Kawahara dynamics in dispersive media

Physica D: Nonlinear Phenomena

Volumn 152-153, Issue , 2001, Pages 384-394

  Schneider, Guido ^a   Wayne, C Eugene ^b  

^a UNIVERSITY OF BAYREUTH   (Germany)

^b Boston University   (United States)

Author keywords

Boussinesq equation; Kawahara equations; Water wave problem

Indexed keywords

EID: 0035873112     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(01)00181-6     Document Type: Article

References (23)

1. ben Youssef W., Colin T. Rigorous derivation of Korteweg-de Vries-type systems from a general class of nonlinear hyperbolic systems. Math. Model. Num. Anal. 34:2000;873-911.
2. M.J. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants à l'Académie des Sciences Inst. France (séries 2) 3 (1877) 1-680.
3. Buffoni B., Groves M.D., Toland J.F. A plethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers. Philos. Trans. R. Soc. Lond. A. 354:1996;575-607.
4. Collet P., Eckmann J.-P. The amplitude equation and the Swift-Hohenberg equation. Commun. Math. Phys. 132:1990;139-153.
5. Craig W. An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Comm. Part. Diff. Eq. 10:1985;787-1003.
6. Th. Gallay, G. Schneider, KP-description of unidirectional long waves: the model case, Proc. Roy. Soc. Edinb. B, to be published.
7. Groves M.D. Solitary-wave solutions to a class of fifth-order model equations. Nonlinearity. 11:1998;341-353.
8. Hunter J.K., Scheurle J. Existence of perturbed solitary wave solutions to a model equation for water waves. Physica D. 32:1988;253-268.
9. Kakutani T., Ono H. J. Phys. Soc. Jpn. 26:1969;1305.
10. Kalyakin L.A. Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium. Math. USSR Sbornik. 60:1988;457-483.
11. Kalyakin L.A. Long wave asymptotics: integrable equations as asymptotic limits of non-linear systems. Russian Math. Surv. 44:1989;3-42.
12. T. Kato, On the Cauchy-problem for the (Generalized) Korteweg-de Vries Equation, Studies in Applied Mathematics, Advances in Mathematics: Supplement Studies, Vol. 8, Academic Press, New York, 1983, pp. 93-128.
13. Kawahara T. Oscillatory solitary waves in dispersive media. J. Phys. Soc. Jpn. 33:1972;260-264.
14. R.L. Pego, M.I. Weinstein, On the strong spectral stability of some Boussinesq solitary waves, in: A. Mielke, K. Kirchgässner. (Eds.), Structure and Dynamics of Nonlinear Waves in Fluids, World Scientific, Singapore, 1995, pp. 370-382.
15. Pierce R.D., Wayne C.E. On the validity of mean-field amplitude equations for counter-propagating wave trains. Nonlinearity. 8:1995;769-779.
16. Schneider G. Error estimates for the Ginzburg-Landau approximation. Zeit. Angew. Math. Phys. 45:1994;433-457.
17. Schneider G. Validity and Limitation of the Newell-Whitehead equation. Math. Nachr. 176:1995;249-263.
18. Schneider G. The long wave limit for a Boussinesq equation. SIAM J. Appl. Math. 58:1998;1237-1245.
19. G. Schneider, C.E. Wayne, Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, in: B. Fiedler, K. Gröger, J. Sprekels. (Eds.), Proceedings of the EQUADIFF'99 International Conference on Differential Equations, World Scientific, Singapore, 2000, pp. 390-404.
20. Schneider G., Wayne C.E. The long wave limit for the water wave problem. I. The case of zero surface tension. Commun. Pure Appl. Math. 53:2000;1475-1535.
21. G. Schneider, C.E. Wayne, The long wave limit for the water wave problem. II. The case of non-zero surface tension, Preprint.
22. van Harten A. On the validity of Ginzburg-Landau's equation. J. Nonlinear Sci. 1:1991;397-422.
23. Yosihara H. Capillary-gravity waves for an incompressible ideal fluid. J. Math. Kyoto Univ. 23:1983;649-694.