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Single peak solitary wave solutions for the osmosis $K\left(2,2\right)$ equation under inhomogeneous boundary condition. (English) Zbl 1196.35180
Summary: The qualitative theory of differential equations is applied to the $K$(2,2) equation with osmosis dispersion. Smooth, peaked and cusped solitary wave solutions of the osmosis $K$(2,2) equation under inhomogeneous boundary condition are obtained. The parametric conditions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for smooth, peaked and cusped solitary wave solutions of the osmosis $K$(2,2) equation.
##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35C08 Soliton solutions of PDE 35B65 Smoothness and regularity of solutions of PDE 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 35B40 Asymptotic behavior of solutions of PDE
##### References:
 [1] Lenells, J.: Traveling wave solutions of the Camassa – Holm and Korteweg – de Vries equations, J. nonlinear math. Phys. 11, 508-520 (2004) · Zbl 1069.35072 · doi:10.2991/jnmp.2004.11.4.7 [2] Camassa, R.; Holm, D.: An integrable shallow wave equation with peaked solitons, Phys. rev. Lett. 71, 1661-1664 (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661 [3] Lenells, J.: Traveling wave solutions of the Camassa – Holm equation, J. differential equations 217, 393-430 (2005) · Zbl 1082.35127 · doi:10.1016/j.jde.2004.09.007 [4] Rosenau, P.; Hyman, J. M.: Compactons: solitons with finite wavelengths, Phys. rev. Lett. 70, 564-567 (1993) · Zbl 0952.35502 · doi:10.1103/PhysRevLett.70.564 [5] Xu, C.; Tian, L.: The bifurcation and peakon for $K\left(2,2\right)$ equation with osmosis dispersion, Chaos solitons fractals 40, 893-901 (2009) · Zbl 1197.35253 · doi:10.1016/j.chaos.2007.08.042 [6] J. Zhou, L. Tian, X. Fan, New exact travelling wave solutions for the K(2,2) equation with osmosis dispersion, Appl. Math. Comput., doi:10.1016/j.amc.2009.04.073, in press [7] Zhou, J.; Tian, L.: Soliton solution of the osmosis $K\left(2,2\right)$ equation, Phys. lett. A 372, 6232-6234 (2008) · Zbl 1225.35194 · doi:10.1016/j.physleta.2008.08.053 [8] Zhou, J.; Tian, L.; Fan, X.: Soliton and periodic wave solutions to the osmosis $K\left(2,2\right)$ equation, Math. probl. Eng. 2009 (2009) [9] Deng, X.; Han, L.: Exact peaked wave solution of the osmosis $K\left(2,2\right)$ equation, Turkish J. Phys. 33, 179-184 (2009) [10] X. Deng, E.J. Parkes, J. Cao, Exact solitary and periodic-wave solutions of the K(2,2) equation (defocusing branch), Appl. Math. Comput. (2009), doi:10.1016/j.amc.2009.06.054, in press [11] Qiao, Z.; Zhang, G.: On peaked and smooth solitons for the Camassa – Holm equation, Europhys. lett. 73, 657-663 (2006) [12] Zhang, G.; Qiao, Z.: Cuspons and smooth solitons of the Degasperis – Procesi equation under inhomogeneous boundary condition, Math. phys. Anal. geom. 10, 205-225 (2007) · Zbl 1153.35385 · doi:10.1007/s11040-007-9027-2 [13] Li, J.; Liu, Z.: Smooth and non-smooth travelling waves in a nonlinearly dispersive equation, Appl. math. Model. 25, 41-56 (2000) · Zbl 0985.37072 · doi:10.1016/S0307-904X(00)00031-7 [14] Li, J.; Dai, H. H.: On the study of singular nonlinear traveling wave equations: dynamical system approach, (2007) [15] Li, J.; Chen, G.: On a class of singular nonlinear traveling wave equations, Internat. J. Bifur. chaos 17, 4049-4065 (2007) · Zbl 1158.35080 · doi:10.1142/S0218127407019858 [16] Li, J.; Zhang, Y.: Exact loop solutions, cusp solutions, solitary wave solutions and periodic wave solutions for the special CH-DP equation, Nonlinear anal. Real world appl. 10, 2502-2507 (2009) · Zbl 1163.76328 · doi:10.1016/j.nonrwa.2008.05.006 [17] Guo, B.; Liu, Z.: Two new types of bounded waves of CH-$\gamma$ equation, Sci. China ser. A 48, 1618-1630 (2005) · Zbl 1217.35161 · doi:10.1360/04ys0205 [18] Tang, M.; Zhang, W.: Four types of bounded wave solutions of CH-$\gamma$ equation, Sci. China ser. A 50, 132-152 (2007) · Zbl 1117.35310 · doi:10.1007/s11425-007-2042-8 [19] Liu, Z.; Guo, B.: Periodic blow-up solutions and their limit forms for the generalized Camassa – Holm equation, Progr. natur. Sci. 18, 259-261 (2008)