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Explicit solutions of the reduced Ostrovsky equation. (English) Zbl 1138.35399
Summary: It is shown that the Vakhnenko equation (VE) and the Ostrovsky-Hunter equation (OHE) are particular forms of the reduced Ostrovsky equation, and that they are related by a simple transformation. Explicit analytic periodic and solitary travelling-wave solutions of the OHE are derived by using a method exploited previously by Vakhnenko and the present author to solve the VE. These exact solutions of the OHE are related to some approximate solutions obtained by {\it J. P. Boyd} [Eur. J. Appl. Math. 16, No. 1, 65--81 (2005; Zbl 1079.35087)].

35Q53KdV-like (Korteweg-de Vries) equations
35C05Solutions of PDE in closed form
35B10Periodic solutions of PDE
Full Text: DOI
[1] Ostrovsky, L. A.: Nonlinear internal waves in a rotating ocean. Oceanology 18, 119-125 (1978)
[2] Stepanyants, Y. A.: On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons. Chaos, solitons & fractals 28, 193-204 (2006) · Zbl 1088.35531
[3] Grimshaw, R. H. J.; Ostrovsky, L. A.; Shrira, V. I.; Stepanyants, Y. A.: Long nonlinear surface and internal gravity waves in a rotating ocean. Surv geophys 19, 289-338 (1998)
[4] Boyd, J. P.: Microbreaking and polycnoidal waves in the Ostrovsky-Hunter equation. Phys lett A 338, 36-43 (2005) · Zbl 1136.74330
[5] Boyd, J. P.: Ostrovsky and Hunter’s generic wave equation for weakly dispersive waves: matched asymptotic and pseudospectral study of the paraboidal travelling waves (corner and near-corner waves). Eur J appl math 15, 1-17 (2005) · Zbl 1079.35087
[6] Parkes, E. J.: The stability of solutions of Vakhnenko’s equation. J phys A math gen 26, 6469-6475 (1993) · Zbl 0809.35086
[7] Vakhnenko, V. A.: High-frequency soliton-like waves in a relaxing medium. J math phys 40, 2011-2020 (1999) · Zbl 0946.35094
[8] Vakhnenko, V. A.: Solitons in a nonlinear model medium. J phys A math gen 25, 4181-4187 (1992) · Zbl 0754.35132
[9] Vakhnenko, V. O.; Parkes, E. J.: Periodic and solitary-wave solutions of the Degasperis-Procesi equation. Chaos, solitons & fractals 20, 1059-1073 (2004) · Zbl 1049.35162
[10] Vakhnenko, V. O.; Parkes, E. J.: The two loop soliton solution of the Vakhnenko equation. Nonlinearity 11, 1457-1464 (1998) · Zbl 0914.35115
[11] Morrison, A. J.; Parkes, E. J.; Vakhnenko, V. O.: The N loop soliton solution of the Vakhnenko equation. Nonlinearity 12, 1427-1437 (1999) · Zbl 0935.35129
[12] Vakhnenko, V. O.; Parkes, E. J.; Michtchenko, A. V.: The Vakhnenko equation from the viewpoint of the inverse scattering method for the KdV equation. Int J diff eqns appl 1, 429-449 (2000)
[13] Vakhnenko, V. O.; Parkes, E. J.: The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method. Chaos, solitons & fractals 13, 1819-1826 (2002) · Zbl 1067.37106
[14] Byrd, P. F.; Friedman, M. D.: Handbook of elliptic integrals for engineers and scientists. (1971) · Zbl 0213.16602
[15] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions. (1972) · Zbl 0543.33001
[16] Boyd, J. P.: A Legendre-pseudospectral method for computing travelling waves with corners (slope discontinuities) in one space dimension with application to Whitham’s equation family. J comput phys 189, 98-110 (2003) · Zbl 1027.65136
[17] Boyd, J. P.: Near-corner waves of the Camassa-Holm equation. Phys lett A 336, 342-348 (2005) · Zbl 1136.35445
[18] Boyd, J. P.: The cnoidal wave/corner wave/breaking wave scenario: a one-sided infinite-dimension bifurcation. Math comput simul 69, 235-242 (2005) · Zbl 1070.35027
[19] Turnbull, H. W.: Theory of equations. (1952)