Oscillatory and regularized shock waves for a modified Serre-Green-Naghdi system

Abstract

The Serre-Green-Naghdi equations of water wave theory have been widely employed to study undular bores. In this study, we introduce a modified Serre-Green-Naghdi system incorporating the effect of an artificial term that results in dispersive and dissipative dynamics. We show that, over sufficiently extended time intervals, effectively approximates the classical Serre-Green-Naghdi equations and admits dispersive-diffusive shock waves as traveling wave solutions. The traveling waves converge to the entropic shock wave solution of the shallow water equations when the dispersion and diffusion approach zero in a moderate dispersion regime. These findings contribute to an understanding of the formation of dispersive shock waves in the classical Serre-Green-Naghdi equations and the effects of diffusion in the generation and propagation of undular bores.

Figures (5)

• Figure 1: A typical potential function $\Phi(z)$
• Figure 2: The phase-space and the corresponding streamlines of the flow for a dissipative dominated flow ($s=2$, $\zeta_l=1$, $w_l=0$, $\varepsilon=0.3$, $\delta=0.02$)
• Figure 3: Amplitudes of undular bores observed in Favre1935, SZ2002 and T1994vs speed $s$ in comparison with the extreme values of the amplitude $A$ as functions of time as estimated by the modified SGN equations
• Figure 4: Oscillatory shock waves in comparison with experimental data of Chanson2010
• Figure 5: Comparison between of oscillatory shock waves of the modified SGN system and a dispersive shock wave of the classical SGN system ($\delta=1/3$)

Theorems & Definitions (25)

• Remark 2.1
• Lemma 2.1
• proof
• Remark 2.2
• Remark 2.3
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4