Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation
The model studied by the authors is a key model in modern nonlinear optics, having increasingly important applications in the telecommunication industry. The authors investigate it in the semiclassical limit, where the initial data have a rapidly decreasing amplitude and a phase function that is rapidly approaching the constant value. This model exhibits both “spatial solitons” envisioned as self-guided beams that can form fundamental components of an all-optical switching system, as well as the envelope pulses known as “temporal solitons” envisioned as robust bits in a digital signal travelling through the fibre. The book is concerned with a deep semiclassical analysis of the inverse-scattering step. The authors develop a method that is a generalization of the variational principle exploited by Lax and Levermore in their study of the zero dispersion limit of the Korteweg-de Vries equation and they present a new generalization of the steepest-descent method introduced by Deift and Zhou. Topics treated in the book include holomorphic Riemann-Hilbert problems for solitons, semiclassical soliton ensembles, asymptotic analysis of the inverse problem, direct construction of the complex phase, the genus-zero Anstatz and the transition to genus two, and variational theory of the complex phase. In the Appendices, the reader can find Hölder theory of local Riemann-Hilbert problem and the near identity Riemann-Hilbert Problem in L2.