On the infinitely many nonperturbative solutions in a transmission eigenvalue problem for Maxwell’s equations with cubic nonlinearity

Abstract

The paper focuses on a transmission eigenvalue problem for Maxwell’s equations with cubic nonlinearity that describes the propagation of transverse magnetic waves along the boundaries of a dielectric layer filled with nonlinear (Kerr) medium. Using an original approach, it is proved that even for small values of the nonlinearity coefficient, the nonlinear problem has infinitely many nonperturbative solutions (eigenvalues and eigenwaves), whereas the corresponding linear problem always has a finite number of solutions. This fact implies the theoretical existence of a novel type of eigenwaves that do not reduce to the linear ones in the limit in which the nonlinear coefficient reduces to zero. Asymptotic distribution of the eigenvalues is found, periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined.

Nonlinear multi-frequency electromagnetic wave propagation phenomena

“…We notice that the theory of 1-frequency transverseelectric (TE) and transverse-magnetic (TM) wave propagation in layered waveguides filled with non-linear media is widely studied [6,7,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. In the 1-frequency case, the GW is characterised by one frequency and one propagation constant (PC).…”

Theory of Nonlinear Guided Electromagnetic Waves in a Plane Two‐Layered Dielectric Waveguide

“…Indeed, it was proven lately that, even in a simpler case of a one-layer waveguide, the Kerr nonlinearity results in the existence of novel guided regimes as well [23][24][25]. Moreover, similar results have recently been found in the case of polynomial nonlinearity [39].…”

“…The second type corresponds to the PCs, which present a novel guiding regime (they do not reduce to any linear solutions in the linear limit; see item (3) in Theorem 3). In the latter case, we call them "purely nonlinear" PCs; see also [23][24][25]. In Figures 4, 6, and 8, the PĈ1 marked with a blue dot corresponds to the first type; the PCŝ2 and̂3 marked with green and brown dots, respectively, correspond to the second type.…”

“…From the mathematical standpoint, similar problems for circle cylindrical waveguides are much more complicated. To the best of our knowledge, the first rigorous formulation of TE and TM wave propagation in plane and circle cylindrical waveguides with nonlinear filling had been proposed in [14], and since then these and similar problems have been studied very intensively [7,10,[15][16][17][18][19][20][21][22][23][24][25]. Nevertheless, key results in the cases of TE and TM wave propagation in a layer with Kerr nonlinearity have been found only recently [23][24][25].…”

How Kerr nonlinearity influences polarized electromagnetic wave propagation

“…As is well known in wave-guiding problems, the spectrum (and its properties) are the most important aspects, as they defi ne the properties of the waveguide. The technique given here was already generalized to the case of transverse-magnetic (TM) waves and Kerr nonlinearity [33]. From the theoretical point of view, for the TM case, the dispersion equation can be expressed in terms of hyperelliptic functions, but no one hardly has the courage to do it.…”