Darboux dressing transformation and superregular breathers for a coupled nonlinear Schrödinger system with the negative coherent coupling in a weakly birefringent fibre

Abstract

For a coupled nonlinear Schrödinger system with the negative coherent coupling, which describes two orthogonally polarized pulses in a weakly birefringent fibre, we construct the Darboux dressing transformation and the N-th-order breather solutions with N as a positive integer. When the retarded time tends to be infinite, limits of the ratios between the N-th-order breather solutions and seed solutions are obtained. For the first-order breathers, we present the condition to distinguish the degenerate and nondegenerate cases. For the nondegenerate breathers, we analyse whether the breathers could be kink-type based on the above limits. For the second-order breathers, superregular breathers (SRBs) are derived, where each SRB could consist of two (1) kink-type, (2) single-hump or (3) double-hump quasi-Akhmediev breathers (quasi-ABs). Before and after the interaction, profiles of two quasi-ABs change for Case (1) but keep unchange for Case (2) or (3).

Keywords:

• Weakly birefringent fibre
• coupled nonlinear Schrödinger system with the negative coherent coupling
• Darboux dressing transformation
• superregular breathers

2010 AMS SUBJECT CLASSIFICATION:

• 35Q55 35B40 35Q60

Acknowledgments

We sincerely thank the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11272023, 11471050 and 11805020, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05) and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Kink-type breathers possess the different background amplitudes on different sides of the breathers [28].

2 Motivated by Ref. [13], for the breathers described via $q_1$ and $q_2$, if the condition $q_1=\cos (\theta )q$ and $q_2=\sin (\theta )q$ holds, where q is the breather solution for a scalar NLS equation, i.e. $i{q}_z+q_{tt}+2|q{|}^{2}q=0$, and θ is a real parameter, then the breathers are defined as the degenerate breathers; if the above condition does not hold, the breathers are defines as the nondegenerate breathers. Definition of the degenerate breather here is different from that in Refs. [9,28], where the breathers with the coalescence of eigenvalues are called the degenerate breathers.

3 As ε is close enough to 0, we call the two breathers as two quasi-ABs instead of two GBs.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 11772017, 11272023, 11471050 and 11805020].