CONTRIBUTION OF HIGHER ORDER TERMS TO THE NONLINEAR SHALLOW WATER WAVES

Abstract

In this work, by utilizing the scaled multiple-space expansion method, we studied the propagation of weakly nonlinear waves in shallow water and obtained the governing evolution equations of various order terms in the perturbation expansion. Seeking a progressive wave solution to these evolution equations we obtained the speed correction terms so as to remove some possible secularities. The result obtained here is exactly the same with that of obtained by the modified reductive perturbation method [12]. We also proposed a method for the evolution equation governing the n th order term in the perturbation expansion. By defining a single time parameter we showed the connection of the modified reductive perturbation method to the scaled multiple-space expansion method.

Keywords

• Shallow water waves,scaled multiple-space expansion,solitary waves

References

1. Davidson, N., (1972), Methods in Nonlinear Plasma Theory, Academic Press, New York.
2. Antar, N. and Demiray, H., (1999), Weakly nonlinear waves in a prestressed thin elastic tube contain- ing a viscous fluid, Int. J. Engr. Sci., 37, 1859-1876.
3. Ichikawa, Y. H., Mitsuhashi, T. and Konno, K., (1976), Contribution of higher order terms in reductive perturbation theory, J. Phys. Soc. Japan, 41, 1382-1386.
4. Aoyama, T. and Ichikawa, Y. H., (1977), Contribution of second order terms of the nonlinear shallow water waves, J. Phys. Soc. Japan, 42, 313-318.
5. Sugimoto, N. and Kakutani, T., (1977), Note on the higher order terms in reductive perturbation theory, J. Phys. Soc. Japan, 43, 1469-1470.
6. Kodama, Y. and Tanuiti, T., (1978), Higher order approximation in the reductive perturbation method
7. I. Weakly dispersive systems, J. Phys. Soc. Japan, 45, 298-310. Kraenkel, R. A. and Manna, M. A., (1995), The Korteweg-de Vries hierarchy and long water-waves, J. Math. Phys., 36, 307-320.
8. Malfliet, M. and Wieers, E., (1996), Theory of ion-acoustic waves revisited, J. Plasma Phys., 56, 450.

9. Demiray, H., (1999), A modified reductive perturbation method as applied to nonlinear ion-acoustic waves, J. Phys. Soc. Japan, 68, 1833-1837.
10. Demiray, H., (2002), Contribution of higher order terms in nonlinear ion-acoustic waves: Strongly dispersive case, J. Phys. Soc. Japan, 71, 1921-1930.
11. Demiray, H., (2011), An application of modified reductive perturbation method to symmetric regularized-long-wave equation, TWMS Appl. and Engr. Math., 1, 49-57.
12. Demiray, H., (2011), An application of the modified reductive perturbation method to long water waves, Int. J. Engr. Sci., 49, 1397-1403.
13. Hilmi Demiray, for a photograph and biography, see TWMS Journal of Applied and Engineering Mathematics, Volume 1, No.1, 2011.