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Kawahara dynamics in dispersive media. (English) Zbl 0977.35123
Summary: In a previous paper it was shown that in the long wave limit the water wave problem without surface tension can be described approximately by two decoupled KdV equations. Here, we study a model problem and prove that in a degenerate case which occurs for the water wave problem with surface tension and near other codimension two points at which the coefficient in front of the leading order dispersive term in the equation of motion vanishes, the long wave limit can be rigorously approximated by two decoupled Kawahara equations.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[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, 873-911 (2000) · Zbl 0962.35152
[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, 575-607 (1996) · Zbl 0861.76012
[4] Collet, P.; Eckmann, J. -P.: The amplitude equation and the Swift--Hohenberg equation. Commun. math. Phys. 132, 139-153 (1990) · Zbl 0756.35096
[5] Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg--de Vries scaling limits. Comm. part. Diff. eq. 10, 787-1003 (1985) · Zbl 0577.76030
[6] Th. Gallay, G. Schneider, KP-description of unidirectional long waves: the model case, Proc. Roy. Soc. Edinb. B, to be published. · Zbl 1015.76015
[7] Groves, M. D.: Solitary-wave solutions to a class of fifth-order model equations. Nonlinearity 11, 341-353 (1998) · Zbl 0994.34034
[8] Hunter, J. K.; Scheurle, J.: Existence of perturbed solitary wave solutions to a model equation for water waves. Physica D 32, 253-268 (1988) · Zbl 0694.35204
[9] Kakutani, T.; Ono, H.: J. phys. Soc. jpn.. 26, 1305 (1969)
[10] Kalyakin, L. A.: Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium. Math. USSR sbornik 60, 457-483 (1988) · Zbl 0699.35135
[11] Kalyakin, L. A.: Long wave asymptotics: integrable equations as asymptotic limits of non-linear systems. Russian math. Surv. 44, 3-42 (1989) · Zbl 0683.35082
[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. · Zbl 0549.34001
[13] Kawahara, T.: Oscillatory solitary waves in dispersive media. J. phys. Soc. jpn. 33, 260-264 (1972)
[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. · Zbl 0872.76042
[15] Pierce, R. D.; Wayne, C. E.: On the validity of mean-field amplitude equations for counter-propagating wave trains. Nonlinearity 8, 769-779 (1995) · Zbl 0833.35128
[16] Schneider, G.: Error estimates for the Ginzburg--Landau approximation. Zeit. angew. Math. phys. 45, 433-457 (1994) · Zbl 0805.35125
[17] Schneider, G.: Validity and limitation of the Newell--Whitehead equation. Math. nachr. 176, 249-263 (1995) · Zbl 0844.35120
[18] Schneider, G.: The long wave limit for a Boussinesq equation. SIAM J. Appl. math. 58, 1237-1245 (1998) · Zbl 0911.35105
[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. · Zbl 0970.35126
[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, 1475-1535 (2000) · Zbl 1034.76011
[21] G. Schneider, C.E. Wayne, The long wave limit for the water wave problem. II. The case of non-zero surface tension, Preprint. · Zbl 1034.76011
[22] Van Harten, A.: On the validity of Ginzburg--Landau’s equation. J. nonlinear sci. 1, 397-422 (1991) · Zbl 0795.35112
[23] Yosihara, H.: Capillary-gravity waves for an incompressible ideal fluid. J. math. Kyoto univ. 23, 649-694 (1983) · Zbl 0548.76018