Blow-up solutions of Helmholtz equation for a Kerr slab with a complex linear and nonlinear permittivity

TL;DR

This paper demonstrates that the Helmholtz equation for TE waves in a Kerr slab with complex permittivity admits blow-up solutions under defocusing conditions, applicable to both homogeneous and inhomogeneous slabs with continuous properties.

Contribution

It establishes the existence of blow-up solutions in Kerr slabs with complex permittivity, extending understanding to inhomogeneous cases with bounded Kerr coefficients.

Findings

Blow-up solutions occur when the real part of the Kerr coefficient is negative.

Existence of solutions in both homogeneous and inhomogeneous Kerr slabs.

In inhomogeneous slabs, solutions exist if the Kerr coefficient's real part is sufficiently negative.

Abstract

We show that the Helmholtz equation describing the propagation of transverse electric waves in a Kerr slab with a complex linear permittivity and a complex Kerr coefficient admits blow-up solutions provided that the real part of the Kerr coefficient is negative, i.e., the slab is defocusing. This result applies to homogeneous as well as inhomogeneous Kerr slabs whose linear permittivity and Kerr coefficient are continuous functions of the transverse coordinate. For an inhomogeneous Kerr slab, blow-up solutions exist if the real part of Kerr coefficient is bounded above by a negative number.