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Dispersive blow-up for nonlinear Schrödinger equations revisited
Authors:J.L. Bona  G. Ponce  J.-C. Saut  C. Sparber
Affiliation:1. Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago, 851 S. Morgan Street MC249, Chicago, IL 60607-7045, United States;2. Department of Mathematics, University of California, Santa Barbara, CA 93106, United States;3. Laboratoire de Mathématiques, UMR 8628, Université Paris-Sud et CNRS, 91405 Orsay, France
Abstract:The possibility of finite-time, dispersive blow-up for nonlinear equations of Schrödinger type is revisited. This mathematical phenomena is one of the conceivable explanations for oceanic and optical rogue waves. In dimension one, the fact that dispersive blow up does occur for nonlinear Schrödinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey–Stewartson and Gross–Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schrödinger equations in spatial dimensions larger than one and for more general power-law nonlinearities. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel's formula is obtained.
Keywords:35B44   35Q55   76B15   78A60   86A05
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