Investigation of third order nonlinearity in propagation of cylindrical waves in homogeneous nonlinear media

Exact solutions to the (1+1)-dimensional nonlinear Maxwell equations in the orthogonal curvilinear coordinates

“…In contrast with the previous studies starting with a nonlinear system directly [5,6,[19][20][21][22][23][24][25][26][27][28], our work begins with a linear system and then transforms it into a nonlinear system through a hodograph transformation [30][31][32]. Because there exists some mathematical difficulties in applying the hodograph transformation on nonlinear system directly.…”

“…When selecting cylindrical coordinate system (r, φ, z), taking r axis as u axis and considering z-axis polarization, these equations are reduced into a model of cylindrical waves which are studied in Refs. [5,6,[19][20][21][22][23][24][25][26][27][28]. For convenience, we introduce the dimensionless variables ρ = u/u 0 , τ = ct/u 0 , E = E v /(N/C), and H = µ 0 cH w /(N/C), where u 0 is a constant associated with the characteristic spatial scale, c is the speed of light in vacuum, and (N/C) is the unit of electric field strength in the international system of units.…”

“…[5] reports that they can obtain an exact solution to the cylindrical electromagnetic wave propagation problem in a medium which has a special form of nonlinearity -with the dielectric function being an exponential function of electric field. Following their lead, a number of related works have been reported recently [6,[19][20][21][22][23][24][25][26][27][28]. To the best of our knowledge, however, serious difficulties have been met in studying other nonlinear forms.…”

“…FIG.3. Frequency spectrum of the electric field of Eqs (28). when two cylindrical waves propagate in a semi infinite nonlinear medium.…”

“…Figure3shows the result of the fast Fourier transform of the electric field determined by Eqs (28). at ρ = 2 by taking α = −1, β = 9, a = 1, A 1 = A 2 = 1, ̟ 1 = 2π 9 and ̟ 2 = π 3 (ω 1 = 9̟1 2π , ω 2 = 9̟2 2π).…”