Henry, David; Ivanov, Rossen One-dimensional weakly nonlinear model equations for Rossby waves. (English) Zbl 1292.35230 Discrete Contin. Dyn. Syst. 34, No. 8, 3025-3034 (2014). Summary: In this study we explore several possibilities for modelling weakly nonlinear Rossby waves in fluid of constant depth, which propagate predominantly in one direction. The model equations obtained include the BBM equation, as well as the integrable KdV and Degasperis-Procesi equations. Cited in 10 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 35Q51 Soliton equations 35Q53 KdV equations (Korteweg-de Vries equations) 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Keywords:Rossby waves; KdV equation; BBM equation; Degasperis-Procesi equation; Camassa-Holm equation; nonlinear waves; long waves; solitons PDFBibTeX XMLCite \textit{D. Henry} and \textit{R. Ivanov}, Discrete Contin. Dyn. Syst. 34, No. 8, 3025--3034 (2014; Zbl 1292.35230) Full Text: DOI arXiv References: [1] B. Alvarez-Samaniego, Large time existence for 3D water-waves and asymptotics,, Invent. Math., 171, 485 (2008) · Zbl 1131.76012 [2] T. B. Benjamin, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Roy. Soc. London Ser. A, 272, 47 (1972) · Zbl 0229.35013 [3] J. L. Bona, A comparison of solutions of two model equations for lonf waves,, Lectures in Applied Mathematics, 20, 235 (1983) [4] A. Boutet de Monvel, Long-time asymptotics for the Camassa-Holm equation,, SIAM J. Math. Anal., 41, 1559 (2009) · Zbl 1204.37073 [5] A. Boutet de Monvel, A Riemann-Hilbert approach for the Degasperis-Procesi equation,, Nonlinearity, 26, 2081 (2013) · Zbl 1291.35326 [6] J. P. Boyd, Equatorial solitary waves. Part I: Rossby solitons,, Journal of Physical Oceanography, 10, 1699 (1980) [7] B. Buffoni, <em>Analytic Theory of Global Bifurcation. An Introduction</em>,, Princeton University Press (2003) · Zbl 1021.47044 [8] R. Camassa, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71, 1661 (1993) · Zbl 0972.35521 [9] A. Constantin, Some three-dimensional nonlinear equatorial flows,, J. Phys. Ocean., 43, 165 (2013) [10] A. Constantin, On equatorial wind waves,, Differential and Integral Equations, 26, 237 (2013) · Zbl 1289.86002 [11] A. Constantin, An exact solution for equatorially trapped waves,, J. Geophys. Res., 117 (2012) [12] A. Constantin, On the modelling of equatorial waves,, Geophys. Res. Lett., 39 (2012) [13] A. Constantin, <em>Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis</em>,, SIAM (2011) · Zbl 1266.76002 [14] A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166, 523 (2006) · Zbl 1108.76013 [15] A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005) · Zbl 1076.35109 [16] A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. Roy. Soc. London A, 457, 953 (2001) · Zbl 0999.35065 [17] A. Constantin, Particle trajectories in solitary water wave,, Bull. Amer. Math. Soc., 44, 423 (2007) · Zbl 1126.76012 [18] A. Constantin, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Mathematica, 181, 229 (1998) · Zbl 0923.76025 [19] A. Constantin, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22, 2197 (2006) · Zbl 1105.37044 [20] A. Constantin, Inverse scattering transform for the Degasperis-Procesi equation,, Nonlinearity, 23, 2559 (2010) · Zbl 1211.37081 [21] A. Constantin, On geodesic exponential maps of the Virasoro group,, Ann. Global Anal. Geom., 31, 155 (2007) · Zbl 1121.35111 [22] A. Constantin, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192, 165 (2009) · Zbl 1169.76010 [23] A. Constantin, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52, 949 (1999) · Zbl 0940.35177 [24] A. Constantin, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57, 481 (2004) · Zbl 1038.76011 [25] A. Constantin, Stability of peakons,, Comm. Pure Appl. Math., 53, 603 (2000) · Zbl 1049.35149 [26] B. Cushman-Roisin, <em>Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects</em>,, Academic (2011) · Zbl 1319.86001 [27] A. Degasperis, A new integrable equation with peakon solutions,, Theoretical and Mathematical Physics, 133, 1461 (2002) [28] A. Degasperis, Asymptotic integrability,, in Symmetry and Perturbation Theory (eds. A. Degasperis and G. Gaeta), 23 (1999) · Zbl 0963.35167 [29] J. Escher, The Degasperis-Procesi equation as a non-metric Euler equation,, Math. Z., 269, 1137 (2011) · Zbl 1234.35220 [30] J. Escher, Global weak solutions blow-up structure for the Degasperis-Procesi equation,, J. Funct. Anal., 241, 457 (2006) · Zbl 1126.35053 [31] A. V. Fedorov, Equatorial waves,, in Encyclopedia of Ocean Sciences (ed. J. Steele), 3679 (2009) [32] C. Gardner, Method for solving the Korteweg-de Vries equation,, Phys. Rev. Letters, 19, 1095 (1967) · Zbl 1061.35520 [33] G. A. Gottwald, The Zakharov-Kuznetsov equation as a two-dimensional model for nonlinear Rossby waves,, preprint [34] D. Henry, An exact solution for equatorial geophysical water waves with an underlying current,, Eur. J. Mech. B Fluids, 38, 18 (2013) · Zbl 1297.86002 [35] D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlinear Anal., 70, 1565 (2009) · Zbl 1170.35509 [36] D. Henry, Compactly supported solutions of a family of nonlinear partial differential equations,, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 15, 145 (2008) · Zbl 1165.35308 [37] D. Henry, Infinite propagation speed for the Degasperis-Procesi equation,, J. Math. Anal. Appl., 311, 755 (2005) · Zbl 1094.35099 [38] D. Henry, Compactly supported solutions of the Camassa-Holm equation,, J. Nonlinear Math. Phys., 12, 342 (2005) · Zbl 1086.35079 [39] D. Holm, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137, 1 (1998) · Zbl 0951.37020 [40] D. Holm, Smooth and peaked solitons of the Camassa-Holm equation and applications,, J. of Geometry and Symmetry in Physics, 22, 13 (2011) · Zbl 1231.35167 [41] R. Ivanov, Water waves and integrability,, Philos. Trans. Roy. Soc.: Ser. A., 365, 2267 (2007) · Zbl 1152.76322 [42] R. S. Johnson, <em>A Modern Introduction to the Mathematical Theory of Water Waves</em>,, Cambridge University Press (1997) · Zbl 0892.76001 [43] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455, 63 (2002) · Zbl 1037.76006 [44] B. Kolev, Some geometric investigations on the Degasperis-Procesi shallow water equation,, Wave Motion, 46, 412 (2009) · Zbl 1231.76044 [45] D. J. Korteweg, On the change of form of long waves advancing in a rectangularchannel, an on a new type of long stationary waves,, Philos. Mag., 39, 422 (1895) · JFM 26.0881.02 [46] Z. Lin, Stability of peakons for the Degasperis-Procesi equation,, Comm. Pure Appl. Math., 62, 125 (2009) · Zbl 1165.35045 [47] A. V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach,, J. Phys. A, 45 (2012) · Zbl 1339.86001 [48] O. Mustafa, A note on the Degasperis-Procesi equation,, J. Nonlinear Math. Phys., 12, 10 (2005) · Zbl 1067.35078 [49] G. W. Owen, Scattering of barotropic Rossby waves by the Antarctic Circumpolar Current,, J. Geophys. Res., 111 (2006) [50] J. Pedlosky, <em>Geophysical Fluid Dynamics</em>,, Springer-Verlag (1979) · Zbl 0429.76001 [51] P. B. Rhines, Lectures in Geophysical Fluid Dynamics,, Lectures in Applied Mathematics, 20, 3 (1983) · Zbl 0543.76058 [52] G. B. Whitham, Variational methods and applications to water waves,, Proc. R. Soc. Lond. A, 299, 6 (1967) · Zbl 0163.21104 [53] G. B. Whitham, <em>Linear and Nonlinear Waves</em>,, Wiley (1974) · Zbl 0373.76001 [54] V. E. Zakharov, <em>Theory of Solitons: The Inverse Scattering Method</em>,, Plenum (1984) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.