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Natural Hazards and Earth System Sciences An interactive open-access journal of the European Geosciences Union
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Volume 14, issue 6 | Copyright

Special issue: Extreme seas and ship operations

Nat. Hazards Earth Syst. Sci., 14, 1431-1440, 2014
© Author(s) 2014. This work is distributed under
the Creative Commons Attribution 3.0 License.

Research article 06 Jun 2014

Research article | 06 Jun 2014

Numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger equation

A. Calini1 and C. M. Schober2 A. Calini and C. M. Schober
  • 1Department of Mathematics, College of Charleston, Charleston, SC, USA
  • 2Department of Mathematics, University of Central Florida, Orlando, FL, USA

Abstract. In this article we conduct a broad numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger (NLS) equation, a widely used model of rogue wave generation and dynamics in deep water. NLS breathers rising over an unstable background state are frequently used to model rogue waves. However, the issue of whether these solutions are robust with respect to the kind of random perturbations occurring in physical settings and laboratory experiments has just recently begun to be addressed. Numerical experiments for spatially periodic breathers with one or two modes involving large ensembles of perturbed initial data for six typical random perturbations suggest interesting conclusions. Breathers over an unstable background with N unstable modes are generally unstable to small perturbations in the initial data unless they are "maximal breathers" (i.e., they have N spatial modes). Additionally, among the maximal breathers with two spatial modes, the one of highest amplitude due to coalescence of the modes appears to be the most robust. The numerical observations support and extend to more realistic settings the results of our previous stability analysis, which we hope will provide a useful tool for identifying physically realizable wave forms in experimental and observational studies of rogue waves.

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