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Physical Review E

Solution of the reduced anisotropic Maxwell-Bloch equations by using the Riemann-Hilbert problem

Alexander A. Zabolotskii

Published Mar 23, 2007• 12 citations
Publisher

Abstract

We develop a method of solution for recently found integrable system of the reduced Maxwell-Bloch equations with two components of polarization and with an anisotropic dipole momentum by using the appropriate modification of the inverse scattering transform. The method is based on solution of the Riemann-Hilbert problem with taking into account symmetry properties of corresponding fundamental solutions. We show that these symmetries lead to some particular forms of the inverse scattering transform equations which may be used for finding as soliton-type as radiation-type solutions.

DOI
10.1103/physreve.75.036612

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We develop a method of solution for recently found integrable system of the reduced Maxwell-Bloch equations with two components of polarization and with an anisotropic dipole momentum by using the appropriate modification of the inverse scattering transform. The method is based on solution of the Riemann-Hilbert problem with taking into account symmetry properties of corresponding fundamental solutions. We show that these symmetries lead to some particular forms of the inverse scattering transform equations which may be used for finding as soliton-type as radiation-type solutions.
We find a general integrable system of reduced Maxwell-Bloch equations describing the interaction of a two-component electromagnetic field with the dipole transition of a two-level medium. The anisotropy of the anisotropic dipole momentum of the transition as a permanent dipole momentum is taken into account. A solution of the model is found for a particular case by using the inverse scattering transform. The method is based on a solution of the Riemann-Hilbert problem taking into account the symmetry properties of the corresponding fundamental solutions.
The dynamics of soliton pulses for use in nonlinear optical devices is mathematically modelled by Maxwell-Bloch systems of equations for the interaction of light with a uniform distribution of quantum-mechanical atoms. We study the Reduced Maxwell-Bloch (RMB) equations occurring when an ensemble of rotationally symmetric 3-level atoms is assumed. The model applies for on and off-resonance conditions and is completely integrable using Inverse Scattering theory, since it arises as the compatibility condition of a 3 x 3 AKNS-system. Furthermore this integrability remains valid for all timescales of the optical field because only the “one-way wave approximation” is required during the derivation. Solutions are constructed in two ways: 1. Darboux-Bäcklund transforms are applied, generating single soliton pulses of ultrashort (< 1ps) duration, and families of elliptically polarised 2-solitons not possible in lower dimensional problems. 2. A general Inverse Scattering scheme is developed and tested. The Direct Scattering Problem is dealt with first to obtain a complete set of scattering data. Subsequently the Inverse Problem is solved both formally and then in explicit closed form for the special case that the reflection coefficients vanish for real values of the spectral parameter. In this case the main result is a determined system of n linear algebraic equations which yield the n-soliton of our RMB-system. It is confirmed that the 1-solitons found by means of Darboux transform are precisely the same as those given by the full mechanism of Inverse Scattering. Finally we calculate the invariants of the motion for the RMB-equations, and derive an evolution equation giving the variation with propagation distance of the invariant functionals when the original RMB-system is modified by an arbitrary perturbing term. As an application dissipative effects on 1-solitons are considered.
The physical generalization of the system of the reduced Maxwell-Bloch equations describing the nonlinear interaction of the laser pulses with the multilevel quantum medium is obtained without using the slowly varying envelope approximation. It is shown that this system is integrable in the frameworks of the inverse scattering transformation method. The soliton and breather solutions in the form of the unipolar and few-cycle pulses are constructed. Their distinctive features caused solely by the multilevel structure of the medium are discussed. It is revealed that the collision of the solitons can lead to an appearance of the large-amplitude short-living pulse, whose dynamics resembles that of rogue waves.