Bi-Solitons on the Surface of a Deep Fluid: An Inverse Scattering Transform Perspective Based on Perturbation Theory.

We investigate theoretically and numerically the dynamics of long-living oscillating coherent structures-bi-solitons-in the exact and approximate models for waves on the free surface of deep water. We generate numerically the bi-solitons of the approximate Dyachenko-Zakharov equation and fully nonlinear equations propagating without significant loss of energy for hundreds of the structure oscillation periods, which is hundreds of thousands of characteristic periods of the surface waves. To elucidate the long-living bi-soliton complex nature we apply an analytical-numerical approach based on the perturbation theory and the inverse scattering transform (IST) for the one-dimensional focusing nonlinear Schrödinger equation model. We observe a periodic energy and momentum exchange between solitons and continuous spectrum radiation resulting in repetitive oscillations of the coherent structure. We find that soliton eigenvalues oscillate on stable trajectories experiencing a slight drift on a scale of hundreds of the structure oscillation periods so that the eigenvalue dynamics is in good agreement with predictions of the IST perturbation theory. Based on the obtained results, we conclude that the IST perturbation theory justifies the existence of the long-living bi-solitons on the surface of deep water that emerge as a result of a balance between their dominant solitonic part and a portion of continuous spectrum radiation.

Fast nonlinear Fourier transform algorithms for optical data processing.

The nonlinear Fourier transform (NFT) is an approach that is similar to a conventional Fourier transform. In particular, NFT allows to analyze the structure of a signal governed by the nonlinear Schrödinger equation (NLSE). Recently, NFT applied to NLSE has attracted special attention in applications of fiber-optic communication. Improving the speed and accuracy of the NFT algorithms remains an urgent problem in optics. We present an approach that allows to find all variants of symmetric exponential splitting schemes suitable for the fast NFT (FNFT) algorithms with low complexity. One of the obtained schemes showed good numerical results in computing the continuous spectrum compared with other fast fourth-order NFT schemes.