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Propagation of long nonlinear waves in a ponderable fluid beneath an ice sheet. (English. Russian original) Zbl 0692.76006
Fluid Dyn. 24, No. 1, 73-79 (1989); translation from Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 1989, No. 1, 88-95 (1989).
Summary: The stability of small-amplitude steady-state periodic solutions of \[ (*)\quad u_ t+uu_ x+\alpha u_{xxx}+u_{xxxxx}=0,\quad \alpha =\pm 1 \] in the neighbourhood of \(k=k_ n\) are investigated. The results of the investigations are consistent with those of S. E. Haupt and J. P. Boyd [Wave Motion 10, 83-98 (1988; Zbl 0628.76023)]. In Sec. 2. The stability of periodic waves not lying in the neighborhood of resonance is considered. It is shown that in the region of instability when \(\alpha =1\) steady-state solutions of the soliton type with oscillatory structure may exist. In Sec. 3. The properties of certain exact solutions - periodic waves and solitons - are studied in relation to the nature of the singular points of the dynamical system derived from (*). In Sec. 4, the evolution of rapidly decreasing Cauchy data is considered.

MSC:
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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References:
[1] S. E. Haupt and J. P. Boyd, ?Modeling nonlinear resonance: A modification to the Stokes perturbation expansion,? J. Wave Motion,10, 83 (1988). · Zbl 0628.76023
[2] J. A. Zufiria, ?Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth,? J. Fluid Mech.,184, 183 (1987). · Zbl 0634.76016
[3] A. V. Marchenko, ?Long waves in a shallow liquid beneath an ice sheet,? Prikl. Mat. Mekh.,52, 230 (1988). · Zbl 0691.76009
[4] V. I. Karpman, Nonlinear Waves in Dispersive Media [in Russian], Nauka, Moscow (1973).
[5] V. P. Korobeinikov and A. T. Il’ichev, ?On some methods of qualitative investigation of one-dimensional evolution equations in continuum mechancs,? in: Contributions to Nonlinear Motion, Longman, London (1988), p. 129.
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