Single peak solitary wave solutions for the Fornberg–Whitham equation

Abstract

The qualitative theory of differential equations is applied to the Fornberg–Whitham equation. Smooth, peaked and cusped solitary wave solutions of the Fornberg–Whitham equation under inhomogeneous boundary condition are obtained. The 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 Fornberg–Whitham equation. The results presented in this article extend and improve the previous results.

Keywords:

• Fornberg–Whitham equation
• peakon
• cuspon
• solitary wave

AMS Subject Classifications::

• 35Q51
• 35Q53

Acknowledgements

This work are supported by the National Natural Science Foundation of China (No. 10961011, No. 60964006, No. 61004101 and No. 11061010). The authors wish to thank the anonymous reviewers for their helpful comments and suggestions.