2024/02/03 by Sitai Li, Li, Sitai · 1 citation
Mathematics · Physics and Astronomy · #35Q15 #35Q51 #35Q60 #37K10 #37K15 #37K40 #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Pattern Formation and Solitons (nlin.PS)
paper · pdf · doi:10.48550/arxiv.2402.02166
openalex publication_date 2024/02/03 · openalex created_date 2024/02/07 · openalex updated_date 2026/07/28
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 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→+∞.