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

TL;DR

This work analyzes the long-time behavior of the reduced Maxwell–Bloch equations in the sharp-line limit by extending the nonlinear $\overline{\partial}$-steepest descent framework to an integrable system with Lax-pair singularities at $z=\pm\mu/2$. The authors construct a properly posed Riemann–Hilbert problem through a modified time evolution of the reflection coefficient, enabling rigorous inverse scattering and reconstruction of the electric field and Bloch components. In the soliton-dominated regime within a fixed cone, the leading order is given by a superposition of $N(\mathcal{J})$ solitons and $N(\mathcal{J})$ kinks, with subleading $t^{-1/2}$ terms arising from soliton–radiation interactions; the remainder decays as $t^{-\gamma_0}$ with $\gamma_0>1/2$. The results establish soliton resolution for RMB in the sharp-line limit and demonstrate the applicability of the $\overline{\partial}$-steepest descent method to integrable systems featuring singular Lax-pair data.

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.

Figures (10)

• Figure 1: The jump contour $\Sigma_{1}$ of RHP \ref{['RHP1']} consists of four semicircular arcs $\Sigma_k^\kappa$ ($k=1,\dots,4$) and three intervals: $(-\infty, -\mu/2 - \kappa]$, $[- \mu/2 + \kappa, \mu/2 - \kappa]$, and $[\mu/2 + \kappa, \infty)$. Each $\Sigma_k^\kappa$ corresponds to the upper or lower semicircle of the circles $|z \pm \mu/2| = \kappa$.
• Figure 2: The signature of $\mathop{\mathrm{Re}}\limits [i\theta(z)]$, the contour $G (\mathop{\mathrm{Re}}\limits z,\mathop{\mathrm{Im}}\limits z)=0$, and the stationary phase points $\pm \zeta_0$ and $\pm \zeta_1$ in the complex plane for ${x}/{t}\in( -{1}/\mu ^{2},0)$ with $\mu \in ( 0,1]$.
• Figure 3: The contours defined by \ref{['eq46.1']} and regions $\Omega_k$ (for $k = 1, \ldots, 13$).
• Figure 4: The jump contour $\Sigma_2$ of the $\overline{\partial}$-RHP \ref{['RHP3']} consists of the segments $\Sigma_k^\kappa$, $\Sigma_k^A$, $\Sigma_k^B$, and $\Sigma_k^C$ for $k = 1, \dots, 4$, together with $\Sigma_k^D$ for $k = 1, \dots, 6$. The exponential factor $e^{it\theta(z)}$ decays along the red contours, whereas $e^{-it\theta(z)}$ decays along the blue contours.
• Figure 5: As $t\to \infty$, uniform decay of $V^{(2)}(z)$ to the identity matrix on black contours, and point-wise convergence on green contours. The red boundaries are $\partial D(\pm \zeta_0,\kappa).$

Theorems & Definitions (34)

• Proposition 1
• Proposition 2
• Remark 1
• Proposition 3
• proof
• Remark 2
• Theorem 1
• proof
• Remark 3
• Theorem 2