Hyperbolicity of the Modulation Equations for the Serre–Green–Naghdi Model

Abstract

Serre-Green-Naghdi equations (SGN equations) is the most simple dispersive model of long water waves having "good" mathematical and physical properties. First, the model is a mathematically justified approximation of the exact water wave problem. Second, the SGN equations are the Euler-Lagrange equations coming from Hamilton's principle of stationary action with a natural approximate Lagrangian. Finally, the equations are Galilean invariant which is necessary for physically relevant mathematical models.We have derived the modulation equations to the SGN model and show that they are strictly hyperbolic for any wave amplitude, i.e., the periodic wave trains are modulationally stable. Numerical tests for the full SGN equations are shown. The results confirm the modulational stability analysis.

Stationary shock-like transition fronts in dispersive systems

“…In the case of SGN system, the Whitham's approach can be shown to produce a hyperbolic problem [60]. The corresponding higher dimensional system generates singularities and we conjecture that at least some of these singularities can be interpreted as the limits of the dispersive shock-like fronts studied in this paper.…”

“…To the best of our knowledge, the emergence in the zero dispersion limit of such generalized shocks, linking homogeneous configurations with measure-valued infinitely fine dynamic mixtures, has not been reported before. To understand stability of such shocks it is necessary to study the associated higher order hyperbolic (Whitham) system [60].…”

Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart

“…The literature on the subject is vast, and we refer to [41] for an overview and references in the case of the Korteweg-de Vries (KdV) equation as a perturbation of the inviscid Burgers (iB) equation, where a complete asymptotic description is available, and to [32] for an introduction to the modulation theory for more general equations and an extensive list of references. Let us also specifically mention [31,75] for a description of the Whitham modulation theory in the case of the SGN equations, and [56,21,65,64,67,39] for some numerical experiments. In contrast to these works, we do not consider here steplike initial data, but study the appearance of modulated oscillations from smooth rapidly decaying initial data, from which solitary wave resolution is the expected large-time asymptotic behavior.…”

Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart

“…The literature on the subject is vast, and we refer to [35] for an overview and references in the case of the Korteweg-de Vries (KdV) equation as a perturbation of the inviscid Burgers (iB) equation, where a complete asymptotic description is available. Let us also mention [26,66] for a description of the Whitham modulation theory in the case of the SGN equations, and [18,57,58] for some numerical experiments. In contrast to [57] we do not consider here steplike initial data, but study the appearence of DSWs from smooth rapidly decaying initial data.…”