Periodic Reporting for period 1 - NWACOMPLEX (Advanced nonlinear wave analysis for complex experimental data)
Période du rapport: 2022-03-01 au 2024-02-29
Today, the fundamental interest in complex nonlinear wave phenomena, and also practical needs of the society, such as dense optical data transmission and prediction of extreme water wave events, bring to agenda the problem of strongly nonlinear waves description, when nonlinearity can no longer be accounted using the perturbative approach. The basic models of nonlinear wave propagation in optics and surface waves hydrodynamics – the one-dimensional nonlinear Schrodinger (NLS) equations – belong to the class of nonlinear integrable mathematical models, which admit the direct and inverse scattering transformations (DST/IST). Combination of integrable models and strongly nonlinear fields leads to dynamics where such nonlinear coherent structures as solitons and breathers play the dominant role. Among the most fascinating strongly nonlinear phenomena described by integrable NLS equation are the modulation instability development, formation of rogue waves and dam-break problem, which are actively discussed in both scientific and engineering contexts. The advanced experimental facilities now allow to investigate the complex nonlinear wave phenomena in extensive physical settings with optics and hydrodynamics providing favourable conditions to observe solitons, breathers and their interactions.
This research program entitled “Advanced nonlinear wave analysis for complex experimental data” (NWACOMPLEX) aims at pioneering studies of nonlinear wave processes in optics and hydrodynamics in collaboration of the Fellow with top-level experimental teams of the host and partners organizations. The Fellow being experienced in the DST and IST techniques, physics of nonlinear coherent structures and various computational methods proposes novel theoretical approaches and numerical tools for advanced analysis of modern experiments on generation, detection and nonlinear propagation of light in optical systems and waves on the surface of water. The innovative idea of the project is to reveal the nature of complex nonlinear phenomena using inverse scattering transform theory employing the most recent numerical advancements.
The concrete objectives of the project are the developing of new stable and efficient DST /IST algorithms for complex nonlinear wavefields, studying of the soliton and breather complexes, novel soliton and breather gases configurations. Analysis of the effects laying beyond integrable models predictions. Applying of the developed DST/ IST algorithms and verifying of the obtained theoretical predictions in the laboratories of the Host and partner organizations. The project is focused on two different branches of wave physics, namely, optics and hydrodynamics. Its multidisciplinary approach allows universal theoretical description on a time scale of seconds for water waves and on a time scale of picoseconds for light waves. The project's other objective is to build a strong collaboration between the Fellow and the project participants, benefiting the host and partner organizations through the two-way transfer of knowledge.
This research program entitled “Advanced nonlinear wave analysis for complex experimental data” (NWACOMPLEX) aims at pioneering studies of nonlinear wave processes in optics and hydrodynamics in collaboration of the Fellow with top-level experimental teams of the host and partners organizations. The Fellow being experienced in the DST and IST techniques, physics of nonlinear coherent structures and various computational methods proposes novel theoretical approaches and numerical tools for advanced analysis of modern experiments on generation, detection and nonlinear propagation of light in optical systems and waves on the surface of water. The innovative idea of the project is to reveal the nature of complex nonlinear phenomena using inverse scattering transform theory employing the most recent numerical advancements.
The concrete objectives of the project are the developing of new stable and efficient DST /IST algorithms for complex nonlinear wavefields, studying of the soliton and breather complexes, novel soliton and breather gases configurations. Analysis of the effects laying beyond integrable models predictions. Applying of the developed DST/ IST algorithms and verifying of the obtained theoretical predictions in the laboratories of the Host and partner organizations. The project is focused on two different branches of wave physics, namely, optics and hydrodynamics. Its multidisciplinary approach allows universal theoretical description on a time scale of seconds for water waves and on a time scale of picoseconds for light waves. The project's other objective is to build a strong collaboration between the Fellow and the project participants, benefiting the host and partner organizations through the two-way transfer of knowledge.
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.
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.
Also, within the NWACOMPLEX project the Fellow participated as a co-author in the work on the perspective article “Soliton Gas: Theory, Numerics and Experiments” and studied bi-solitons on the surface of a deep fluid. This work is available at arXiv and has been accepted for publication in Physical Review E and Physical Review Letters journals.