Math @ Duke

Publications [#244171] of Stephanos Venakides
Papers Published
 Buckingham, R; Venakides, S, Longtime asymptotics of the nonlinear Schrödinger equation shock problem,
Communications on Pure & Applied Mathematics, vol. 60 no. 9
(2007),
pp. 13491414, ISSN 00103640 [MR2337507], [doi]
(last updated on 2017/12/18)
Abstract: The longtime asymptotics of two colliding plane waves governed by the focusing nonlinear Schrödinger equation are analyzed via the inverse scattering method. We find three asymptotic regions in spacetime: a region with the original wave modified by a phase perturbation, a residual region with a onephase wave, and an intermediate transition region with a modulated twophase wave. The leadingorder terms for the three regions are computed with error estimates using the steepestdescent method for RiemannHilbert problems. The nondecaying initial data requires a new adaptation of this method. A new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions. Also, the effect of the collision is felt in the planewave state well beyond the shock front at large times. © 2007 Wiley Periodicals, Inc.


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