|
On the stability of solitary waves for the Ostrovsky equation
Author(s):
Yue
Liu
Journal:
Quart. Appl. Math.
65
(2007),
571-589.
MSC (2000):
Primary 35Q53, 35B60, 76B25
Posted:
July 26, 2007
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
Considered herein is the stability of solitary-wave solutions of the Ostrovsky equation which is an adaptation of the Korteweg-de Vries equation widely used to describe the effect of rotation on the surface and internal solitary waves or the capillary waves. It is shown that the ground state solitary waves are global minimizers of energy functionals with the constrained variational problem and are deduced to be nonlinearly stable for the small effect of rotation. The analysis makes frequent use of the variational properties of the ground states.
References:
-
- [Be]
- Benilov, E. S., On the surface waves in a shallow channel with an uneven bottom, Stud. Appl. Math., 87(1992), 1-14. MR 1165932 (93b:76009)
- [BrLi]
- Brézis, H. and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1993), 437-477. MR 699419 (84e:28003)
- [BoSa]
- de Bouard, A. and Saut, J. C., Solitary waves of generalized Kadomtsev-Petviashvili equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, A14(1997), 211-236. MR 1441393 (98a:35115)
- [CaLi]
- Cazenave, T. and Lions, P. L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. MR 677997 (84i:81015)
- [FrLiLo]
- Fröhlich, J., Lieb, E. H., and Loss, M., Stability of coulomb systems with magnetic fields I. The one electron atom, Commun. Math. Phys., 104(1986), 251-270. MR 836003 (88a:81023a)
- [GaSt]
- Galkin, V. N. and Stepanyants, Y. A., On the existence of stationary solitary waves in a rotating field, J. Appl. Math. Mech., 55 (1991), 939-943. MR 1150448 (92j:76096)
- [GiGrSt]
- Gilman, O. A., Grimshaw, R., and Stepanyants, Y. A., Approximate and numerical solutions of the stationary Ostrovsky equation, Stud. Appl. Math., 95(1995), 115-126. MR 1342073 (96c:76018)
- [Gr]
- Grimshaw, R., Evolution equations for weakly nonlinear long internal waves in a rotating fluid, Stud. Appl. Math., 73(1985), 1-33. MR 797556 (87a:76029)
- [KaPe]
- Kadomtsev, B. B. and Petviashvili, V. I., On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl., 15(6)(1970), 539-541.
- [Lie]
- Lieb, E. H., On the lowest eigenvalue of Laplacian for the intersection of two domains, Invent. Math., 74(1983), 441-448. MR 724014 (85e:35090)
- [Lio]
- Lions, P. L., The concentration compactness principle in the calculus of variations. The locally compact case, Part 1 and Part 2, Ann. Inst. H. Poincaré, Anal. Nonlinéaire, 1(1984), 109-145, 223-283. MR 778970 (87e:49035a); MR 0778974 (87e:49035b)
- [LiVa]
- Liu, Yue and Varlamov, V., Stability of solitary waves and weak rotation limit for the Ostrovsky equation, J. Differential Equations, 203(2004), 159-183. MR 2070389 (2005f:76022)
- [MiGaKr]
- Miura, R. M., Gardner, C. S., and Kruskal, M. D., Korteweg-de Vries equation and generalizations II, Existence conservation laws and constant of motion, J. Math. Phys., 9(1968), 1204-1209. MR 0252826 (40:6042b)
- [Os]
- Ostrovsky, L. A., Nonlinear internal waves in a rotating ocean, Oceanologia, 18(1978), 181-191.
- [OsSt]
- Ostrovsky, L. A. and Stepanyants, Y. A., Nonlinear surface and internal waves in rotating fluids, Research Reports in Physics, Nonlinear Waves 3, Springer, Berlin, Heidelberg, 1990.
- [VaLi]
- Varlamov, V. and Liu, Yue, Cauchy problem for the Ostrovsky equation, Discrete Dynam. Systems, 10(2004), 731-751. MR 2018877 (2004j:35235)
Similar Articles:
Retrieve articles in Quarterly of Applied Mathematics
with MSC
(2000):
35Q53, 35B60, 76B25
Retrieve articles in all Journals with MSC
(2000):
35Q53, 35B60, 76B25
Additional Information:
Yue
Liu
Affiliation:
Department of Mathematics, University of Texas, Arlington, Texas 76019
Email:
yliu@uta.edu
PII:
S0033-569X-07-01065-8
Keywords:
Ostrovsky equation,
solitary waves,
stability,
weak rotation
Received by editor(s):
November 20, 2006
Posted:
July 26, 2007
Copyright of article:
Copyright
2007,
Brown University
The copyright for this article reverts to public domain after 28 years from publication.
|
