The first result of the NWACOMPLEX project represents the development and experimental demonstration of the novel theoretical approach called management of breather interactions. The management approach allows for the adjustment of the initial positions and phases of more than two NLSE breathers to observe various desired wave states at controllable moments of wave evolution. Our theoretical framework relies on exact multibreather solutions to the NLSE and asymptotic expressions describing shifts of breather positions and phases acquired by them in mutual collisions. As proof of principle, we considered a couple of separated pairs of breathers initially synchronized in a small-amplitude pattern; meanwhile, our approach can be generalized to other breather types and wave states. We obtained an explicit expression for the separation interval between the pairs so that the interactions of the breathers from the neighboring patterns lead to the formation of an extreme amplitude wave or recurrence to the initial small-amplitude state. Finally, we carried out experiments on a light wave platform with a nearly conservative optical fiber system, which accurately reproduces the predicted dynamics and proves the viability of our nonlinear wave theory. We published the results of the breather management in “Physical Review Research” journal. The project Fellow presented the breather management concept at “Advanced Photonics Congress 2022” (online talk in Maastricht, Netherlands), “International Workshop on Rogue Waves 2022” (invited online talk in Vermont, USA) and “European Optical Society Annual Meeting 2023” (talk in Dijon, France).
Then we considered theoretically theory of rogue waves formation in the framework of the NLE model. Nowadays, breather solutions of the NLS equation are generally accepted models of the rogue waves. However, breathers exist on a finite background and therefore are not localized, while wavefields in nature can generally be considered as localized due to the limited sizes of physical domain. Hence, the theory of rogue waves needs to be supplemented with localized solutions, which evolve locally as breathers. We present a universal method for constructing such solutions from exact multisoliton solutions of the NLS equation. The method represents replacing the plane wave in the IST dressing construction of the breathers with a specific exact N-soliton solution converging asymptotically to the plane wave at large number of solitons N. We published our theoretical concept of the rogue waves description in the journal “Studies in Applied Mathematics”. The project Fellow presented these results at “International Workshop on Extreme Waves 2023” (talk in Dresden, Germany) and “Dynamics Days Europe” conference (invited talk in Naples, Italy).
We also contributed to the broadening of the DST numerical algorithms family. We studied the problem of the scattering data numerical computation for a broad class of the NLS breathers localized in space. Such DST procedure requires a numerical solution of the auxiliary Zakharov-Shabat system with boundary conditions corresponding to the continuous wave background. To find the solution we computed the transfer matrix using the second-order Boffetta-Osborne approach and recently developed high-order numerical schemes based on the Magnus expansion. To recover the scattering data of breathers, we derived analytical relations between the scattering coefficients and the transfer matrix elements. We used localized single- and multi-breather solutions and verify the developed numerical approach by recovering the complete set of the scattering data with the built-in accuracy providing the information about the amplitude, velocity, phase, and position of each breather. This work has been published in Proceedings of the Royal Society A journal.
