Solitary wave-mean flow interaction in strongly nonlineardispersive shallow water waves

Abstract

The interaction of a solitary wave and a slowly varying mean background or flow for the Serre-Green-Naghdi (SGN) equations is studied using Whitham modulation theory. The exact form of the three SGN-Whitham modulation equations -- two for the mean horizontal velocity and depth decoupled from one for the solitary wave amplitude field -- are obtained exactly in the solitary wave limit. Although the three equations are not diagonalizable, the restriction of the full system to simple waves for the mean equations is diagonalized in terms of Riemann invariants. The Riemann invariants are used to analytically describe the head-on and overtaking interactions of a solitary wave with a rarefaction wave and dispersive shock wave (DSW), leading to scenarios of solitary wave trapping or transmission by the mean flow. The analytical results for overtaking interactions prove that a simpler, approximate approach based on the DSW fitting method is accurate to the second order in solitary wave amplitude, beyond the first order accurate Korteweg-de Vries approximation. The analytical results also accurately predict the SGN DSW's solitary wave edge amplitude and speed. The analytical results are favourably compared with corresponding numerical solutions of the full SGN equations. Because the SGN equations model the bi-directional propagation of strongly nonlinear, long gravity waves over a flat bottom, the analysis presented here describes large amplitude solitary wave-mean flow interactions in shallow water waves.

References in corpus (19)

• Dispersive shock waves and modulation theory
• Resolution of a shock in hyperbolic systems modified by weak dispersion
• Unsteady undular bores in fully nonlinear shallow-water theory
• The theory of optical dispersive shock waves in photorefractive media
• Dispersive Shock Waves in Viscously Deformable Media
• Solitonic dispersive hydrodynamics: theory and observation
• Hydrodynamic Optical Soliton Tunneling
• Shock waves in dispersive Eulerian fluids
• Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory
• Soliton-mean field interaction in Korteweg-de Vries dispersive hydrodynamics
• A Kinematic Conservation Law in Free Surface Flow
• Soliton trapping, transmission and wake in modulationally unstable media
• Dispersive Riemann problem for the Benjamin-Bona-Mahony equation
• Motion of dark solitons in a non-uniform flow of Bose-Einstein condensate
• Solitary wave fission of a large disturbance in a viscous fluid conduit
• Solitons, the Korteweg-de Vries equation with step boundary values and pseudo-embedded eigenvalues
• Oblique interactions between solitons and mean flows in the Kadomtsev-Petviashvili equation
• Propagation of generalized Korteweg-de Vries solitons along large-scale waves
• Observation of Traveling Breathers and Their Scattering in a Two-Fluid System