Uzunov Ivan M, Nikolov Svetoslav G, Arabadzhiev Todor N, Georgiev Zhivko D
Department of Applied Physics, Faculty of Applied Mathematics and Informatics, <a href="https://ror.org/052prhs50">Technical University of Sofia</a>, 8 Kliment Ohridski Boulevard, Sofia 1000, Bulgaria.
<a href="https://ror.org/03x78r446">Institute of Mechanics</a>, <a href="https://ror.org/01x8hew03">Bulgarian Academy of Sciences</a>, Academy Georgi Bonchev Strasse, Building 4, 1113 Sofia, Bulgaria.
Phys Rev E. 2024 Aug;110(2-1):024214. doi: 10.1103/PhysRevE.110.024214.
We have found long-living periodic solutions of the complex cubic-quintic Ginzburg-Landau equation (CCQGLE) perturbed with intrapulse Raman scattering. To achieve this we have applied a model system of ordinary differential equations (SODE). A set of the fixed points of the system has been described. A complete phase portrait as well as phase portraits near the fixed points have been built for a proper choice of parameters. The behavior of the model system near the fixed points has been determined. We have presented a detailed description of the subcritical Poincaré-Andronov-Hopf bifurcation due to the intrapulse Raman scattering that appears at one of the fixed points. We have established that there appears an unstable limit cycle in the SODE. To check the validity of the obtained results from the model system we have compared them with the results of the numerical solution of the CCQGLE perturbed with intrapulse Raman scattering. There has been found a remarkable correspondence between the obtained numerical results for the amplitude and frequency of the soliton pulses and the results for these parameters of the bifurcation theory. We have observed that the numerical characteristics of the propagating solitonlike pulses-amplitude, frequency, width, and position-periodically change if we change the distance with a period determined by the bifurcation analysis.
我们发现了受脉冲内拉曼散射扰动的复立方 - 五次金兹堡 - 朗道方程（CCQGLE）的长寿命周期解。为实现这一点，我们应用了常微分方程的模型系统（SODE）。描述了该系统的一组不动点。针对适当的参数选择，构建了完整的相图以及不动点附近的相图。确定了模型系统在不动点附近的行为。我们详细描述了在其中一个不动点出现的由脉冲内拉曼散射引起的亚临界庞加莱 - 安德罗诺夫 - 霍普夫分岔。我们确定在SODE中出现了一个不稳定极限环。为检验从模型系统获得的结果的有效性，我们将它们与受脉冲内拉曼散射扰动的CCQGLE的数值解结果进行了比较。在孤子脉冲的幅度和频率的数值结果与分岔理论的这些参数的结果之间发现了显著的对应关系。我们观察到，如果我们以分岔分析确定的周期改变距离，传播的类孤子脉冲的数值特征——幅度、频率、宽度和位置——会周期性地变化。
