For current nonlinear physical systems, nonlinear optical fibers serve as a mature nonlinear experimental platform in experimental science. As a type of nonlinear wave with periodic evolution or periodic distribution structure, breathers have become one of the research hotspots in nonlinear optical systems. As the demand for long-distance and high-capacity fiber optic communication increases, the dynamic properties of breathers are receiving increasing attention. Studying the breather solutions for the AB system is of great significance for better understanding long-distance transmission without shape changing in fiber optic communication. In the context of the periodic solution of the AB system, we focus on the breathers of the system. By studying the interactions between two breathers, it is found that the collision between breathers is elastic, which means that breathers can be transported over long distances without changing their shapes. The results obtained in this article will help to understand the dynamics and interactions of breathers under periodic backgrounds in nonlinear optics.

Via the Darboux transformation method in soliton theory, multi-breather solutions for the AB system were constructed under the elliptic function background. With the help of Matlab software, the spatiotemporal structure of the breathers was plotted, and the nonlinear dynamic characteristics of these breathers were further analyzed. Firstly, elliptic function solutions of the AB system were solved by the modified squared wave (MSW) function approach and the traveling wave transformation. Then, we obtained the basic solution to the Lax pair corresponding to the seed solution to the Jacobi elliptic function. Based on the elliptic function transformation formulas and the integral formulas, the potential function solution could be expressed in terms of the Weierstrass elliptic function. Secondly, by the once-iterated Darboux transformation, three types of breather solutions under the elliptic function background were constructed including the general breather (GB), the Kuznetsov-Ma breather (KMB), and the Akhmediev breather (AB). In addition, we analyzed the dynamic behaviors of these three kinds of breathers and presented their three-dimensional spatiotemporal structures. By the twice-iterated Darboux transformation, the spatiotemporal structure of the interaction between a GB and a KMB under the dn background was investigated, as well as the interaction between two GBs under the cn background.

As an important integrable model, the AB system can be used to describe various nonlinear phenomena in many physical fields such as the quantum field theory, weak nonlinear dispersive water wave, and nonlinear optics. It is meaningful to solve various types of solutions of this model to describe the propagation of nonlinear waves. As far as we know, the breather solutions for the AB system have not been constructed under the elliptic function background. In the context of the periodic solution to the elliptic function in the AB system, the basic solution to the Lax pair of the system is obtained using the MSW function. Using the Darboux transformation method, multiple breathers are constructed under the elliptic function background. Based on the expressions of the breather solutions, the dynamic characteristics of three types of breathers are discussed, including the GB, the KMB, and the AB (Figs. 1 and 2). Finally, the spatiotemporal structure of the interaction between a GB and a KMB under the dn background is investigated (Fig. 3), as well as the interaction between two GBs under the cn background (Fig. 4). It is found that collisions between breathers are elastic, which means that breathers can be transmitted over long distances without changing their shapes. These theoretical research results contribute to exploring the practical physical significance and applications of breathers in nonlinear optics.

Based on the elliptic function formulas, we derive the explicit expressions of the first- and second-order breather solutions under the backgrounds of the dn and cn elliptic functions using the Darboux iteration algorithm. By analyzing the dynamic characteristics of three types of breathers and studying the spatiotemporal structure of multi-breather interactions under the dn and cn backgrounds, we find that the collision of GBs and the collision between GB and KMB in the AB system are both elastic, and the breathers do not undergo any shape change during their propagation. This discovery is of great significance for understanding the propagation characteristics of breathers and further elucidating their ability to complete long-distance transmission without changing their shapes. This research will help to understand the dynamics and interactions of breathers under the periodic background from fluid dynamics to nonlinear optics.