Solitons in Bragg gratings with saturable nonlinearities

Abstract

We introduce two different systems of coupled-mode equations to describe the interaction of two waves coupled by the Bragg reflection in the presence of saturable nonlinearity. The basic model assumes the ordinary linear coupling between the modes. It may be realized as a photorefractive waveguide, with a Bragg lattice permanently written in its cladding. We demonstrate the presence of a cutoff point in the system's bandgap, with gap solitons existing only on one side of it. Close to this point, the soliton's norm diverges with power −3 / 2. The soliton family between the cutoff point and the edge of the bandgap is stable. In this model, stationary bound states of two in-phase solitons are found too, but they are unstable, transforming themselves into breathers. Another model assumes a photoinduced longitudinal bulk grating, with the corresponding intermode coupling subject to saturation along with the nonlinearity. In that model, another cutoff point is found, with the soliton's norm diverging near it with power −2. Solitons are stable in this model too (while it does not give rise to twosoliton bound states). Collisions between moving solitons are always quasi-elastic, in either model.

The virial theorem and ground state energy estimates of nonlinear Schrödinger equations in $$\mathbb {R}^2$$ R 2 with square root and saturable nonlinearities in nonlinear optics

“…(I) square-root nonlinearity: f (s) = 1 − 1 √ 1+s for s > 0, (II) saturable nonlinearity: f (s) = 1 − 1 1+s for s > 0, which describe narrow-gap semiconductors ( [18,21]) and photorefractive media ( [6,7,8,13,14,15,19] Besides, the total energy E = K + P can be denoted as the sum of the kinetic energy K and the potential energy P , where the kinetic energy is 2) and the potential energy is…”

“…(I) square-root nonlinearity: f (s) = 1 − 1 √ 1+s for s > 0, (II) saturable nonlinearity: f (s) = 1 − 1 1+s for s > 0, which describe narrow-gap semiconductors ( [18,21]) and photorefractive media ( [6,7,8,13,14,15,19]), respectively. Equation (1.1) can be represented as i ∂A ∂z = δE [A] δA , where…”

Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities

“…In physics literature, the most popular nonlinearity is the so-called Kerr, or cubic, nonlinearity which is a representative of the wide class of superlinear at infinity nonlinearities. However, in recent years one can see a growing interest to saturable, i.e., asymptotically linear at infinity, nonlinearities (see [1][2][3]5,[8][9][10][11]17]). Such nonlinearities serve certain models of photorefractive media.…”

Multiple positive solutions of saturable nonlinear Schrödinger equations with intensity functions

“…Physically, φ = φ (x, z) is the electromagnetic field along the propagation coordinate z in a photorefractive material equipped with the photonic lattice described by an intensity (distribution) function I = I (x) (cf. [9,10,12,18,27]). Besides,…”