Journal Article

A nonlinear Gibbs-type phenomenon for the defocusing nonlinear Schrödinger equation

Jeffery C. DiFranco and Kenneth T.-R. McLaughlin

in International Mathematics Research Papers

Volume 2005, issue 8, pages 403-459
Published in print January 2005 | ISSN: 1687-3017
Published online January 2005 | e-ISSN: 1687-3009 | DOI: http://dx.doi.org/10.1155/IMRP.2005.403
A nonlinear Gibbs-type phenomenon for the defocusing nonlinear Schrödinger equation

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We analyze the Cauchy problem for the defocusing nonlinear Schrödinger equation with the particular initial potential, [math], that is zero for [math] and 1 otherwise. For t > 0, the jump discontinuities of [math] are regularized by the onset of rapid oscillations. The behavior of these oscillations is realized as a nonlinear Gibbs Phenomenon. We prove that when t is small, these oscillations can be described by an asymptotic expansion in powers of t1/2 where the leading order term is given by an explicit special function. The analysis relies on the fact that the solutions to Cauchy problems for the defocussing nonlinear Schrödinger equation can be obtained via an associated Riemann-Hilbert problem. We analyze the solution to this associated Riemann-Hilbert problem following the steepest-descent/ stationary-phase method for oscillatory Riemann-Hilbert problems introduced by Deift and Zhou in 1993, recognizing the new distinct feature that the solution depends on the behavior of the reflection coefficient at infinity rather than at a finite point or on an interval.

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Subjects: Mathematics

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