On Zero-Background Solitons of the Sharp-Line Maxwell–Bloch Equations

Abstract

This work is devoted to systematically study general N-soliton solutions possibly containing multiple degenerate soliton groups (DSGs), in the context of the sharp-line Maxwell–Bloch equations with a zero background. We also show that results can be readily migrated to other integrable systems, with the same non-self-adjoint Zakharov–Shabat scattering problem or alike. Results for the focusing nonlinear Schrödinger equation and the complex modified Korteweg–De Vries equation are obtained as explicit examples for demonstrative purposes. A DSG is a localized coherent nonlinear traveling-wave structure, comprised of inseparable solitons with identical velocities. Hence, DSGs are generalizations of single solitons (considered as 1-DSGs), and form fundamental building blocks of solutions of many integrable systems. We provide an explicit formula for an N-DSG and its center. With the help of the Deift–Zhou’s nonlinear steepest descent method, we prove the localization of DSGs, and calculate the long-time asymptotics for an arbitrary N-soliton solutions. It is shown that the solution becomes a linear combination of multiple DSGs in the distant past and future, with explicit formulæ for the asymptotic phase shift for each DSG. Other generalizations of a single soliton are also discussed, such as Nth-order solitons and soliton gases. We prove that every Nth-order soliton can be obtained by fusion of eigenvalues of N-soliton solutions, with proper rescalings of norming constants, and demonstrate that soliton-gas solution can be considered as limits of N-soliton solutions as \(N\rightarrow +\infty \).