Long-time behavior of the reduced Maxwell-Bloch equations in the sharp-line limit

TL;DR

This paper analyzes the long-time behavior of solutions to the reduced Maxwell-Bloch equations using inverse scattering and Riemann-Hilbert techniques, revealing soliton-dominated asymptotics and extending the nonlinear steepest descent method.

Contribution

It introduces a modified inverse scattering transform for the equations, handling singularities and deriving detailed long-time asymptotics within a fixed cone.

Findings

Asymptotic solutions are dominated by solitons within the cone.

Soliton-radiation interactions appear in lower-order asymptotics.

The method extends the nonlinear steepest descent approach to singular Lax pairs.

Abstract

We study the Cauchy problem for the reduced Maxwell-Bloch equations with initial data for the electric field in weighted Sobolev spaces, assuming that all atoms initially reside in their ground state. Using the d-bar steepest descent method, we derive long-time asymptotic expansions of the solutions, including both the electric field and the components of the Bloch vector, within any fixed cone. In particular, we formulate the inverse scattering transform as a properly posed Riemann-Hilbert problem, avoiding singularities in the scattering data by modifying the time evolution of the reflection coefficient. Under assumptions that allow only soliton generation, the leading-order asymptotics are determined by solitons inside the cone, while soliton-radiation interactions appear in lower-order terms. These results extend the applicability of the nonlinear steepest descent method to integrable systems with singularities in the associated Lax pair.