On the propagation of tsunami waves, with emphasis on the tsunami of 2004

Discrete and Continuous Dynamical Systems - Series B

Volumn 12, Issue 3, 2009, Pages 525-537

  Adrian, Constantin ^a  

^a UNIVERSITY OF VIENNA   (Austria)

Author keywords

Dispersion; Shallow water; Soliton; Tsunami

Indexed keywords

EID: 70350677172     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2009.12.525     Document Type: Article

References (58)

1. B. Alvarez-Samaniego and D. Larmes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541.
2. C. J. Amick and J. F. Toland, On solitary water-waves of finite amplitude, Arch. Ration. Mech. Anal., 76 (1981), 9-95.
3. T. P. Barnett and K. E. Kenyon, Recent advances in the study of wind waves, Rep. Prog. Phys., 38 (1975), 667-729.
4. T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London A, 328 (1972), 153-183.
5. T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.
6. A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
7. A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
8. E. Bryant, "Tsunami: The Underrated Hazard," Springer, Heidelberg, 2008.
9. R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
10. A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
11. A. Constantin, Solitons from the Lagrangian perspective, Discrete Contin. Dyn. Syst., 19 (2007), 469-481.
12. A. Constantin, On the relevance of soliton theory to tsunami modelling, Wave Motion, DOI: 10.1016/j.wavemoti.2009.05.002
13. A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603.
14. A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243.
15. A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.
16. A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the CamassaHolm equation, Inverse Problems, 22 (2006), 2197-2207.
17. A. Constantin and D. Henry, Solitons and tsunamis, Z. Naturforsch., 64a (2009), 65-68.
18. A. Constantin and R. S. Johnson, Modelling tsunamis, J. Phys. A, 39 (2006), L215-L217.
19. A. Constantin and R. S. Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonl. Math. Phys., 15 (2008), 58-73.
20. A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res., 40 (2008), 175-211.
21. A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
22. A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.
23. A. Constantin and W. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-422.
24. A. Constantin and W. Strauss, Exact periodic water waves with vorticity, C. R. Math. Acad. Sci. Paris, 335 (2002), 797-800.
25. A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.
26. A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950.
27. A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Phil. Trans. Roy. Soc. London A, 365 (2007), 2227-2239.
28. W. Craig, Surface water waves and tsunamis, J. Dynam. Differential Equations, 18 (2006), 525-549.
29. W. Craig, Non-existence of solitary water waves in three dimensions, Phil. Trans. Roy. Soc. London Ser. A, 360 (2002), 2127-2135.
30. W. Craig and P. Sternberg, Symmetry of solitary waves, Comm. Partial Differential Equations, 13 (1988), 603-633.
31. A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302.
32. P. G. Drazin and R. S. Johnson, "Solitons: An Introduction," Cambridge University Press, Cambridge, 1989.
33. 1-localized solutions and stability of the train of N solitary waves for the Camassa-Holm equation, Philos. Trans. Roy. Soc. London A, 365 (2007), 2313-2331.
34. A. T. Filippov, "The Versatile Soliton," Birkhäuser Boston, 2000.
35. A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246.
36. R. Grimshaw, Solitary waves propagating over variable topography, in "Tsunami and Nonlinear Waves" (ed. A. Kundu), Springer, (2007), 51-64.
37. J. L. Hammack, A note on tsunamis: their generation and propagation in an ocean of uniform depth, J. Fluid Mech., 60 (1973), 769-799.
38. J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves, J. Fluid Mech., 84 (1978), 337-358.
39. S. Israwi, Variable depth KDV equations and generalizations to more nonlinear regimes, preprint, arXiv:0901.3201v1.
40. R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves," Cambridge University Press, Cambridge, 1997.
41. M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, in "Tsunami and Nonlinear Waves," (ed. A. Kundu), Springer, (2007), 31-49.
42. J. Lighthill, "Waves in Fluids," Cambridge University Press, Cambridge, 1978.
43. L.-F. Liu Philip, P. Lynett et al., Observations by the international Tsunami survey team in Sri Lanka, Science, 308 (2005), 1595.
44. J. H. Maddocks and R. L. Sachs, On the stability of KdV multi-solitons, Comm. Pure Appl. Math., 46 (1993), 867-901.
45. P. A. Madsen, D. R. Fuhrman and H. A. Schaeffer, On the solitary wave paradigm for tsunamis, J. Geophys. Res. - Oceans, 113 (2008), Art. No. C12012
46. Y. Martel, F. Merle and T. Mizumachi, Description of the inelastic collision of two solitary waves for the BBM equation, preprint, arXiv:0803.4020v2.
47. Y. Matsuno, Nonlinear evolutions of surface gravity waves on fluid of finite depth, Phys. Rev. Lett., 69 (1992), 609-611.
48. P. J. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Camb. Phil. Soc., 85 (1979), 143-160.
49. D. H. Peregine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.
50. R. V. Sarma, M. V. Ramana et al., Morphological features in the Bay of Bengal, J. Ind. Geophys. Union, 4 (2000), 185-190.
51. G. Schneider and C. E. Wayne, On the validity of 2D-surface water wave models, GAMM Mitt. Ges. Angew. Math. Mech., 25 (2002), 127-151.
52. H. Segur, Integrable models of waves in shallow water, in "Probability, Geometry and Integrable Systems," (eds. M. Pinsky and B. Birnir), Math. Sci. Res. Inst. Publ., Cambridge University Press, 2008, 345-371.
53. H. Segur, Waves in shallow water, with emphasis on the tsunami of 2004, in "Tsunami and Nonlinear Waves," (ed. A. Kundu), 2007, Springer, 3-29.
54. F. Serre, Contribution à l'étude des écoulements permanents et variables dans les canaux, La Houille Blanche, 3 (1953), 374-388.
55. J. J. Stoker, "Water Waves. The Mathematical Theory with Applications," Wiley-Interscience, New York, 1992.
56. S. S. Voit, Tsunamis, Ann. Rev. Fluid Mech., 19 (1987), 217-236.
57. G. B. Whitham, "Linear and Nonlinear Waves," John Wiley & Sons, Inc., New York, 1999.
58. D. H. Zhang, T. L. Yip and C.-O. Ng, Predicting tsunami arrivals: Estimates and policy implications, Marine Policy, 33 (2009), 643-650.