Higher-dimensional extended shallow water equations and resonant soliton radiation

Theodoros P. Horikis, Dimitrios J. Frantzeskakis, Timothy R. Marchant, and Noel F. Smyth
Phys. Rev. Fluids 6, 104401 – Published 25 October 2021

Abstract

The higher order corrections to the equations that describe nonlinear wave motion in shallow water are derived from the water wave equations. In particular, the extended cylindrical Korteweg–de Vries and Kadomtsev-Petviashvili equations—which include higher order nonlinear, dispersive, and nonlocal terms—are derived from the Euler system in (2+1) dimensions, using asymptotic expansions. It is thus found that the nonlocal terms are inherent only to the higher dimensional problem, both in cylindrical and Cartesian geometry. Asymptotic theory is used to study the resonant radiation generated by solitary waves governed by the extended equations, with an excellent comparison obtained between the theoretical predictions for the resonant radiation amplitude and the numerical solutions. In addition, resonant dispersive shock waves (undular bores) governed by the extended equations are studied. It is shown that the asymptotic theory, applicable for solitary waves, also provides an accurate estimate of the resonant radiation amplitude of the undular bore.

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  • Received 12 July 2021
  • Accepted 7 October 2021

DOI:https://doi.org/10.1103/PhysRevFluids.6.104401

©2021 American Physical Society

Physics Subject Headings (PhySH)

  1. Research Areas
  1. Physical Systems
Fluid DynamicsNonlinear Dynamics

Authors & Affiliations

Theodoros P. Horikis1, Dimitrios J. Frantzeskakis2, Timothy R. Marchant3,4, and Noel F. Smyth5,3

  • 1Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
  • 2Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
  • 3School of Mathematics and Applied Statistics, University of Wollongong, Northfields Avenue, Wollongong, 2522 New South Wales, Australia
  • 4Australian Mathematical Sciences Institute, University of Melbourne, Melbourne, 3052 Victoria, Australia
  • 5School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, Scotland, United Kingdom

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Issue

Vol. 6, Iss. 10 — October 2021

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