Oscillatory and regularized shock waves for a dissipative Peregrine–Boussinesq system

• Published in: IMA journal of applied mathematics Vol. 88; no. 4; pp. 602 - 631
• Main Authors: Brudvik-Lindner, Larkspur, Mitsotakis, Dimitrios, Tzavaras, Athanasios E
• Format: Journal Article
• Language: English
• Published: Oxford University Press 28.12.2023
• Subjects: positive surge; diffusive-dispersive shock wave; 35Q35; 92C35; Boussinesq system; existence; undular bore; convergence 2000 Mathematics Subject Classification; 35C07
• ISSN: 0272-4960; 1464-3634
• DOI: 10.1093/imamat/hxad030

Abstract We consider a dissipative, dispersive system of the Boussinesq type, which describes wave phenomena in scenarios where dissipation plays a significant role. Examples include undular bores in rivers or oceans, where turbulence-induced dissipation significantly influences their behavior. In this study, we demonstrate that the proposed system admits traveling wave solutions known as diffusive-dispersive shock waves. These solutions can be categorized as oscillatory and regularized shock waves, depending on the interplay between dispersion and dissipation effects. By comparing numerically computed solutions with laboratory data, we observe that the proposed model accurately captures the behavior of undular bores over a broad range of phase speeds. Traditionally, undular bores have been approximated using the original Peregrine system, which, even though it doesn’t possess these as traveling wave solutions, tends to offer accurate approximations within suitable time scales. To shed light on this phenomenon, we demonstrate that the discrepancy between the solutions of the dissipative Peregrine system and the non-dissipative counterpart is proportional to the product of the dissipation coefficient and the observation time.