Quasi-periodic breathers and their dynamics to the Fokas system in nonlinear optics

The Fokas system is widely applied in nonlinear optics which can be used to describe the propagation behavior of optical solitons. An effective method for constructing the quasi-periodic breathers of the Fokas system is presented by combining the Hirota’s bilinear method with the theta function. The solvable problem of the quasi-periodic breathers is successfully transformed into a least squares problem whose numerical solutions ultimately are obtained through the Gauss–Newton method and the Levenberg–Marquardt method. Theoretical inference and numerical results show that when the real part of the diagonal elements of the Riemann matrix tends to positive infinity, the quasi-periodic breathers can be reduced to regular breathers. By analyzing the propagation characteristics of the quasi-periodic breathers, these quasi-periodic breathers are divided into three categories, general quasi-periodic breathers, quasi-periodic approximate Kuznetsov–Ma breathers and quasi-periodic Akhmediev breathers. Furthermore, by using an analytical method related to the characteristic lines for the quasi-periodic breathers, the dynamic characteristics including the periods and wave velocities of the quasi-periodic breathers are analyzed.