Asymptotic models for internal waves

Journal des Mathematiques Pures et Appliquees

Volumn 89, Issue 6, 2008, Pages 538-566

  Bona, J L ^a   Lannes, D ^b   Saut, J C ^c  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^b UNIVERSITÉ DE BORDEAUX   (France)

^c UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Asymptotic models; Internal waves; Water waves

Indexed keywords

EID: 43749120480     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matpur.2008.02.003     Document Type: Article

References (36)

1. Albert J.P., Bona J.L., and Restrepo J.M. Solitary-wave solutions of the Benjamin equation. SIAM J. Appl. Math. 59 (1999) 2139-2161
2. Alvarez-Samaniego B., and Lannes D. Large time existence for 3D water-waves and asymptotics. Invent. Math. 171 (2008) 485-541
3. Barros R., Gavrilov S.L., and Teshukov V.M. Dispersive nonlinear waves in two-layer flows with free surface. I. Model derivation and general properties. Stud. Appl. Math. 119 (2007) 191-211
4. Beale J.T., Hou T.Y., and Lowengrub J.S. Growth rates for the linearized motion of fluid interfaces away from equilibrium. Comm. Pure Appl. Math. 46 (1993) 1269-1301
5. Benjamin T.B. Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29 (1967) 559-592
6. Benjamin T.B., and Bridges T.J. Reappraisal of the Kelvin-Helmholtz problem. Part 1. Hamiltonian structure. J. Fluid Mech. 333 (1997) 301-325
7. Benjamin T.B., Bona J.L., and Mahony J.J. Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A 272 (1972) 47-78
8. Bona J.L., Chen M., and Saut J.-C. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Part I. Derivation and linear theory. J. Nonlinear Sci. 12 (2002) 283-318
9. Bona J.L., Chen M., and Saut J.-C. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Part II. Nonlinear theory. Nonlinearity 17 (2004) 925-952
10. Bona J.L., Colin T., and Lannes D. Long wave approximations for water-waves. Arch. Rational Mech. Anal. 178 (2005) 373-410
11. Bona J.L., and Kalisch H. Models for internal waves in deep water. Discrete Cont. Dyn. Systems 6 (2000) 1-20
12. Camassa R., Choi W., Michallet H., Rusas P.-O., and Sveen J.K. On the realm of validity of strongly nonlinear asymptotic approximations for internal waves. J. Fluid Mech. 549 (2006) 1-23
13. Chazel F. Influence of bottom topography on water-waves in the long-wave regime. M2AN 41 (2007) 771-799
14. Choi W., and Camassa R. Weakly nonlinear internal waves in a two-fluid system. J. Fluid Mech. 313 (1996) 83-103
15. Choi W., and Camassa R. Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396 (1999) 1-36
16. Constantin P. On the Euler equations of incompressible fluids. Bull. Amer. Math. Soc. 44 4 (2007) 603-621
17. Craig W., Guyenne P., and Kalisch H. Hamiltonian long-wave expansions for free surfaces and interfaces. Comm. Pure Appl. Math. 58 (2005) 1587-1641
18. Craig W., Schanz U., and Sulem C. The modulational regime of three-dimensional water waves and the Davey-Stewartson system. Ann. Inst. H. Poincaré, Anal. Non Linéaire 14 (1997) 615-667
19. Craig W., Sulem C., and Sulem P.-L. Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5 (1992) 497-522
20. Helfrich K.R., and Melville W.K. Long nonlinear internal waves. Annual Review of Fluid Mechanics 38 (2006) 395-425
21. Iguchi T., Tanaka N., and Tani A. On the two-phase free boundary problem for two-dimensional water waves. Math. Ann. 309 (1997) 199-223
22. Jackson C.R. An atlas of internal solitary-like waves and their properties. http://www.internalwaveatlas.com/Atlas2_index.html
23. Joseph R. Solitary waves in a finite depth fluid. J. Phys. A: Math. Gen. 10 (1977) 225-227
24. Koop C.G., and Butler G. An investigation of internal solitary waves in a two-fluid system. J. Fluid Mech. 112 (1981) 225-251
25. Kubota T., Ko D.R.S., and Dobbs L. Propagation of weakly nonlinear internal waves in a stratified fluid of finite depth. J. Hydronautics 12 (1978) 157-165
26. Lannes D. Well-posedness of the water-waves equations. J. Amer. Math. Soc. 18 (2005) 605-654
27. Lannes D. Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators. J. Funct. Anal. 232 (2006) 496-539
28. Matsuno Y. A unified theory of nonlinear wave propagation in two-layer fluid systems. J. Phys. Soc. Japan 62 (1993) 1902-1916
29. H.Y. Nguyen, F. Dias, A Boussinesq system for two-way propagation of interfacial waves, Physica D (2008), in press
30. K. Ohi, T. Iguchi, A two-phase problem for capillary-gravity waves and the Benjamin-Ono equation, preprint, Disc. Cont. Dyn. Syst. (2008), in press
31. Segur H., and Hammack J.L. Soliton models of long internal waves. J. Fluid Mech. 118 (1982) 285-304
32. Sulem C., and Sulem P.-L. Finite time analyticity for the two- and three-dimensional Rayleigh-Taylor instability. Trans. Amer. Math. Soc. 287 (1985) 127-160
33. Sulem C., Sulem P.-L., Bardos C., and Frisch U. Finite time analyticity for the two- and three-dimensional Kelvin-Helmholtz instability. Comm. Math. Phys. 80 (1981) 485-516
34. Taylor M. Partial Differential Equations. I. Basic Theory. Applied Mathematical Sciences vol. 115 (1996), Springer-Verlag, New York
35. Walker L.R. Interfacial solitary waves in a two-fluid medium. Phys. Fluids 16 (1973) 1796-1804
36. Zakharov V.E. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2 (1968) 190-194