Hamiltonian description and traveling waves of the spatial Dysthe equations

Francesco Fedele (1) , Denys Dutykh (2)

1 School of Civil and Environmental Engineering [Atlanta]
2 LAMA - Laboratoire de Mathématiques

The spatial version of the fourth-order Dysthe equations describe the evolution of weakly nonlinear narrowband wave trains in deep waters. For unidirectional waves, the hidden Hamiltonian structure and new invariants are unveiled by means of a gauge transformation to a new canonical form of the evolution equations. A highly accurate Fourier-type spectral scheme is developed to solve for the equations and validate the new conservation laws, which are satisfied up to machine precision. Further, traveling waves are numerically investigated using the Petviashvili method. It is found that their collision appears inelastic, suggesting the non-integrability of the Dysthe equations.

Mots clés: Dysthe equation; deep water; Hamiltonian; ground state; traveling wave; solitary wave

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

Francesco Fedele, Denys Dutykh. Hamiltonian description and traveling waves of the spatial Dysthe equations. 2012. ⟨hal-00632862v3⟩