Speaker
Description
The evolution of the complex envelopes of signals in fiber-optics communications is described by the focusing nonlinear Schr\"odinger equation. This equation is integrable by the inverse scattering method, which has three versions corresponding to rapidly decaying, periodic and local holomorphic signals respectively. We mention the remarkable recent advances in the study of the pointwise convergence and time-frequency localization properties of the direct scattering transform (in the first and second versions), which is regarded as the nonlinear Fourier transform. We use the third version to prove the global-in-time real-analytic solvability of the Cauchy problem for any real-analytic initial data and discuss the natural isomorphism between the third version and each of the first two versions in the case of real-analytic signals satisfying the corresponding boundary conditions.
