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Dispersive blow-up II. Schrödinger-type equations, optical and oceanic rogue waves

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Abstract

Addressed here is the occurrence of point singularities which owe to the focusing of short or long waves, a phenomenon labeled dispersive blow-up. The context of this investigation is linear and nonlinear, strongly dispersive equations or systems of equations. The present essay deals with linear and nonlinear Schrödinger equations, a class of fractional order Schrödinger equations and the linearized water wave equations, with and without surface tension. Commentary about how the results may bear upon the formation of rogue waves in fluid and optical environments is also included.

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Author information

Correspondence to Jerry L. Bona.

Additional information

Dedicated to Professor Roger Temam with Friendship and Admiration

Project supported by the Agence Nationale de la Recherche, France (No. ANR-07-BLAN-0250), the University of Illinois at Chicago, the Wolfgang Pauli Institute in Vienna, the University of Illinois at Chicago and the Université de Paris 11.

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Bona, J.L., Saut, J. Dispersive blow-up II. Schrödinger-type equations, optical and oceanic rogue waves. Chin. Ann. Math. Ser. B 31, 793–818 (2010). https://doi.org/10.1007/s11401-010-0617-0

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Keywords

  • Rogue waves
  • Dispersive blow-up
  • Nonlinear dispersive equations
  • Nonlinear Schrödinger equation
  • Water wave equations
  • Propagation in optical cables
  • Weak turbulence models

2000 MR Subject Classification

  • 35Q35
  • 35Q51
  • 35Q53
  • 35Q55
  • 35Q60
  • 35Q86
  • 76B03
  • 76B15
  • 76B45
  • 76F99
  • 78A60