Nonuniform continuity of the osmosis \(K(2, 2)\) equation. (English) Zbl 1470.35357

Chen, Aiyong; Ding, Yong; Huang, Wentao

Abstr. Appl. Anal. 2013, Article ID 717042, 8 p. (2013).

Summary: The qualitative theory of differential equations is applied to the osmosis \(K(2, 2)\) equation. The parametric conditions of existence of the smooth periodic travelling wave solutions are given. We show that the solution map is not uniformly continuous by using the theory of Himonas and Misiolek. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

MSC:

35Q92	PDEs in connection with biology, chemistry and other natural sciences
35Q53	KdV equations (Korteweg-de Vries equations)
35C07	Traveling wave solutions

References:

[1]	Lenells, J., Traveling wave solutions of the Camassa-Holm and Korteweg-de Vries equations, Journal of Nonlinear Mathematical Physics, 11, 4, 508-520 (2004) · Zbl 1069.35072 · doi:10.2991/jnmp.2004.11.4.7
[2]	Bona, J. L.; Smith, R., The initial-value problem for the Korteweg-de Vries equation, Philosophical Transactions of the Royal Society A, 278, 1287, 555-601 (1975) · Zbl 0306.35027 · doi:10.1098/rsta.1975.0035
[3]	Bona, J., On the stability theory of solitary waves, Proceedings of the Royal Society A, 344, 1638, 363-374 (1975) · Zbl 0328.76016 · doi:10.1098/rspa.1975.0106
[4]	Angulo Pava, J.; Bona, J. L.; Scialom, M., Stability of cnoidal waves, Advances in Differential Equations, 11, 12, 1321-1374 (2006) · Zbl 1147.35079
[5]	Camassa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons, Physical Review Letters, 71, 11, 1661-1664 (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[6]	Lenells, J., Traveling wave solutions of the Camassa-Holm equation, Journal of Differential Equations, 217, 2, 393-430 (2005) · Zbl 1082.35127 · doi:10.1016/j.jde.2004.09.007
[7]	Qiao, Z. J.; Zhang, G. P., On peaked and smooth solitons for the Camassa-Holm equation, Europhysics Letters, 73, 5, 657-663 (2006) · doi:10.1209/epl/i2005-10453-y
[8]	Liu, Z.; Qian, T., Peakons of the Camassa-Holm equation, Applied Mathematical Modelling, 26, 3, 473-480 (2002) · Zbl 1018.35061 · doi:10.1016/S0307-904X(01)00086-5
[9]	Liu, Z.; Li, Q.; Lin, Q., New bounded traveling waves of Camassa-Holm equation, International Journal of Bifurcation and Chaos, 14, 10, 3541-3556 (2004) · Zbl 1063.35138 · doi:10.1142/S0218127404011521
[10]	Liu, Z.; Guo, B., Periodic blow-up solutions and their limit forms for the generalized Camassa-Holm equation, Progress in Natural Science, 18, 3, 259-266 (2008) · doi:10.1016/j.pnsc.2007.11.004
[11]	Guo, B.; Liu, Z., Two new types of bounded waves of CH-\( \gamma\) equation, Science in China. Series A, 48, 12, 1618-1630 (2005) · Zbl 1217.35161 · doi:10.1360/04ys0205
[12]	Himonas, A. A.; Misiołek, G., High-frequency smooth solutions and well-posedness of the Camassa-Holm equation, International Mathematics Research Notices, 51, 3135-3151 (2005) · Zbl 1092.35085 · doi:10.1155/IMRN.2005.3135
[13]	Himonas, A. A.; Kenig, C.; Misiołek, G., Non-uniform dependence for the periodic CH equation, Communications in Partial Differential Equations, 35, 6, 1145-1162 (2010) · Zbl 1193.35189 · doi:10.1080/03605300903436746
[14]	Zhang, G.; Qiao, Z., Cuspons and smooth solitons of the Degasperis-Procesi equation under inhomogeneous boundary condition, Mathematical Physics, Analysis and Geometry, 10, 3, 205-225 (2007) · Zbl 1153.35385 · doi:10.1007/s11040-007-9027-2
[15]	Liu, Y.; Yin, Z., Global existence and blow-up phenomena for the Degasperis-Procesi equation, Communications in Mathematical Physics, 267, 3, 801-820 (2006) · Zbl 1131.35074 · doi:10.1007/s00220-006-0082-5
[16]	Christov, O.; Hakkaev, S., On the Cauchy problem for the periodic \(b\)-family of equations and of the non-uniform continuity of Degasperis-Procesi equation, Journal of Mathematical Analysis and Applications, 360, 1, 47-56 (2009) · Zbl 1178.35081 · doi:10.1016/j.jmaa.2009.06.035
[17]	Rosenau, P.; Hyman, J. M., Compactons: solitons with finite wavelength, Physical Review Letters, 70, 5, 564-567 (1993) · Zbl 0952.35502 · doi:10.1103/PhysRevLett.70.564
[18]	Xu, C.; Tian, L., The bifurcation and peakon for \(K(2, 2)\) equation with osmosis dispersion, Chaos, Solitons & Fractals, 40, 2, 893-901 (2009) · Zbl 1197.35253 · doi:10.1016/j.chaos.2007.08.042
[19]	Zhou, J.; Tian, L.; Fan, X., New exact travelling wave solutions for the \(K(2, 2)\) equation with osmosis dispersion, Applied Mathematics and Computation, 217, 4, 1355-1366 (2010) · Zbl 1203.35258 · doi:10.1016/j.amc.2009.04.073
[20]	Zhou, J.; Tian, L., Soliton solution of the osmosis \(K(2, 2)\) equation, Physics Letters A, 372, 41, 6232-6234 (2008) · Zbl 1225.35194 · doi:10.1016/j.physleta.2008.08.053
[21]	Deng, X.; Han, L., Exact peaked wave solution of the osmosis \(K(2, 2)\) equation, Turkish Journal of Physics, 33, 3, 179-184 (2009) · doi:10.3906/fiz-0901-3
[22]	Deng, X.; Parkes, E. J.; Cao, J., Exact solitary and periodic-wave solutions of the \(K(2, 2)\) equation (defocusing branch), Applied Mathematics and Computation, 217, 4, 1566-1576 (2010) · Zbl 1202.35185 · doi:10.1016/j.amc.2009.06.054
[23]	Chen, A.; Li, J., Single peak solitary wave solutions for the osmosis \(K(2, 2)\) equation under inhomogeneous boundary condition, Journal of Mathematical Analysis and Applications, 369, 2, 758-766 (2010) · Zbl 1196.35180 · doi:10.1016/j.jmaa.2010.04.018
[24]	Li, J.; Liu, Z., Smooth and non-smooth traveling waves in a nonlinearly dispersive equation, Applied Mathematical Modelling, 25, 1, 41-56 (2000) · Zbl 0985.37072 · doi:10.1016/S0307-904X(00)00031-7
[25]	Li, J.; Chen, G., On a class of singular nonlinear traveling wave equations, International Journal of Bifurcation and Chaos, 17, 11, 4049-4065 (2007) · Zbl 1158.35080 · doi:10.1142/S0218127407019858