Deng, Xijun; Parkes, E. J.; Cao, Jinlong Exact solitary and periodic-wave solutions of the \(K(2,2)\) equation (defocusing branch). (English) Zbl 1202.35185 Appl. Math. Comput. 217, No. 4, 1566-1576 (2010). Summary: An auxiliary elliptic equation method is presented for constructing exact solitary and periodic travelling-wave solutions of the \(K(2,2)\) equation (defocusing branch). Some known results in the literature are recovered more efficiently, and some new exact travelling-wave solutions are obtained. Also, new stationary-wave solutions are obtained. Cited in 5 Documents MSC: 35Q51 Soliton equations 35C07 Traveling wave solutions 35C08 Soliton solutions 35B10 Periodic solutions to PDEs Keywords:auxiliary elliptic equation method; solitary wave solutions; periodic-wave solutions; \(K(2, 2)\) equation; defocusing branch PDFBibTeX XMLCite \textit{X. Deng} et al., Appl. Math. Comput. 217, No. 4, 1566--1576 (2010; Zbl 1202.35185) Full Text: DOI Link References: [1] Rosenau, P.; Hyman, J. M., Compactons: solitons with finite wavelength, Phys. Rev. Lett., 70, 564-567 (1993) · Zbl 0952.35502 [2] Rosenau, P., On nonanalytic solitary waves formed by a nonlinear dispersion, Phys. Lett. A, 230, 305-318 (1997) · Zbl 1052.35511 [3] Biswas, A., 1-soliton solution of the K(m,n) equation with generalized evolution, Phys. Lett. A, 372, 4601-4602 (2008) · Zbl 1221.35099 [4] Lenells, J., Traveling wave solutions of the Camassa-Holm equation, J. Diff. Eq., 217, 393-430 (2005) · Zbl 1082.35127 [5] Boyd, J. P., Near-corner waves of the Camassa-Holm equation, Phys. Lett. A, 336, 342-348 (2005) · Zbl 1136.35445 [6] Wazwaz, A. M., Exact special solutions with solitary patterns for the nonlinear dispersive \(K(m, n)\) equations, Chaos Soliton Fract., 13, 161-170 (2002) · Zbl 1027.35115 [7] Xu, C. H.; Tian, L. X., The bifurcation and peakon for \(K(2, 2)\) equation with osmosis dispersion, Chaos Soliton Fract. (2007) [8] Zhou, J. B.; Tian, L. X., Soliton solution of the osmosis \(K(2, 2)\) equation, Phys. Lett. A, 372, 6232-6234 (2008) · Zbl 1225.35194 [9] Vakhnenko, V. O.; Parkes, E. J., Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos Soliton Fract., 20, 1059-1073 (2004) · Zbl 1049.35162 [10] Stepanyants, Y. A., On stationary solutions of the reduced Ostrovsky equation: periodic-waves, compactons and compound solitons, Chaos Soliton Fract., 28, 193-204 (2006) · Zbl 1088.35531 [11] Li, J. B., Dynamical understanding of loop soliton solution for several nonlinear wave equations, Sci. China Ser. A, 50, 773-785 (2007) · Zbl 1139.35076 [12] Zhang, G. P.; Qiao, Z. J., Cuspons and smooth solitons of the Degasperis-Procesi equation under inhomogeneous boundary condition, Math. Phys. Anal. Geom., 10, 205-225 (2007) · Zbl 1153.35385 [13] Parkes, E. J., The stability of solutions of Vakhnenko’s equation, J. Phys. A: Math. Gen., 26, 6469-6475 (1993) · Zbl 0809.35086 [14] Parkes, E. J.; Vakhnenko, V. O., Explicit solutions of the Camassa-Holm equation, Chaos Soliton Fract., 26, 1309-1316 (2005) · Zbl 1072.35156 [15] Byrd, P. F.; Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists (1971), Springer: Springer Berlin · Zbl 0213.16602 [16] Lenells, J., Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306, 72-82 (2005) · Zbl 1068.35163 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.