Abstract
An approximate analytical solution of the non-integrable problem of steady-state adiabatic interaction of a cnoidal wave with a breather is obtained. The solving algorithm is described by the example of one-dimensional problem of steady-state interaction of a plane cnoidal wave with circular polarization (the "information" signal) with orthogonally polarized rational soliton (the "control" signal) in an isotropic nonlinear gyrotropic medium with Kerr nonlinearity and second-order group-velocity dispersion. It is shown that such the interaction results in a strong amplitude and frequency modulation of the information signal and this modulation is localized in the region where intensity of the control signal changes.
Highlights
Features of interaction of self-consistent solutions of nonlinear problems have recently attracted an increasing interest [2,3,4,5,6,7,8,9,10,11]
Following the adiabatic approach [21], we firstly solve Eq (3) on the assumption that | A+ |2 ≅ const. This enables us to write down the solution of Eq (3) in well-known forms of possible cnoidal waves, for example, in the form r− (t) = B− cn(ν −t, μ− ) [2,21], where cn(x, μ) is Jacobi elliptic function [25], B−2
The used algorithm have been described by the example of one-dimensional problem of steadystate interaction of a plane cnoidal wave with circular polarization with orthogonally polarized rational soliton (a “control” signal) in an isotropic nonlinear gyrotropic medium with Kerr nonlinearity and second-order group-velocity dispersion
Summary
Features of interaction of self-consistent solutions of nonlinear problems (breathers, solitons, and cnoidal waves [1]) have recently attracted an increasing interest [2,3,4,5,6,7,8,9,10,11]. Solution of the interaction problem provides the answer to a crucial question about possibility or impossibility of simultaneous transmission of multiple streams of undistorted optical information through the common channel Due to such processes we can control characteristics of radiation, passed through a nonlinear medium, by means of optical signals [15,16,17]. The adiabatic approximation enables one to describe nonlinear interactions between “fast” and “slow” subsystems (see [21] and reference in it) is very popular both in the quantum, classical and semi-classical descriptions of nonlinear dynamics of quite different systems [22] Combining this approximation with a procedure of separation of variables has recently enabled obtaining the approximate analytical solutions of non-integrable problem of interaction of two cnoidal waves [21]. It is shown that in this case interaction with the control signal results in appreciable amplitude and frequency modulation of the information signal and this modulation is localized in a region where the control signal intensity changes
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