Camassa, R.; Zenchuk, A. I. On the initial value problem for a completely integrable shallow water wave equation. (English) Zbl 1002.76012 Phys. Lett., A 281, No. 1, 26-33 (2001). Summary: We present an algorithm for solving the initial value problem on the infinite line for a wave equation arising in the study of long waves at the free surface of water. The algorithm exploits a link between the hierarchy of shallow water equations and that of the modified Korteweg-de Vries equations. Cited in 8 Documents MSC: 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 35Q53 KdV equations (Korteweg-de Vries equations) 35Q35 PDEs in connection with fluid mechanics Keywords:long waves; free water surface; modified Korteweg-de Vries equations; initial value problem; wave equation; shallow water equations PDF BibTeX XML Cite \textit{R. Camassa} and \textit{A. I. Zenchuk}, Phys. Lett., A 281, No. 1, 26--33 (2001; Zbl 1002.76012) Full Text: DOI References: [1] Benjamin, T. B.; Bona, J. L.; Mahoney, J. J., Philos. Trans. R. Soc. London A, 227, 47 (1972) [2] Camassa, R.; Holm, D. D., Phys. Rev. Lett., 71, 1661 (1993) [3] Fuchssteiner, B., Physica D, 95, 229 (1996) [4] Fuchssteiner, B., Prog. Theor. Phys., 78, 1082 (1982) [5] Fuchssteiner, B.; Fokas, A. S., Physica D, 4, 47 (1981) [6] Fokas, A. S., Physica D, 87, 145 (1995) [7] Alber, M. S.; Camassa, R.; Holm, D. D.; Marsden, J. E., Lett. Math. Phys., 32, 137 (1994) [8] Li, Y. A.; Olver, P. J., Discrete Cont. Dynam. Syst., 3, 419 (1997) [9] Li, Y. A.; Olver, P. J., Discrete Cont. Dynam. Syst., 4, 159 (1998) [10] Zenchuk, A., JETP Lett., 68, 715 (1998) [11] Zakharov, V. E.; Shabat, A. B., Sov. Phys. JETP, 37, 823 (1973) [12] Camassa, R., (Boiti, M.; etal., Nonlinearity, Integrability and All That: Twenty Years After NEEDS ’79 (2000), World Scientific: World Scientific Singapore) [13] Kraenkel, R. A.; Zenchuk, A. I., J. Phys. A: Math. Gen., 32, 4733 (1999) [14] Whitham, G. B., Linear and Nonlinear Waves (1974), Wiley: Wiley New York · Zbl 0373.76001 [15] Zakharov, V. E.; Manakov, S. V.; Novikov, S. P.; Pitaevsky, L. P., Theory of Solitons. The Inverse Problem Method (1984), Plenum Press: Plenum Press New York [16] Ablowitz, M. J.; Segur, H., Solitons and Inverse Scattering Transform (1981), SIAM: SIAM Philadelphia · Zbl 0299.35076 [17] Wadati, M., J. Phys. Soc. Jpn., 32, 1681 (1972) 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.