Tian, Kai; Gao, Binfang; Liu, Q. P.; Chen, Chen On Kupershmidt’s extended equation of dispersive water waves. (English) Zbl 1412.35074 Appl. Math. Lett. 92, 121-127 (2019). Summary: An extended equation of dispersive water waves, proposed by Kupershmidt, is considered. By an ansatz on eigenfunctions of its recursion operator, a linear spectral problem, which turns to be type of the energy dependent Schrödinger, is constructed for the system. As by-products, modified systems are presented. Furthermore, a new bi-Hamiltonian system is obtained and shown to be related to a three component generalization of the Camassa-Holm equation under a Miura-type transformation. Cited in 1 Document MSC: 35G55 Initial value problems for systems of nonlinear higher-order PDEs 35Q35 PDEs in connection with fluid mechanics 35P05 General topics in linear spectral theory for PDEs Keywords:recursion operator; linear spectral problem; Miura-type transformation; Camassa-Holm equation PDFBibTeX XMLCite \textit{K. Tian} et al., Appl. Math. Lett. 92, 121--127 (2019; Zbl 1412.35074) Full Text: DOI References: [1] Kupershmidt, B. A., Extended equations of long waves, Stud. Appl. Math., 116, 4, 415-434 (2006) · Zbl 1145.37328 [2] Tamizhmani, K. M.; Ilangovane, R.; Dubrovin, B., Symmetries and casimir of an extended classical long wave system, Pramana-J. Phys., 80, 4, 559-569 (2013) [3] Kupershmidt, B. A., Mathematics of dispersion water waves, Comm. Math. Phys., 99, 1, 51-73 (1985) · Zbl 1093.37511 [4] Broer, L. J.F., Approximate equations for long water waves, Appl. Sci. Res., 31, 5, 377-395 (1975) · Zbl 0326.76017 [5] Kaup, D. J., A higher-order water-wave equation and the method for solving it, Progr. Theoret. Phys., 54, 2, 396-408 (1975) · Zbl 1079.37514 [6] Leo, R. A.; Mancarella, G.; Soliani, G., On the Broer-Kaup hydrodynamical system, J. Phys. Soc. Japan, 57, 3, 753-756 (1988) [7] Alonso, L. M.; Reus, E. M., Soliton interaction with change of form in the classical Boussinesq system, Phys. Lett. A, 167, 4, 370-376 (1992) [8] Aratyn, H.; Ferreira, L. A.; Gomes, J. F.; Zimerman, A. H., On two-current realization of KP hierarchy, Nuclear Phys. B, 402, 1-2, 85-117 (1993) · Zbl 0941.37525 [9] Bonora, L.; Xiong, C. S., Matrix models without scaling limit, Internat. J. Modern Phys. A, 8, 17, 2973-2992 (1993) · Zbl 0984.81533 [10] Fuchssteiner, B., Application of spectral-gradient methods to nonlinear soliton equations (1980), unpublished, http://fuchssteiner.info/papers/29.pdf [11] Fokas, A. S.; Anderson, R. L., On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems, J. Math. Phys., 23, 6, 1066-1073 (1982) · Zbl 0495.58016 [12] Antonowicz, M.; Fordy, A. P., Coupled KdV equations with multi-Hamiltonian structures, Physica D, 28, 3, 345-357 (1987) · Zbl 0638.35079 [13] Antonowicz, M.; Fordy, A. P., A family of completely integrable multi-Hamiltonian systems, Phys. Lett. A, 122, 2, 95-99 (1987) [14] Antonowicz, M.; Fordy, A. P., Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems, Comm. Math. Phys., 124, 3, 465-486 (1989) · Zbl 0696.35172 [15] Olver, P. J.; Rosenau, P., Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53, 2, 1900-1906 (1996) [16] Chen, M.; Liu, S. Q.; Zhang, Y. J., Hamiltonian structures and their reciprocal transformations for the \(r\)-KdV-CH hierarchy, J. Geom. Phys., 59, 9, 1227-1243 (2009) · Zbl 1172.37318 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.