Abstract

We report a fundamentally new approach to interpreting modulation instability (MI) dynamics in optical fiber propagation using the machine-learning technique of data-driven dominant balance [1]. Our aim is to automate the task of identifying which particular physical processes drive propagation in different regimes, a task usually performed using intuition and/or asymptotic limits [2]. In particular, we study MI in the nonlinear Schrödinger equation (NLSE) written in normalized form as: iu_ξ + ½ u_ττ + |u|²u = 0, where ξ is distance, τ is comoving time, and u is the field envelope. Data-driven dominant balance involves several steps: (i) firstly determining the evolution map for u(ξ,τ); (ii) analysing the map in its “equation space,” to identify clusters where different combinations of the NLSE terms (u_ξ, u_ττ, |u|²u) dominate the interaction; (iii) re-mapping these clusters back onto the (ξ,τ) space for comparison with the evolution map. Figures 1(a) and (b) apply the technique to the analytic u(ξ,τ) solutions for the Peregrine soliton (PS) and the Akhmediev breather (AB) respectively. These are localized structures well-known to emerge from MI [3]. In each case, subfigure (i) shows the evolution map, subfigure (ii) shows an example of equation space cluster detection, and subfigure (iii) shows the clusters remapped to (ξ,τ) for comparison with the evolution. Color scales are on the bottom right. The key physics apparent in subfigures (iii) is that the blue regions are associated with dominant (u_ξ, |u|²u) interactions, whilst orange regions are associated with all terms (u_ξ, u_ττ, |u|²u). These results show that dispersive effects do not play a dominant role in the blue “continuous wave” region where temporal localization is not present. In contrast, with strong temporal localization, the identified orange region shows how dispersion contributes comparably with nonlinearity. Figure 1(c) shows evolution for chaotic MI seeded by a noisy continuous wave. Even for this much more complex case, dominant balance successfully determines regions of only dominant nonlinearity (blue) and combined nonlinearity and dispersion (orange). Moreover, we identify characteristic signatures of PS and AB structures as marked.

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