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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...
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... λ−(ξ) → Re E 0 , λ 0 (ξ) → 0, λ+(ξ) → 0 as ξ → 0. Applying MAPLE to Eq. (3.49), we find the signature table of Im g presented in Fig. 4. Here, an oval Cg and the curve˜γcurve˜ curve˜γ = γg ⋃ γ g , which connects points E 0 , E 0 through λ−, are those curves where Im g = 0. We deform [E 0 , E 0 ] into new cut γg ⋃ γ g , where Im g = 0. This cut depends on ξ, and 0 < ξ ≤ ξ 0 . Denote by Σ (1) = R ⋃ γg ⋃ γ g the new conjugation contour with the orientation depicted in ...
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... in Fig. 4. Here, an oval Cg and the curve˜γcurve˜ curve˜γ = γg ⋃ γ g , which connects points E 0 , E 0 through λ−, are those curves where Im g = 0. We deform [E 0 , E 0 ] into new cut γg ⋃ γ g , where Im g = 0. This cut depends on ξ, and 0 < ξ ≤ ξ 0 . Denote by Σ (1) = R ⋃ γg ⋃ γ g the new conjugation contour with the orientation depicted in Fig. 4. We make the first (analytically equivalent) deformation of the RH problem by replacing θ(z) with g(z, ξ), Mathematical Physics ...
Citations
... In [5] the IBVP (1.7)-(1.9) have been studied in the case when n(λ) is the Dirac deltafunction, i.e. without inhomogeneous broadening n(λ). ...
... In general, n(λ) can be a singular generalized function, for example, a delta function. The case when n(λ) = δ(λ) was considered in [5]. ...
We consider the problem of the propagation of electric field generated by periodic pumping in a stable medium of two-level atoms as the initial-boundary value problem (mixed problem) for the Maxwell-Bloch equations with an arbitrary inhomogeneous 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 problem for this initial-boundary value problem that allows one to obtain a solution for the Maxwell-Bloch equations in terms of the solution of the associated Riemann-Hilbert problem.
... Later, the Deift-Zhou nonlinear steepest descent method was applied to a related problem [28], but only initially-stable media were considered and the results were somewhat incomplete in the sense that (i) error estimates were omitted (although in principle they are accessible via the methodology employed) and (ii) the leading-order term was given implicitly in terms of the solution of a singular integral equation that is difficult to compare with the Riemann-Hilbert characterization we offer below. Very recently, assuming periodic incident pulses injected into an initially-stable medium, the large-t asymptotic problem was revisited and analyzed by the nonlinear steepest descent method [29]. ...
... This is why the moments and Taylor expansion of r(λ) about λ = 0 are of primary importance in our analysis. 29 FIGURE 16. Numerical study of incident pulse (d) in the medium-bulk regime for propagation in an initially-stable medium (D − = −1). ...
We study the (characteristic) Cauchy problem for the Maxwell-Bloch equations of light-matter interaction via asymptotics, under assumptions that prevent the generation of solitons. Our analysis clarifies some features of the sense in which physically-motivated initial/boundary conditions are satisfied. In particular, we present a proper Riemann-Hilbert problem that generates the unique causal solution to the Cauchy problem, that is, the solution vanishes outside of the light cone. Inside the light cone, we relate the leading-order asymptotics to self-similar solutions that satisfy a system of ordinary differential equations related to the Painlev\'e-III (PIII) equation. We identify these solutions and show that they are related to a family of PIII solutions recently discovered in connection with several limiting processes involving the focusing nonlinear Schr\"odinger equation. We fully explain a resulting boundary layer phenomenon in which, even for smooth initial data (an incident pulse), the solution makes a sudden transition over an infinitesimally small propagation distance. At a formal level, this phenomenon has been described by other authors in terms of the PIII self-similar solutions. We make this observation precise and for the first time we relate the PIII self-similar solutions to the Cauchy problem. Our analysis of the asymptotic behavior satisfied by the optical field and medium density matrix reveals slow decay of the optical field in one direction that is actually inconsistent with the simplest version of scattering theory. Our results identify a precise generic condition on an optical pulse incident on an initially-unstable medium sufficient for the pulse to stimulate the decay of the medium to its stable state.
