On the nonlinear dynamics of the traveling-wave solutions of the Serre equations

Dimitrios Mitsotakis (1, 2) , Denys Dutykh (3) , John D. Carter (4)

1 UC Merced - University of California [Merced]
2 School of Natural Sciences
3 EDPs2
4 Mathematics Department

In this paper, we study numerically nonlinear phenomena related to the dynamics of the traveling wave solutions of the Serre equations including their stability, their persistence, resolution into solitary waves, and wave breaking. Other forms of solutions such as DSWs, are also considered. Some differences between the solutions of the Serre equations and the full Euler equations are also studied. Euler solitary waves propagate without large variations in shape when they are used as initial conditions in the Serre equations. The nonlinearities seem to play a crucial role in the generation of small-amplitude waves and appear to cause a recurrence phenomenon in linearly unstable solutions. The numerical method used in the paper utilizes a high order FEM with smooth, periodic splines in space and explicit Runge-Kutta methods in time. The solutions of the Serre system are compared with the corresponding ones of the asymptotically-related Euler system whenever is possible.

Mots clés: Finite element methods; solitary waves; cnoidal waves; Green--Naghdi system; stability

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

Dimitrios Mitsotakis, Denys Dutykh, John D. Carter. On the nonlinear dynamics of the traveling-wave solutions of the Serre equations. 2014. ⟨hal-00984035v2⟩