## A type of bounded traveling wave solutions for the Fornberg-Whitham equation.(English)Zbl 1146.35025

Summary: By using bifurcation method, we successfully find the Fornberg-Whitham equation $u_t - u_{xxt}+u_x=uu_{xxx} - uu_x+3u_xu_{xx}$ has a type of traveling wave solutions called kink-like wave solutions and antikink-like wave solutions. They are defined on some semifinal bounded domains and possess properties of kink waves and anti-kink waves. Their implicit expressions are obtained. For some concrete data, the graphs of the implicit functions are displayed, and the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation.

### MSC:

 35G25 Initial value problems for nonlinear higher-order PDEs
Full Text:

### 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 [2] Dey, B., Domain wall solutions of KdV like equations with higher order nonlinearity, J. phys. A, 19, L9-L12, (1986) · Zbl 0624.35070 [3] Lenells, J., Traveling wave solutions of the camassa – holm equation, J. differential equations, 217, 393-430, (2005) · Zbl 1082.35127 [4] Liu, Z.; Chen, C., Compactons in a general compressible hyperelastic rod, Chaos solitons fractals, 22, 627-640, (2004) · Zbl 1116.74374 [5] Lenells, J., Traveling wave solutions of the degasperis – procesi equation, J. math. anal. appl., 306, 72-82, (2005) · Zbl 1068.35163 [6] Nickel, J., Travelling wave solutions to the kuramoto – sivashinsky equation, Chaos solitons fractals, 33, 1376-1382, (2007) · Zbl 1137.35063 [7] Liu, Z.; Li, Q.; Lin, Q., New bounded traveling waves of camassa – holm equation, Internat. J. bifur. chaos, 14, 3541-3556, (2004) · Zbl 1063.35138 [8] Guo, B.; Liu, Z., Two new types of bounded waves of CH-γ equation, Sci. China ser. A, 48, 1618-1630, (2005) · Zbl 1217.35161 [9] Tang, M.; Zhang, W., Four types of bounded wave solutions of CH-γ equation, Sci. China ser. A, 50, 132-152, (2007) · Zbl 1117.35310 [10] Chen, C.; Tang, M., A new type of bounded waves for degasperis – procesi equation, Chaos solitons fractals, 27, 698-704, (2006) · Zbl 1082.35044 [11] Liu, Z.; Yao, L., Compacton-like wave and kink-like wave of GCH equation, Nonlinear anal. real world appl., 8, 136-155, (2007) · Zbl 1106.35065 [12] Whitham, G.B., Variational methods and applications to water wave, Proc. R. soc. lond. ser. A, 299, 6-25, (1967) · Zbl 0163.21104 [13] 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 [14] Luo, D., Bifurcation theory and methods of dynamical systems, (1997), World Scientific Publishing Co. London
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.