On the stability of solitary waves for the Ostrovsky equation

Liu, Yue

doi:10.1090/S0033-569X-07-01065-8

Author: Yue Liu
Journal: Quart. Appl. Math. 65 (2007), 571-589
MSC (2000): Primary 35Q53, 35B60, 76B25
DOI: https://doi.org/10.1090/S0033-569X-07-01065-8
Published electronically: July 26, 2007
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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

Eugene S. Benilov, On the surface waves in a shallow channel with an uneven bottom, Stud. Appl. Math. 87 (1992), no. 1, 1–14. MR 1165932, DOI https://doi.org/10.1002/sapm19928711
Haïm Brézis and Elliott Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. MR 699419, DOI https://doi.org/10.1090/S0002-9939-1983-0699419-3
Anne de Bouard and Jean-Claude Saut, Solitary waves of generalized Kadomtsev-Petviashvili equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 2, 211–236 (English, with English and French summaries). MR 1441393, DOI https://doi.org/10.1016/S0294-1449%2897%2980145-X
T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), no. 4, 549–561. MR 677997
Jürg Fröhlich, Elliott H. Lieb, and Michael Loss, Stability of Coulomb systems with magnetic fields. I. The one-electron atom, Comm. Math. Phys. 104 (1986), no. 2, 251–270. MR 836003
V. M. Galkin and Yu. A. Stepan′yants, On the existence of stationary solitary waves in a rotating fluid, Prikl. Mat. Mekh. 55 (1991), no. 6, 1051–1055 (Russian); English transl., J. Appl. Math. Mech. 55 (1991), no. 6, 939–943 (1992). MR 1150448, DOI https://doi.org/10.1016/0021-8928%2891%2990148-N
O. A. Gil′man, R. Grimshaw, and Yu. A. Stepan′yants, Approximate analytical and numerical solutions of the stationary Ostrovsky equation, Stud. Appl. Math. 95 (1995), no. 1, 115–126. MR 1342073, DOI https://doi.org/10.1002/sapm1995951115
R. Grimshaw, Evolution equations for weakly nonlinear, long internal waves in a rotating fluid, Stud. Appl. Math. 73 (1985), no. 1, 1–33. MR 797556, DOI https://doi.org/10.1002/sapm19857311

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.

Elliott H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983), no. 3, 441–448. MR 724014, DOI https://doi.org/10.1007/BF01394245

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145 (English, with French summary). MR 778970

Yue Liu and Vladimir Varlamov, Stability of solitary waves and weak rotation limit for the Ostrovsky equation, J. Differential Equations 203 (2004), no. 1, 159–183. MR 2070389, DOI https://doi.org/10.1016/j.jde.2004.03.026

Robert M. Miura, Clifford S. Gardner, and Martin D. Kruskal, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Mathematical Phys. 9 (1968), 1204–1209. MR 252826, DOI https://doi.org/10.1063/1.1664701

Ostrovsky, L. A., Nonlinear internal waves in a rotating ocean, Oceanologia, 18(1978), 181-191.

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.

V. Varlamov and Yue Liu, Cauchy problem for the Ostrovsky equation, Discrete Contin. Dyn. Syst. 10 (2004), no. 3, 731–753. MR 2018877, DOI https://doi.org/10.3934/dcds.2004.10.731