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

TL;DR

This work develops a comprehensive inverse-scattering framework for the zero-background sharp-line Maxwell-Bloch equations, enabling explicit N-soliton solutions that may consist of degenerate soliton groups (DSGs). By formulating equivalent residue and jump Riemann-Hilbert problems and applying Deift-Zhou steepest descent, the authors derive sharp localization, DSG centers, and rigorous long-time asymptotics, showing that general N-soliton solutions decompose into sums of DSGs along lines z = V_j t with explicit phase shifts. The paper also extends these results to high-order solitons via eigenvalue fusion and to soliton gases as N rrow ty, and demonstrates straightforward generalization to the focusing NLS and complex mKdV equations through corresponding dispersion relations. The results advance explicit, scalable understanding of multi-soliton interactions in zero-background integrable systems and have potential implications for nonlinear optics and related AKNS hierarchies. Finally, the work provides explicit formulas for DSG centers, asymptotic shifts, and Nth-order soliton solutions, enabling precise predictions of complex soliton dynamics and their limiting behaviors.

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 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 $N\to+\infty$.

Figures (13)

• Figure 1: Left: an illustration of the pole distribution in RHP \ref{['rhp:N-soliton-residue-form']}. Right: an illustration of the jump configuration in RHP \ref{['rhp:N-soliton-jump-form']} which is equivalent to the left plot.
• Figure 2: Plots of exact $1$-soliton solutions in an initially stable medium with $r_1 = 1$, $\xi_{1,1} = 0$ and $\phi_{1,1} = \pi/2$, and $\alpha_{1,1} = \pi/7$ or $\alpha_{1,1} = \pi/24$ from Corollary \ref{['thm:1-soliton']}. The two solutions have identical velocities, but different amplitudes determined by $\alpha_{1,1}$.
• Figure 3: Plots of exact symmetric $2$-DSG in an initially stable medium with $r_1 = 1$, $\xi_{1,1} = \phi_{1,1} = 0$, and $\alpha = \pi/10$ or $\alpha = \pi/30$ from Corollary \ref{['thm:2-DSG-symmetric']}. Since $q(t,z)$ is purely imaginary, the right column only shows the imaginary parts. The two cases have identical velocity. The parameter $\alpha$ affects amplitudes and oscillatory behavior.
• Figure 4: Density plots of $|q(t,z)|$ of exact $N$-soliton solutions in stable media from Theorem \ref{['thm:N-soliton-formula']} with different setups. Left: A $4$-DSG with parameters $r_{1} = 1$, $\alpha_{1,k}= \pi/5,\pi/3,\pi/2,2\pi/3$, and $\omega_{1,k} = 1$ for all $1\le k \le 4$. The red dashed line corresponds to the center in Theorem \ref{['thm:N-DSG']}. Center: A $3$-soliton solution with two DSGs. The parameters are $r_1 = 0.4$, $r_2 = 1$, $\alpha_{1,1} = 2\pi/5$, $\alpha_{2,1} = \pi/3$, $\alpha_{2,2} = \pi/2$, $\omega_{1,1} = \mathrm{e}^{14}$, $\omega_{2,1} = 1$, and $\omega_{2,2} = \mathrm{i}$. The red dashed lines correspond to the asymptotic centers in Corollary \ref{['thm:asymptotic-shifts']}. Right: A $5$-soliton solution with three DSGs. The parameters are $r_1 = 0.4$, $r_2 = 1$, $r_3 = 2$, $\alpha_{1,k} = \pi/3,\pi/2$, $\alpha_{2,k} = \pi/3,2\pi/3$, $\alpha_{3,1} = \pi/4$, $\omega_{1,1} = \omega_{1,2} = \mathrm{e}^{15}$, $\omega_{2,1} = \mathrm{e}^3$, $\omega_{2,2} = \mathrm{e}^4$, and $\omega_{3,1} = \mathrm{e}^{-10}$. Again, the red dashed lines correspond to the asymptotic centers in Corollary \ref{['thm:asymptotic-shifts']}.
• Figure 5: Density plots of $|q(t,z)|$ of three exact $N$th-order soliton solutions in stable media from Theorem \ref{['thm:N-order-soliton:solution-formula']}. All three plots have similar parameters $\lambda_\circ = \mathrm{i}$, $\omega_{\circ,0} = 1$ and $\omega_{\circ,k} = 0$ for $k\ge1$, but with different values of $N$. Left: $N = 3$. Center: $N = 4$. Right: $N = 5$.

Theorems & Definitions (27)

• Remark 1: On notations
• Definition 1: Eigenvalues and norming constants
• Remark 2
• Lemma 1: Reconstruction formula
• Remark 3
• Remark 4
• Theorem 1: General $N$-soliton solution formula
• Theorem 2: Degenerate $N$-soliton solution
• Remark 5
• Corollary 1: One-soliton solution