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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 $$u_t - u_{xxt}+u_x+uu_x=uu_{xxx}+3u_xu_{xx}.$$ 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.

35Q51Soliton-like equations
35C07Traveling wave solutions of PDE
35C08Soliton solutions of PDE
35B10Periodic solutions of PDE
37K50Bifurcation problems (infinite-dimensional systems)
35B40Asymptotic behavior of solutions of PDE
Full Text: DOI
[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) · Zbl 0961.37015