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Mathematical Methods in the Applied Sciences

Multi‐soliton solutions of the three‐level coupled Schrödinger Maxwell‐Bloch equations via the Riemann‐Hilbert approach

Peng Li, Huanhe Dong, Xiangrong Wang

Published Feb 4, 2024• 0 citations
Publisher

Abstract

In this paper, a study of the coupled Schrödinger Maxwell‐Bloch equations, which described the propagation of two optical pulses in an optical medium with coherent three‐level atoms, is presented via the Riemann‐Hilbert approach. First, we performed a spectral analysis on the basis of Lax pair with the negative flow, and then the Jost function, scattering matrix, and their analytic and symmetric properties are given. Second, the Riemann‐Hilbert problem is established successfully through a standard dressing procedure via the Riemann‐Hilbert approach, and then the potential function related to the solution of the Riemann‐Hilbert problem is reconstructed. By introducing the special matrix functions, we can transform the irregular Riemann‐Hilbert problem into the regular one, which can be solved by the Plemelj formula. Finally, some applications are given to solve the Riemann‐Hilbert problem without reflection for the coupled Schrödinger Maxwell‐Bloch equation, and the multi‐soliton solutions are obtained explicitly. Moreover, the dynamic behaviors of some soliton solutions are discussed graphically by choosing appropriate parameters.

DOI
10.1002/mma.9879

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In this paper, a study of the coupled Schrödinger Maxwell‐Bloch equations, which described the propagation of two optical pulses in an optical medium with coherent three‐level atoms, is presented via the Riemann‐Hilbert approach. First, we performed a spectral analysis on the basis of Lax pair with the negative flow, and then the Jost function, scattering matrix, and their analytic and symmetric properties are given. Second, the Riemann‐Hilbert problem is established successfully through a standard dressing procedure via the Riemann‐Hilbert approach, and then the potential function related to the solution of the Riemann‐Hilbert problem is reconstructed. By introducing the special matrix functions, we can transform the irregular Riemann‐Hilbert problem into the regular one, which can be solved by the Plemelj formula. Finally, some applications are given to solve the Riemann‐Hilbert problem without reflection for the coupled Schrödinger Maxwell‐Bloch equation, and the multi‐soliton solutions are obtained explicitly. Moreover, the dynamic behaviors of some soliton solutions are discussed graphically by choosing appropriate parameters.
We propose the system of generalized inhomogeneous coupled Hirota–Maxwell–Bloch equations which describes propagation of an optical soliton in an inhomogeneous erbium-doped fiber with two-level resonant atoms. For this system, higher-order dispersion, self-steepening and self-Raman scattering are assumed to be inhomogeneous, like the group velocity dispersion. The exact analytical multisoliton solutions are obtained by employing the simple, straightforward Darboux transformation based on the obtained 3 × 3 Lax pair. As an example, the one- and two-soliton solutions in explicit forms are generated. Then, we give the relative numerical evolution and discuss the propagation characteristics of the pulse.
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.
ABSTRACT We study an initial‐boundary‐value problem (IBVP) for a system of coupled Maxwell–Bloch equations (CMBE) that model two colors or polarizations of light resonantly interacting with a degenerate, two‐level, active optical medium with an excited state and a pair of degenerate ground states. We assume that the electromagnetic field approaches nonvanishing plane waves in the far past and future. This type of interaction has been found to underlie nonlinear optical phenomena including electromagnetically induced transparency, slow light, stopped light, and quantum memory. Under the assumptions of unidirectional, lossless propagation of slowly modulated plane waves, the resulting CMBE become completely integrable in the sense of possessing a Lax pair. In this paper, we formulate an inverse scattering transform (IST) corresponding to these CMBE and their Lax pair, allowing for the spectral line of the atomic transitions in the active medium to have a finite width. The scattering problem for this Lax pair is the same as for the Manakov system. The main advancement in this IST for CMBE is calculating the nontrivial spatial propagation of the spectral data and determining the state of the optical medium in the distant future from that in the distant past, which is needed for the complete formulation of the IBVP. The Riemann–Hilbert problem is used to extract the spatio‐temporal dependence of the solution from the evolving spectral data. We further derive and analyze several types of solitons and determine their velocity and stability, as well as find dark states of the medium, which fail to interact with a given pulse.
Exact soliton solutions of a coupled system of Maxwell-Bloch equations are obtained analytically within the inverse scattering transform scheme. The equations describe a light pulse coherently propagating through a mixed ensemble of two-level atoms characterized by two slightly different detuning parameters. The solution illustrates the information about the detailed atomic properties which are impressed on both phase and velocity of the pulse during propagation, with potential applications in high resolution spectroscopy.
We consider the problem of the propagation of an electric field generated by periodic pumping in a stable medium of two-level atoms as the mixed problem for the Maxwell–Bloch equations without spectrum broadening. An approach to the study of such a problem is proposed. We use the inverse scattering transform method in the form of the matrix Riemann–Hilbert (RH) problem, using simultaneous spectral analysis of both the Lax equations. The proposed matrix RH problem solves the problem of the propagation of a sinusoidal signal in an unperturbed stable medium (attenuator). It is proved that this RH problem provides the causality principle for the region t < x, and for the region of the light cone, 0 < x < t allows us to find the asymptotics of the transmitted signal. First, we study the asymptotics of the RH problem for large times, and then, we obtain asymptotic formulas for the mixed problem solution of the Maxwell–Bloch equations when the attenuator is long enough. Three sectors are obtained in the light cone where the asymptotics have essentially different behaviors.