On zero-background solitons of the sharp-line Maxwell-Bloch equations

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 Schr\"{o}dinger 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 formulae for the asymptotic phase shift for each DSG. Other generalizations of a single soliton are also discussed, such as $N$th-order solitons and soliton gases. We prove that every $N$th-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 $N\to+\infty$.