Some special solutions to the Hyperbolic NLS equation

Laurent Vuillon (1) , Denys Dutykh (1, 2, 3) , Francesco Fedele (4)

1 LAMA - Laboratoire de Mathématiques
2 School of Mathematical Sciences [Dublin]
3 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
4 ECE GeorgiaTech - School of Electrical and Computer Engineering - Georgia Insitute of Technology

The Hyperbolic Nonlinear Schrodinger equation (HypNLS) arises as a model for the dynamics of three-dimensional narrowband deep water gravity waves. In this study, the Petviashvili method is exploited to numerically compute bi-periodic time-harmonic solutions of the HypNLS equation. In physical space they represent non-localized standing waves. Non-trivial spatial patterns are revealed and an attempt is made to describe them using symbolic dynamics and the language of substitutions. Finally, the dynamics of a slightly perturbed standing wave is numerically investigated by means a highly acccurate Fourier solver.

Mots clés: deep water waves; wave patterns; Hyperbolic equations; NLS equation

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Equations aux dérivées partielles [math.AP]; Formation de Structures et Solitons [nlin.PS]; Dynamique des Fluides [physics.flu-dyn]; Physique Atmosphérique et Océanique [physics.ao-ph]; Physique Numérique [physics.comp-ph]; Systèmes Solubles et Intégrables [nlin.SI]; Géophysique [physics.geo-ph]; Géophysique [physics.geo-ph]; Milieux et Changements globaux; Analyse numérique [math.NA]

Laurent Vuillon, Denys Dutykh, Francesco Fedele. Some special solutions to the Hyperbolic NLS equation. Communications in Nonlinear Science and Numerical Simulation, 2018, 57, pp.202-220. ⟨10.1016/j.cnsns.2017.09.018⟩. ⟨hal-00846801v3⟩