Yin, Jiuli; Tian, Lixin Classification of the travelling waves in the nonlinear dispersive KdV equation. (English) Zbl 1209.34038 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 2, 465-470 (2010). The traveling wave solutions of the Korteweg-de Vries-like equation (\(K(2,2)\)) \[ u_t + a(u^2)_x + (u^2)_{xxx} =0 \]modeling the formation of pattern in nonlinear dispersion, are investigated. Besides the known compacton solutions (compactly supported solitons with nonsmooth fronts), the equation is shown to admit other different types of solutions too, such as cuspons, peakons, loopons, stumpons and fractal-like waves. A qualitative analysis of the traveling waves, as well as their classification, is carried on. Finally, some new explicit solutions are given. Reviewer: Cristina Marcelli (Ancona) Cited in 1 Document MSC: 34B40 Boundary value problems on infinite intervals for ordinary differential equations 35Q53 KdV equations (Korteweg-de Vries equations) 35C07 Traveling wave solutions 35C08 Soliton solutions 34A05 Explicit solutions, first integrals of ordinary differential equations Keywords:traveling wave solutions; nonlinear dispersive equations; pattern formation; solitons PDFBibTeX XMLCite \textit{J. Yin} and \textit{L. Tian}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 2, 465--470 (2010; Zbl 1209.34038) Full Text: DOI References: [1] Rosenau, P.; Hyman, M., Phys. Rev. Lett., 70, 564-567 (1993) [2] Mustafa Inc, Physica A, 375, 447-456 (2007) [3] Wazwaz, A. M., Chaos Solitons Fractals, 13, 321-330 (2002) [4] Domairry, Ganji, Phys. Lett. A, 368, 266-270 (2007) [5] He, J. H.; Wu, X. H., Chaos Solitons Fractals, 30, 3, 700-708 (2006) [6] Lai, Shaoyong; Wu, Y. H.; Wiwatanapataphee, B., J. Comput. Appl. Math., 212, 291-299 (2008) [7] Wazwaz, A. M., Appl. Math. Comput., 215, 1548-1552 (2009) [8] Wazwaz, A. M., Chaos Solitons Fractals, 13, 161-170 (2002) [9] Triki, H.; Wazwaz, A. M., Appl. Math. Comput., 214, 370-373 (2009) [10] Biswas, A., Phys. Lett. A, 372, 4601-4602 (2008) [11] Tian, L. X.; Yin, J. L., J. Comput. Appl. Math., 207, 1, 46-52 (2007) [12] Fan, Xinghua; Tian, Lixin, Int. J. Nonlinear Sci., 1, 105-110 (2006) · Zbl 1394.35392 [13] Matsuno, Y., Phys. Lett. A, 359, 451-457 (2006) [14] He, Ji-Huan; Zhang, Li-Na, Phys. Lett. A, 372, 1044-1047 (2008) [15] Guo, Boling; Liu, Zhengrong, Chaos Solitons Fractals, 23, 1451-1463 (2005) · Zbl 1068.35103 [16] Tian, Lixin; Yin, Jiuli, Chaos Solitons Fractals, 24, 353-363 (2005) [17] Ju, Lin, Int. J. Nonlinear Sci., 1, 43-48 (2006) [18] Wang, Lixia; Zhou, Jiangbo, Int. J. Nonlinear Sci., 1, 58-64 (2006) [19] Lenells, J., J. Differential Equations, 217, 393-430 (2005) 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.