For a nonlinear system, the excitation and growth of modes from perturbations constitute intriguing issues in dynamics. Such scenarios occur frequently in optics, as well as other disciplines like fluids and plasma. When a small amplitude fundamental mode is introduced, high-order modes are generated due to nonlinearity and can be amplified rapidly. Instead of equal energy distribution among all the modes in a dynamic equilibrium, amplitudes of a few selected modes may grow to their peak values, and then decay back to the initial states. Such a process can occur recursively. These issues have been considered in a broad sense since the 1950s, and are commonly known under the name of Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomenon [[1], [2], [3]].

The initial stage of FPUT is closely linked with modulation instability (MI) of the nonlinear systems [[4], [5], [6]], as MI propels growth of small disturbance from the interplay of dispersion and nonlinearity. In optical physics, Kerr nonlinearity may have the same (opposite) sign of dispersion in the anomalous (normal) regime and lead to presence (absence) of MI. The generation and growth of the high-order modes have been under intensive scrutiny, but the dynamics are still not fully understood. We just mention the ‘cascading mechanism’ as one possible candidate [6,7]. Small amplitude higher-order modes grow exponentially at a rate larger than that of the fundamental mode. Eventually all modes attain the same magnitude. A ‘breather’ is formed at that time instant (or spatial location). Breather subsequently decays and MI resumes at a sufficiently small amplitude. Recurring cycles occur and are taken as a manifestation of FPUT.

FPUT dynamics and properties have been studied in several fields of physical science, e.g., optics and fluids [[8], [9], [10], [11], [12], [13], [14], [15], [16], [17]]. In optics, FPUT was probably first observed experimentally in 2001, using a picosecond square-pulse laser source and a two-wavelength configuration of a nonlinear optical loop mirror [18]. The subsequent literature is vast and only a few representative works will be mentioned. The existence of the Kuznetsov-Ma breather was demonstrated experimentally [19]. Recently, four FPUT cycles were successfully observed by using ultra-low loss fiber and an active loss compensation system [20]. Features of the maximum compression points and sideband amplitude were clarified [21]. FPUT was also demonstrated using a recirculating fiber loop with periodic amplification as a model for long-haul transmission system [22].

There are indeed many valuable theoretical and experimental perspectives from these FPUT studies, e.g., the cascading mechanism [6] and matched asymptotic expansions [23]. Examples of other important factors include higher-order MI [24], third-order dispersion [25,26], noise-driven thermalization and transition to irreversible behavior [13]. In terms of technology, distributed measurement of square pulses [27], as well as programmable phase and amplitude shaping [28], allow measurements of FPUT over a large range of input parameters. Similarly, FPUT can be employed for controllable ultrashort pulse generation and new laser source of wide band, by considering the multi-frequency components generated in the propagation.

We now turn to the main objective of the present work, namely, FPUT for a two-core fiber (TCF). These TCFs are indeed valuable in optical engineering. To enhance the capacity for information transmission, new technologies based on the multimode and multicore optical fibers have been implemented, e.g., space-division multiplexing (SDM) or mode-division multiplexing (MDM) [[29], [30], [31], [32], [33], [34]]. Understanding the underlying scientific principles is beneficial.

There is a remarkable correlation between MI and FPUT, even though the former is a linear approximation while the latter is intrinsically a nonlinear feature of the system. For a single core fiber governed by the nonlinear Schrödinger equation, there is a vast literature [[18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]]. FPUT is absent in the normal dispersion regime as there is no MI. There are comparatively fewer studies on multi-core/multi-mode fibers. For fibers with negligible birefringence leading to ‘integrable’, coherently coupled Schrödinger equations, MI can again be used to predict the FPUT patterns generated [7]. For TCF governed by ‘non-integrable’, linearly coupled Schrödinger equations, MI can still serve as an illuminating indicator.

TCFs possess a remarkable property. A pulse launched initially in one core can periodically transfer energy to the other core due to the linear coupling between the two cores. The linear coupling coefficient depends on the fiber design and the wavelength [[35], [36], [37]]. The wavelength dependence of the linear coupling coefficient is termed ‘coupling coefficient dispersion’ (CCD) [37]. As the various wavelength components in an optical pulse may exhibit different coupling properties between the two cores, CCD can cause serious pulse distortion or even pulse splitting [[35], [36], [37]]. Theoretically, CCD is equivalent to the group-delay difference between the even and odd modes of a TCF, namely the intermodal dispersion (IMD) in a TCF [[35], [36], [37]].

Recently, we have studied the effects of linear coupling on the FPUT of TCFs systematically [38]. The FPUT patterns depend crucially on the ratio of the coupling coefficient to the power of the optical wave. Here we extend our study to incorporate CCD, where MI again provides an instructive direction to understand the optical physics. Earlier studies confirm that CCD does not affect the MI gain spectra for the symmetric or antisymmetric continuous-wave (CW) state [39], as only the even or odd mode is excited. In the present study, we show that CCD can dramatically influence the FPUT dynamics in a TCF for the symmetric or antisymmetric CW state, depending on the nature of the perturbations applied to the input CW state. Similar scenarios occur in other coupled waveguides too. For fibers with negligible birefringence and coherently coupled nonlinear Schrödinger equations, asymmetric perturbations destroy the symmetry of the two waveguides [7].

The organization of the paper can now be explained. Firstly, the analytic formulation in terms of the coupled nonlinear Schrödinger equations (NLSEs), the dispersion relation including CCD effects, and the appropriate boundary conditions for the simulations are given (Section 2). Secondly, the impact of CCD on the FPUT dynamics in the anomalous and normal dispersion regimes is discussed (Section 3). To build a theoretical foundation for understanding the effect of CCD, Floquet analysis is performed for the linearized stability equations obtained from the FPUT dynamics in the normal dispersion regime (Section 4). Finally, conclusions are drawn (Section 5).