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Solitons, peakons and periodic cusp wave solutions for the Fornberg-Whitham equation. (English) Zbl 1181.35222

Summary: We employ the bifurcation method to dynamical systems to investigate the exact travelling wave solutions for the Fornberg-Whitham equation

${u}_{t}-{u}_{xxt}+{u}_{x}+u{u}_{x}=u{u}_{xxx}+3{u}_{x}{u}_{xx}·$

The implicit expression for solitons is given. The explicit expressions for peakons and periodic cusp wave solutions are also obtained. Further, we show that the limits of soliton solutions and periodic cusp wave solutions are peakons.

##### MSC:
 35Q51 Soliton-like equations 35C07 Traveling wave solutions of PDE 35C08 Soliton solutions of PDE 35B10 Periodic solutions of PDE 37K50 Bifurcation problems (infinite-dimensional systems) 35B40 Asymptotic behavior of solutions of PDE
##### References:
 [1] Whitham, G. B.: Variational methods and applications to water wave, Proc. R. Soc. lond. Ser. A 299, 6-25 (1967) · Zbl 0163.21104 · doi:10.1098/rspa.1967.0119 [2] Ivanov, R.: On the integrability of a class of nonlinear dispersive wave equations, J. nonlinear math. Phys. 1294, 462-468 (2005) · Zbl 1089.35522 · doi:10.2991/jnmp.2005.12.4.2 [3] Fornberg, B.; Whitham, G. B.: A numerical and theoretical study of certain nonlinear wave phenomena, Philos. trans. R. soc. Lond. ser. A 289, 373-404 (1978) · Zbl 0384.65049 · doi:10.1098/rsta.1978.0064 [4] Zhou, J. B.; Tian, L. X.: A type of bounded traveling wave solutions for the fornberg–Whitham equation, J. math. Anal. appl. 346, 255-261 (2008) · Zbl 1146.35025 · doi:10.1016/j.jmaa.2008.05.055 [5] Luo, D.: Bifurcation theory and methods of dynamical systems, (1997)