# Hyperbolicity of the modulation equations for the Serre-Green-Naghdi model

## TL;DR

This paper derives modulation equations for the Serre-Green-Naghdi water wave model and demonstrates their strict hyperbolicity, indicating stable wave train behavior, supported by numerical tests.

## Contribution

The paper introduces the derivation of modulation equations for the SGN model and proves their strict hyperbolicity for all wave amplitudes, confirming modulational stability.

## Key findings

- Modulation equations for SGN are strictly hyperbolic at all amplitudes.
- Numerical tests confirm the modulational stability of wave trains.
- SGN equations are a justified approximation of water wave dynamics.

## 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.