Model Equations and Traveling Wave Solutions for Shallow-Water Waves with the Coriolis Effect

In the present study, we start by formally deriving the simplified phenomenological models of long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the Earth’s rotation. These new model equations are analogous to the Green–Naghdi equations, the first-order approximations of the KdV-, or BBM type, respectively. We then justify rigorously that in the long-wave limit, unidirectional solutions of a class of KdV- or BBM type are well approximated by the solutions of the Camassa–Holm equation in a rotating setting. The modeling and analysis of those mathematical models then illustrate that the Coriolis forcing in the propagation of shallow-water waves can not be neglected. Indeed, the CH-approximation with the Coriolis effect captures stronger nonlinear effects than the nonlinear dispersive rotational KdV type. Furthermore, we demonstrate nonexistence of the Camassa–Holm-type peaked solution and classify various localized traveling wave solutions to the Camassa–Holm equation with the Coriolis effect depending on the range of the rotation parameter.

[1]  J. Escher,et al.  Wave breaking for nonlinear nonlocal shallow water equations , 1998 .

[2]  Yi Li.  A shallow‐water approximation to the full water wave problem , 2006 .

[3]  J. Bona,et al.  Comparisons between the BBM equation and a Boussinesq system , 2006, Advances in Differential Equations.

[4]  Jonatan Lenells,et al.  Classification of all travelling-wave solutions for some nonlinear dispersive equations , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[5]  B. Fuchssteiner.  The Lie Algebra Structure of Degenerate Hamiltonian and Bi-Hamiltonian Systems , 1982 .

[6]  A. Constantin,et al.  The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations , 2007, 0709.0905.

[7]  J. Moser.  A rapidly convergent iteration method and non-linear partial differential equations - I , 1966 .

[8]  J. Lenells.  Traveling wave solutions of the Camassa-Holm equation , 2005 .

[9]  Athanassios S. Fokas,et al.  Symplectic structures, their B?acklund transformation and hereditary symmetries , 1981 .

[10]  A. Constantin.  Existence of permanent and breaking waves for a shallow water equation: a geometric approach , 2000 .

[11]  Darryl D. Holm,et al.  An integrable shallow water equation with peaked solitons. , 1993, Physical review letters.

[12]  B. Fuchssteiner.  Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation , 1996 .

[13]  Tosio Kato,et al.  Commutator estimates and the euler and navier‐stokes equations , 1988 .

[14]  Peter J. Olver,et al.  Wave-Breaking and Peakons for a Modified Camassa–Holm Equation , 2013 .

[15]  P. M. Naghdi,et al.  A derivation of equations for wave propagation in water of variable depth , 1976, Journal of Fluid Mechanics.

[16]  L. Saint-Raymond,et al.  Chapter 5 - On the influence of the Earth's Rotation on Geophysical Flows , 2007 .

[17]  F. Serre,et al.  CONTRIBUTION À L'ÉTUDE DES ÉCOULEMENTS PERMANENTS ET VARIABLES DANS LES CANAUX , 1953 .

[18]  V. Shrira,et al.  Long Nonlinear Surface and Internal Gravity Waves in a Rotating Ocean , 1998 .

[19]  R. Johnson,et al.  Camassa–Holm, Korteweg–de Vries and related models for water waves , 2002, Journal of Fluid Mechanics.

[20]  A. Constantin,et al.  On the modelling of equatorial waves , 2012 .

[21]  Guilong Gui,et al.  A Nonlocal Shallow-Water Model Arising from the Full Water Waves with the Coriolis Effect , 2018, Journal of Mathematical Fluid Mechanics.

[22]  David Lannes,et al.  Large time existence for 3D water-waves and asymptotics , 2007, math/0702015.

[23]  R. Salmon,et al.  Shallow water equations with a complete Coriolis force and topography , 2005 .

[24]  R. Grimshaw.  Evolution equations for weakly nonlinear, long internal waves in a rotating fluid , 1985 .

[25]  K. A. Semendyayev,et al.  Handbook of mathematics , 1985 .

[26]  Darryl D. Holm,et al.  A New Integrable Shallow Water Equation , 1994 .

[27]  Guilong Gui,et al.  On a shallow-water approximation to the Green–Naghdi equations with the Coriolis effect , 2018, Advances in Mathematics.