2000/12/16 by S. Kamvissis, Kamvissis, S., K. T-R McLaughlin +5 · 3 citations
Physics and Astronomy · #Advanced Fiber Laser Technologies #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #nlin.SI
paper · pdf · doi:10.48550/arxiv.nlin/0012034
215 Pages. Submitted to Annals of Mathematics Studies
arxiv created 2000/12/16 · openalex publication_date 2000/12/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a new generalization of the steepest descent method introduced by Deift and Zhou for matrix Riemann-Hilbert problems and use it to study the semiclassical limit of the focusing nonlinear Schroedinger equation with real analytic, even, bell-shaped initial data. We provide explicit strong locally uniform asymptotics for a sequence of exact solutions corresponding to initial data that has been modified in an asymptotically small sense. We call this sequence of exact solutions a semiclassical soliton ensemble. Our asymptotics are valid in regions of the (x,t) plane where a certain scalar complex phase function can be found. We characterize this complex phase function directly by a finite-gap ansatz and also via the critical point theory of a certain functional; the latter provides the correct generalization of the variational principle exploited by Lax and Levermore in their study of the zero-dispersion limit of the Korteweg-de Vries equation.