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Stability of solitary waves and weak rotation limit for the Ostrovsky equation. (English) Zbl 1064.35148
This paper is concerned with the Ostrovsky equation, which can be written in the form \((u_t - \beta u_{xxx} + (u^2)_x)_x = \gamma u\) for some constants \(\beta, \gamma \in \mathbb R\) with \(\gamma >0\). This equation describes the propagation of long internal surface waves in shallow water in the presence of a rotation. The authors consider a model where dispersion is considered but dissipation is neglected. They classify the existence and nonexistence of the dispersion parameter and prove that the set of solitary waves is stable with respect to perturbation in the case of positive dispersion. They also investigate the issue of passing to the limit as the rotation parameter tends to zero for solutions of the Cauchy problem on a bounded time interval.

35Q35 PDEs in connection with fluid mechanics
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35K35 Initial-boundary value problems for higher-order parabolic equations
35Q51 Soliton equations
76B55 Internal waves for incompressible inviscid fluids
Full Text: DOI
[1] Ablowitz, M; Clarkson, P, Solitons, nonlinear evolution equations and inverse scattering, (1991), Cambridge University Press Cambridge · Zbl 0762.35001
[2] Benilov, E.S, On the surface waves in a shallow channel with an uneven bottom, Stud. appl. math., 87, 1-14, (1992) · Zbl 0749.76007
[3] de Bouard, A; Saut, J.C, Solitary waves of generalized kadomtsev – petviashvili equations, Ann. inst. H. Poincaré anal. non linéaire, 14, 211-236, (1997) · Zbl 0883.35103
[4] F. Dias, G. Iooss, Water waves as a spatial dynamical system, Handbook of Mathematical Fluid Dynamics, North Holland, Amsterdam, Vol. II, 2003, pp. 443-499. · Zbl 1183.76630
[5] Fróhlich, J; Lieb, E.H; Loss, M, Stability of Coulomb systems with magnetic fields I. the one-electron atom, Commun. math. phys., 104, 251-270, (1986) · Zbl 0595.35098
[6] Galkin, V.N; Stepanyants, Yu.A, On the existence of stationary solitary waves in a rotating fluid, J. appl. math. mech., 55, 6, 939-943, (1991) · Zbl 0786.76016
[7] Gilman, O.A; Grimshaw, R; Stepanyants, Yu.A, Approximate and numerical solutions of the stationary Ostrovsky equation, Stud. appl. math., 95, 115-126, (1995) · Zbl 0843.76008
[8] Grimshaw, R, Evolution equations for weakly nonlinear long internal waves in a rotating fluid, Stud. appl. math., 73, 1-33, (1985) · Zbl 0572.76102
[9] Grimshaw, R; Iooss, G, Solitary waves of a coupled korteweg – de Vries system, Math. comput. simulation, 62, 31-40, (2003) · Zbl 1013.35073
[10] Kadomtsev, B.B; Petviashvili, V.I, On the stability of solitary waves in weakly dispersive media, Sov. phys. dokl., 15, 6, 539-541, (1970) · Zbl 0217.25004
[11] Kato, T, On the Cauchy problem for the (generalized) korteweg – de Vries equation, Stud. appl. math., adv. math. suppl. stud., 8, 93-126, (1983)
[12] Kato, T; Ponce, G, Commutator estimates and the Euler and navier – stokes equations, Commun. pure appl. math., 41, 891-907, (1988) · Zbl 0671.35066
[13] Levandosky, S.P, Stability and instability of fourth-order solitary waves, J. dyn. differential equations, 10, 1, 151-188, (1998)
[14] Lieb, E.H, On the lowest eigenvalue of Laplacian for the intersection of two domains, Invent. math., 74, 441-448, (1983) · Zbl 0538.35058
[15] P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Parts 1 and 2. Ann. Inst. H. Poincaré, Anal. Non Linéaire 1 (1984) 109-145, 223-283. · Zbl 0541.49009
[16] F. Linares, M. Scialom, On generalized Benjamin-type equations, preprint. · Zbl 1086.35094
[17] Y. Liu, Stability of solitary waves for the Ostrovsky equation with weak rotation, in preparation.
[18] Ostrovsky, L.A, Nonlinear internal waves in a rotating Ocean, Okeanologia, 18, 2, 181-191, (1978)
[19] Ostrovsky, L.A; Stepanyants, Yu.A, Nonlinear surface and internal waves in rotating fluids, () · Zbl 0719.76021
[20] Shatah, J, Stable standing waves of nonlinear klein – gordon equations, Commun. math. phys., 91, 3, 313-327, (1983) · Zbl 0539.35067
[21] Varlamov, V; Liu, Y, Cauchy problem for the Ostrovsky equation, Discrete continuous dyn. systems, 10, 3, 731-753, (2004) · Zbl 1059.35035
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