Factorization of integro-differential equations of optics of dispersive anisotropic media and tensor integral operators of wave packet velocities

Factorization of abstract operators into two second degree operators and its applications to integro-differential equations

“…Integro-differential equations (IDEs) are used to model many problems in science, engineering, economics, medicine, control theory, micro-inhomogeneous media and viscoelasticity [1][2][3][4][5][6][7][8][9]. Very important tools in solving of Boundary Value Problems (BVPs) with IDEs are the Parametrization Method [10] and the Factorization (Decomposition) method, but the applicability of the last method is confined to certain kinds of integro-differential operators, corresponding to BVPs and cannot be universal for all problems.…”

Fedorov’s beam tensor in solitonic conservation laws

“…The refractive index tensor determines fully the evolution of a randomly polarized harmonic wave in a complex medium. It was demonstrated [5,6] that Maxwell's equations themselves lead to the concept of operator rates of wave packets. Taking into account dispersion and anisotropy, the tensor integral rate operator describes the movement of a random wave.…”

“…For a rather rapid decay of E as z → ±∞, there exist two sets of solutions for system (5), which are called Jost functions: φ i (z, ξ) and ψ i (z, ξ), i = 1, 2, 3. The complex parameter ξ differs from λ only by a constant multiplier (see below).…”

“…The first step is to find the scattering matrix for system (5) and the quantities c 12 and c 13 in each null α 1,1 (ξ) by using instead of E(z) in L l and A l the initial condition E(z, 0). Values of scattering data at subsequent time points are determined by Eq.…”

“…where c (1) and c (2) are components of the soliton unit polarization vector c = c (1) e 1 + c (2) e 2 , e 1 and e 2 are vectors of an orthonormalized basis; and ξ 1 is the intrinsic transform value (5). Let us formulate the MIST so that all physical vectors appear in the formulas as whole ones.…”