×

Soliton and periodic wave solutions to the osmosis \(K(2,2)\) equation. (English) Zbl 1182.37038

Summary: Two types of traveling wave solutions to the osmosis \(K(2,2)\) equation \(u_t+(u^2)_x- (u^2)_{xxx}=0\) are investigated. They are characterized by two parameters. The expresssions for the soliton and periodic wave solutions are obtained.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] P. Rosenau and J. M. Hyman, “Compactons: solitons with finite wavelength,” Physical Review Letters, vol. 70, no. 5, pp. 564-567, 1993. · Zbl 0952.35502 · doi:10.1103/PhysRevLett.70.564
[2] A. M. Wazwaz, “Compactons and solitary patterns structures for variants of the KdV and the KP equations,” Applied Mathematics and Computation, vol. 139, no. 1, pp. 37-54, 2003. · Zbl 1029.35200 · doi:10.1016/S0096-3003(02)00120-0
[3] J.-H. He, “Homotopy perturbation method for bifurcation of nonlinear problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 207-208, 2005. · Zbl 1401.65085
[4] J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700-708, 2006. · Zbl 1141.35448 · doi:10.1016/j.chaos.2006.03.020
[5] J.-H. He and X.-H. Wu, “Construction of solitary solution and compacton-like solution by variational iteration method,” Chaos, Solitons & Fractals, vol. 29, no. 1, pp. 108-113, 2006. · Zbl 1147.35338 · doi:10.1016/j.chaos.2005.10.100
[6] J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141-1199, 2006. · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[7] L. Xu, “Variational approach to solitons of nonlinear dispersive K(m,n) equations,” Chaos, Solitons & Fractals, vol. 37, no. 1, pp. 137-143, 2008. · Zbl 1143.35361 · doi:10.1016/j.chaos.2006.08.016
[8] A. M. Wazwaz, “General compactons solutions and solitary patterns solutions for modified nonlinear dispersive equations mK(n,n) in higher dimensional spaces,” Mathematics and Computers in Simulation, vol. 59, no. 6, pp. 519-531, 2002. · Zbl 0996.35065 · doi:10.1016/S0378-4754(01)00439-6
[9] A. M. Wazwaz, “Compact and noncompact structures for a variant of KdV equation in higher dimensions,” Applied Mathematics and Computation, vol. 132, no. 1, pp. 29-45, 2002. · Zbl 1031.35128 · doi:10.1016/S0096-3003(01)00173-4
[10] Y. Chen, B. Li, and H. Q. Zhang, “New exact solutions for modified nonlinear dispersive equations mK(m,n) in higher dimensions spaces,” Mathematics and Computers in Simulation, vol. 64, no. 5, pp. 549-559, 2004. · Zbl 1073.35052 · doi:10.1016/j.matcom.2003.10.005
[11] B. He, Q. Meng, W. Rui, and Y. Long, “Bifurcations of travelling wave solutions for the mK(n,n) equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 10, pp. 2114-2123, 2008. · Zbl 1221.35336 · doi:10.1016/j.cnsns.2007.06.006
[12] Z. Y. Yan, “Modified nonlinearly dispersive mK(m,n,k) equations. I. New compacton solutions and solitary pattern solutions,” Computer Physics Communications, vol. 152, no. 1, pp. 25-33, 2003. · Zbl 1196.35192 · doi:10.1016/S0010-4655(02)00794-4
[13] Z. Y. Yan, “Modified nonlinearly dispersive mK(m,n,k) equations. II. Jacobi elliptic function solutions,” Computer Physics Communications, vol. 153, no. 1, pp. 1-16, 2003. · Zbl 1196.35194 · doi:10.1016/S0010-4655(02)00851-2
[14] A. Biswas, “1-soliton solution of the K(m,n) equation with generalized evolution,” Physics Letters A, vol. 372, no. 25, pp. 4601-4602, 2008. · Zbl 1221.35099 · doi:10.1016/j.physleta.2008.05.002
[15] Y. G. Zhu, K. Tong, and T. C. Lu, “New exact solitary-wave solutions for the K(2,2,1) and K(3,3,1) equations,” Chaos, Solitons & Fractals, vol. 33, no. 4, pp. 1411-1416, 2007. · Zbl 1137.35427 · doi:10.1016/j.chaos.2006.01.090
[16] C. Xu and L. Tian, “The bifurcation and peakon for K(2,2) equation with osmosis dispersion,” Chaos, Solitons & Fractals, vol. 40, no. 2, pp. 893-901, 2009. · Zbl 1197.35253 · doi:10.1016/j.chaos.2007.08.042
[17] J. Zhou and L. Tian, “Soliton solution of the osmosis K(2,2) equation,” Physics Letters A, vol. 372, no. 41, pp. 6232-6234, 2008. · Zbl 1225.35194 · doi:10.1016/j.physleta.2008.08.053
[18] V. O. Vakhnenko and E. J. Parkes, “Explicit solutions of the Camassa-Holm equation,” Chaos, Solitons & Fractals, vol. 26, no. 5, pp. 1309-1316, 2005. · Zbl 1072.35156 · doi:10.1016/j.chaos.2005.03.011
[19] V. O. Vakhnenko and E. J. Parkes, “Periodic and solitary-wave solutions of the Degasperis-Procesi equation,” Chaos, Solitons & Fractals, vol. 20, no. 5, pp. 1059-1073, 2004. · Zbl 1049.35162 · doi:10.1016/j.chaos.2003.09.043
[20] E. J. Parkes, “The stability of solutions of Vakhnenko’s equation,” Journal of Physics A, vol. 26, no. 22, pp. 6469-6475, 1993. · Zbl 0809.35086 · doi:10.1088/0305-4470/26/22/040
[21] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Die Grundlehren der Mathematischen Wissenschaften, vol. 67, Springer, Berlin, Germany, 2nd edition, 1971. · Zbl 0213.16602
[22] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, USA, 1972. · Zbl 0543.33001
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.