zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Higher-order equations of the KdV type are integrable. (English) Zbl 1206.35220
Summary: We show that a nonlinear equation that represents third-order approximation of long wavelength, small amplitude waves of inviscid and incompressible fluids is integrable for a particular choice of its parameters, since in this case it is equivalent to an integrable equation which has recently appeared in the literature. We also discuss the integrability of both second- and third-order approximations of additional cases.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35Q35PDEs in connection with fluid mechanics
37K05Hamiltonian structures, symmetries, variational principles, conservation laws
WorldCat.org
Full Text: DOI EuDML
References:
[1] G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1974. · Zbl 0373.76001
[2] D. J. Korteweg and G. de Vries, “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary Waves,” Philosophical Magazine, vol. 39, pp. 422-443, 1895. · Zbl 26.0881.02
[3] N. J. Zabusky and M. D. Kruskal, “Interaction of “solitons” in a collisionless plasma and the recurrence of initial states,” Physical Review Letters, vol. 15, no. 6, pp. 240-243, 1965. · Zbl 1201.35174 · doi:10.1103/PhysRevLett.15.240 · http://staff.ustc.edu.cn/~jzheng/PhysRevLett_15_0240.pdf
[4] P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Communications on Pure and Applied Mathematics, vol. 21, no. 5, pp. 467-490, 1968. · Zbl 0162.41103 · doi:10.1002/cpa.3160210503
[5] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-de Vries equation,” Physical Review Letters, vol. 19, no. 19, pp. 1095-1097, 1967. · Zbl 1103.35360 · doi:10.1103/PhysRevLett.19.1095
[6] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, vol. 4 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1981. · Zbl 0472.35002
[7] A. S. Fokas, “On a class of physically important integrable equations,” Physica D, vol. 87, no. 1-4, pp. 145-150, 1995. · Zbl 1194.35363 · doi:10.1016/0167-2789(95)00133-O
[8] A. S. Fokas and Q. M. Liu, “Asymptotic integrability of water waves,” Physical Review Letters, vol. 77, no. 12, pp. 2347-2351, 1996. · Zbl 0982.76511 · doi:10.1103/PhysRevLett.77.2347
[9] V. Marinakis and T. C. Bountis, “On the integrability of a new class of water wave equations,” in Proceedings of the Conference on Nonlinear Coherent Structures in Physics and Biology, D. B. Duncan and J. C. Eilbeck, Eds., Heriot-Watt University, Edinburgh, UK, July 1995. · Zbl 1084.76511
[10] V. Marinakis and T. C. Bountis, “Special solutions of a new class of water wave equations,” Communications in Applied Analysis, vol. 4, no. 3, pp. 433-445, 2000. · Zbl 1084.76511
[11] E. Tzirtzilakis, M. Xenos, V. Marinakis, and T. C. Bountis, “Interactions and stability of solitary waves in shallow water,” Chaos, Solitons & Fractals, vol. 14, no. 1, pp. 87-95, 2002. · Zbl 1068.76011 · doi:10.1016/S0960-0779(01)00211-9
[12] E. Tzirtzilakis, V. Marinakis, C. Apokis, and T. Bountis, “Soliton-like solutions of higher order wave equations of the Korteweg-de Vries type,” Journal of Mathematical Physics, vol. 43, no. 12, pp. 6151-6165, 2002. · Zbl 1060.35127 · doi:10.1063/1.1514387
[13] S. A. Khuri, “Soliton and periodic solutions for higher order wave equations of KdV type. I,” Chaos, Solitons & Fractals, vol. 26, no. 1, pp. 25-32, 2005. · Zbl 1070.35062 · doi:10.1016/j.chaos.2004.12.027
[14] W.-P. Hong, “Dynamics of solitary-waves in the higher order Korteweg-de Vries equation type (I),” Zeitschrift für Naturforschung, vol. 60, no. 11-12, pp. 757-767, 2005.
[15] J. Li, W. Rui, Y. Long, and B. He, “Travelling wave solutions for higher-order wave equations of KdV type. III,” Mathematical Biosciences and Engineering, vol. 3, no. 1, pp. 125-135, 2006. · Zbl 1136.35449 · doi:10.3934/mbe.2006.3.125
[16] Y. Long, J. Li, W. Rui, and B. He, “Traveling wave solutions for a second order wave equation of KdV type,” Applied Mathematics and Mechanics, vol. 28, no. 11, pp. 1455-1465, 2007. · Zbl 1231.35035 · doi:10.1007/s10483-007-1105-y
[17] V. Marinakis, “New solutions of a higher order wave equation of the KdV type,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 4, pp. 519-525, 2007. · Zbl 1170.35518 · doi:10.2991/jnmp.2007.14.4.2
[18] V. Marinakis, “New solitary wave solutions in higher-order wave equations of the Korteweg-de Vries type,” Zeitschrift für Naturforschung, vol. 62, no. 5-6, pp. 227-230, 2007. · Zbl 1203.35237 · http://www.znaturforsch.com/aa/v62a/c62a.htm
[19] J. Li, “Exact explicit peakon and periodic cusp wave solutions for several nonlinear wave equations,” Journal of Dynamics and Differential Equations, vol. 20, no. 4, pp. 909-922, 2008. · Zbl 1178.34002 · doi:10.1007/s10884-008-9114-5
[20] W. Rui, Y. Long, and B. He, “Some new travelling wave solutions with singular or nonsingular character for the higher order wave equation of KdV type (III),” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 11, pp. 3816-3828, 2009. · Zbl 1167.34006 · doi:10.1016/j.na.2008.07.040
[21] J. Li, J. Wu, and H. Zhu, “Traveling waves for an integrable higher order KdV type wave equations,” International Journal of Bifurcation and Chaos, vol. 16, no. 8, pp. 2235-2260, 2006. · Zbl 1192.37100 · doi:10.1142/S0218127406016033
[22] J. Li, “Dynamical understanding of loop soliton solution for several nonlinear wave equations,” Journal Science in China Series A, vol. 50, no. 6, pp. 773-785, 2007. · Zbl 1139.35076 · doi:10.1007/s11425-007-0039-y
[23] Z. Qiao and L. Liu, “A new integrable equation with no smooth solitons,” Chaos, Solitons & Fractals, vol. 41, no. 2, pp. 587-593, 2009. · Zbl 1198.35209 · doi:10.1016/j.chaos.2007.11.034
[24] J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522-526, 1983. · Zbl 0531.35069 · doi:10.1063/1.525875
[25] J. Weiss, “The Painlevé property for partial differential equations. II. Bäcklund transformation, Lax pairs, and the Schwarzian derivative,” Journal of Mathematical Physics, vol. 24, no. 6, pp. 1405-1413, 1983. · Zbl 0531.35069 · doi:10.1063/1.525875
[26] A. Ramani, B. Dorizzi, and B. Grammaticos, “Painlevé conjecture revisited,” Physical Review Letters, vol. 49, no. 21, pp. 1539-1541, 1982. · doi:10.1103/PhysRevLett.49.1539
[27] C. Gilson and A. Pickering, “Factorization and Painlevé analysis of a class of nonlinear third-order partial differential equations,” Journal of Physics A, vol. 28, no. 10, pp. 2871-2888, 1995. · Zbl 0830.35127 · doi:10.1088/0305-4470/28/10/017