On zero-background solitons of the sharp-line Maxwell-Bloch equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929687230414848 |
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| author | Li, Sitai |
| author_facet | Li, Sitai |
| contents | 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 $N\to+\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_02166 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On zero-background solitons of the sharp-line Maxwell-Bloch equations Li, Sitai Exactly Solvable and Integrable Systems Analysis of PDEs Pattern Formation and Solitons 35Q60, 37K40, 35Q15, 37K10, 35Q51, 37K15 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 $N\to+\infty$. |
| title | On zero-background solitons of the sharp-line Maxwell-Bloch equations |
| topic | Exactly Solvable and Integrable Systems Analysis of PDEs Pattern Formation and Solitons 35Q60, 37K40, 35Q15, 37K10, 35Q51, 37K15 |
| url | https://arxiv.org/abs/2402.02166 |