing nonlinearity.…”