Soliton interactions and Yang–Baxter maps for the complex coupled short‐pulse equation
Soliton interactions and Yang–Baxter maps for the complex coupled short‐pulse equation
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DOI：
10.1111/sapm.12580
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发表时间：
2022-10
影响因子：
2.7
通讯作者：
V. Caudrelier;Aikaterini Gkogkou;B. Prinari
中科院分区：
文献类型：
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作者：
V. Caudrelier;Aikaterini Gkogkou;B. Prinari
复杂耦合短脉冲方程（ccSPE）描述了超短光脉冲在非线性双折射光纤中的传播。该系统允许各种向量孤子解：基本孤子，基本呼吸子，复合呼吸子（一般或非一般），以及所谓的自对称复合孤子。在这项工作中，我们使用修饰方法和对应于各种类型孤子的达布矩阵来研究聚焦ccSPE中孤子的相互作用。本研究结合了一些有理环群生成元的重构问题，这些生成元的长时间渐近性，以及导致重构映射和矢量孤子相互作用的Yang-Baxter性质的修饰因子的主要重构定理。在本文所得到的结果中，我们推导出了基孤子偏振位移的显式公式，这些公式类似于众所周知的Manakov系统中矢量孤子相互作用的公式。我们的研究还表明，在与一个基本呼吸子相互作用时，一个基本孤子变成一个基本呼吸子，相反，两个基本呼吸子的相互作用通常产生两个具有偏振位移的基本呼吸子，但也可能产生一个基本孤子和一个基本呼吸子。得到了表征基本呼吸子的系数及其偏振矢量的显式公式。其他类型孤子的相互作用也得到了推导，并进行了详细的讨论和图解。在此过程中获得了新的Yang-Baxter图。
The complex coupled short‐pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so‐called self‐symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long‐time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well‐known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process.