The long-wave limit for the water wave problem I. The case of zero surface tension

Communications on Pure and Applied Mathematics

Volumn 53, Issue 12, 2000, Pages 1475-1535

  Schneider, Guido ^a   Wayne, C Eugene ^b  

^a UNIVERSITY OF BAYREUTH   (Germany)

^b BOSTON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034364719     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V     Document Type: Article

References (41)

1. Ablowitz, M. J.; Segur, H. Solitons and the inverse scattering transform. SIAM Studies in Applied Mathematics, 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981.
2. Airy, G. B. Tides and waves. Encyclopedia Metropolitana 5 (1845), 241-396, London.
3. Beale, J. T.; Hou, T. Y.; Lowengrub, J. S. Growth rates for the linearized motion of fluid interfaces away from equilibrium. Comm. Pure Appl. Math. 46 (1993), no. 9, 1269-1301.
4. Benjamin, T. B.; Bona, J. L.; Mahony, J. J. Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A 272 (1972), no. 1220, 47-78.
5. Boussinesq, M. J. Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. C. R. Acad. Sci. Paris Sér. A-B 72 (1871 ), 755-759.
6. Boussinesq, M. J. Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17 (1872), no. 2, 55-108.
7. Boussinesq, M. J. Essai sur la théorie des eaux courantes. Mémoires présentés par divers savants à l'Académie des Sciences Inst. France (séries 2) 23 (1877), 1-680.
8. Byatt-Smith, J. G. B. An integral equation for unsteady surface waves and a comment on the Boussinesq equation. J. Fluid Mech. 49 (1971), 625-633.
9. Collet, P.; Eckmann, J.-P. The time dependent amplitude equation for the Swift-Hohenberg problem. Comm. Math. Phys. 132 (1990), no. 1, 139-153.
10. Craig, W. An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Comm. Partial Differential Equations 10 (1985), no. 8, 787-1003.
11. Eckhaus, W. The Ginzburg-Landau manifold is an attractor. J. Nonlinear Sci. 3 (1993), no. 3, 329-348.
12. Eckhaus, W.; van Harten, A. The inverse scattering transformation and the theory of solitons. An introduction. North-Holland Mathematics Studies, 50. North-Holland, Amsterdam-New York, 1981.
13. Friedrichs, K. O.; Hyers, D. H. The existence of solitary waves. Comm. Pure Appl. Math. 7 (1954), 517-550.
14. Iooss, G. Capillary-gravity water-waves problem as a dynamical system. Proceedings of the IU-TAM/ISIMM Symposium on Structure and Dynamics of Nonlinear Waves in Fluids (Hannover, 1994), 42-57. Advanced Series in Nonlinear Dynamics, 7. World Scientific, River Edge, N.J., 1995.
15. Kano, T.; Nishida, T. A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves. Osaka J. Math. 23 (1986), no. 2, 389-413.
16. Kato, T. Quasi-linear equations of evolution, with applications to partial differential equations. Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), 25-70. Lecture Notes in Mathematics, 448. Springer, Berlin, 1975.
17. Kato, T. On the Korteweg-de Vries equation. Manuscripta Math. 28 (1979), 89-99.
18. Kato, T. On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Studies in applied mathematics, 93-128. Advances in Mathematics, Supplementary Studies, 8. Academic, New York-London, 1983.
19. Kenig, C. E.; Ponce, G.; Vega, L. Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Amer. Math. Soc. 4 (1991), no. 2, 323-347.
20. Kirrmann, P.; Schneider, G.; Mielke, A. The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), no. 1-2, 85-91.
21. Korteweg, D. G.; de Vries, G. On the change of form of long waves advancing in the rectangular canal and a new type of long stationary waves. Phil. Mag. 39 (1895), 422-443.
22. Landau, L. D.; Lifshitz, E. M. Lehrbuch der theoretischen Physik. Band 6. Hydrodynamik. Akademie, Berlin, 1991.
23. Long, R. R. The initial-value problem for long waves of finite amplitude. J. Fluid. Mech. 20 (1964), 161-170.
24. Mattioli, F. Decomposition of the Boussinesq equations for shallow-water waves into a set of coupled Korteweg-de Vries equations. Phys. Fluids A 3 (1991), no. 10, 2355-2359.
25. Meyer, R. E. On the meaning of the Boussinesq and Korteweg-de Vries equations. Bull. Calcutta Math. Soc. 71 (1979), no. 3, 121-130.
26. Nalimov, V. I. The Cauchy-Poisson problem. Dinamika Splošn. Sredy Vypo 18 Dinamika Zidkost. so Svobod. Granicami (1974), 104-210, 254.
27. Pelinovsky, E. N. Models of Tsunami waves. Computational techniques and applications: CTAC95. Proceedings of the Seventh Biennial Conference held at Swinburne University of Technology, Melbourne, July 3-5, 1995, 619-624. Edited by Robert L. May and Alan K. Easton. World Scientific, River Edge, N.J., 1996.
28. Pierce, R. D.; Wayne, C. E. On the validity of mean-field amplitude equations for counterpropagating wavetrains. Nonlinearity 8 (1995), no. 5, 769-779.
29. Rayleigh, L. On waves. Philos. Mag. 5 (1876), no. 1.
30. Russell, J. S. Report on waves. Rep. 14th Meet. Brit. Assoc. Adv. Sci., York, London, John Murray, (1844) 311-390.
31. Schneider, G. Error estimates for the Ginzburg-Landau approximation. Z. Angew. Math. Phys. 45 (1994), no. 3, 433-457.
32. Schneider, G. Limits for the Korteweg-de Vries approximation. Z. Angew. Math. Mech. 76 (1996), supp. 2, 341-344.
33. Schneider, G. The long wave limit for a Boussinesq equation. SIAM J. Appl. Math. 58 (1998), no. 4, 1237-1245.
34. Schneider, G.; Wayne, C. E. The long wave limit for the water wave problem. II. With surface tension. Preprint, 2000.
35. Shinbrot, M. The initial value problem for surface waves under gravity. I. The simplest case. Indiana Univ. Math. J. 25 (1976), no. 3, 281-300.
36. van Harten, A. On the validity of the Ginzburg-Landau equation. J. Nonlinear Sci. 1 (1991), no. 4, 397-422.
37. Wu, S. Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130 (1997), no. 1, 39-72.
38. Wu, S. Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc. 12 (1999), no. 2, 445-495.
39. Yosihara, H. Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci. 18 (1982), no. 1, 49-96.
40. Yosihara, H. Capillary-gravity waves for an incompressible ideal fluid. J. Math. Kyoto Univ. 23 (1983), no. 4, 649-694.
41. Zabusky, N. J.; Kruskal, M. D. Interactions of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15 (1965), 240-243.