RSS feeds
РусскийEnglish
Issue 11, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

1995

 / 

December

←  →

Physics of our days


Nonlinear long waves on water and solitons


Universite des Sciences et Technologies de Lille1, U.F.R.de Mathematiques Pures et Appliquees, Department de Mecanique Fondamentale, Batiment M 3, Villeneuve d’Ascq Cedex, 59655, France

The water wave problem has been pivotal in the history of nonlinear wave theory. This problem is one of the most interesting and successful applications of nonlinear hydrodynamics. Waves on the free surface of a body of water (perfect liquid) have always been a fascinating subject, for they represent a familiar yet complex phenomenon, easy to observe but very difficult to describe! The archetypical model equations of Kordeweg and de Vries and of Boussinesq, for example, were originally derived as approximations for water waves, and research into the problem has been sustained vigorously up to the present day. In the present paper, the derivation of the model equations is given in depth and rational use is made of asymptotic methods. Indeed, it is important to understand that in some cases the derivation of these approximate equations is intuitive and heuristic. In fact, it is not clear how to insert the model equation under consideration into a hierarchy of rational approximations, which in turn result from the exact formulation of the selected water wave problem.

Fulltext pdf (1.4 MB)
Fulltext is also available at DOI: 10.1070/PU1995v038n12ABEH000124
PACS: 47.10.+g, 47.20.Ky, 92.10.Hm, 43.25.Rq (all)
DOI: 10.1070/PU1995v038n12ABEH000124
URL: https://ufn.ru/en/articles/1995/12/d/
A1995TR07000005
Citation: Zeytounian R Kh "Nonlinear long waves on water and solitons" Phys. Usp. 38 1333–1381 (1995)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Зейтунян Р Х «Нелинейные длинные волны на поверхности воды и солитоны» УФН 165 1403–1456 (1995); DOI: 10.3367/UFNr.0165.199512f.1403

References (142) ↓ Cited by (27) Similar articles (2)

  1. Batchelor G K An Introduction To Fluid Dynamics (Cambridge: Cambridge University Press, 1970)
  2. Crapper G D Introduction To Water Waves (Chichester, England: Ellis Horwood, 1984)
  3. Dai Shi-quiang Appl. Math. Mech. (Wuhan, China) 3 ((6)) 777 (1982), (English ed.)
  4. Hemlholtz H Philos. Mag. (1868), Nov.
  5. Itoh N I Proc. R. Soc. London Ser. A 375 565 (1981)
  6. Keller J B J. Fluid Mech. 4 607 (1958)
  7. Leibovich S, Seebass A R (Eds.) Nonlinear Waves (Ithaca, NY: Cornell University Press, 1974)
  8. Miles J W Phys. Fluids 19 1063 (1976)
  9. Nayfeh A H Introduction To Perturbation Techniques (New York: Wiley, 1981)
  10. Rayleigh (Lord) Proc. London Math. Soc. (1) x. 4 (1879), (Papers i.361)
  11. Riabouchinsky D C. R. Acad. Sci. 195 998 (1932)
  12. Scorer J J Water Waves (New York: Interscience, 1953)
  13. Shrira V I Izv. AN SSSR Fiz. Atm. Okeana 17 ((1)) 76 (1981)
  14. Usher J R, Craik A D D J. Fluid Mech. 66 209 (1974)
  15. Rankine W J H Philos. Trans. 48 1 (1863)
  16. Boussinesq M J C. R. Acad. Sci. 72 755 (1871)
  17. Boussinesq M J C. R. Acad. Sci. 73 256 (1871)
  18. Boussinesq M J J. Math. Pures Et Appl. 17 ((2)) 55 (1872)
  19. Boussinesq M J Essai Sur La Theorie Des Eaux Courantes. Memoires Presentes Par Divers Savants A L’Academie Des Sciences-Institut De France (serie 2) 23 (1) (1877); Boussinesq M J Essai Sur La Theorie Des Eaux Courantes. 24 1 (1877)
  20. de Saint-Venant B Theorie Du Mouvement Non Permanent Des Eaux. Institut De France, Acad. Des Sci. De Paris 73(3) 147 (1871); de Saint-Venant B Theorie Du Mouvement Non Permanent Des Eaux. 73 ((4)) 237 (1871)
  21. Russell J S ""Report on waves"" Report Of The Fourteenth Meeting Of The British Association For The Advancement Of Science, York, 1844 (London: John Murray, 1844) p. 311
  22. Miles J W Annu. Rev. Fluid Mech. 12 11 (1980)
  23. Ursell F Proc. Cambridge Philos. Soc. 49 685 (1953)
  24. Korteweg D J, de Vries G Philos. Mag. 39 422 (1895)
  25. Zabusky N S, Kruskal M D Phys. Rev. Lett. 15 240 (1965)
  26. Stokes G G Philos. Mag. 31 314 (1891); Stokes G G Mathematics And Physics Papers Vol. Vol. 5 (Cambridge University Press, 1891) p. 160
  27. Newell A C Solitons In Mathematics And Physics (Philadelphia: SIAM, 1985)
  28. Kruskal M D Proceedings IBM Science Computing Symposium On Large-Scale Problems In Physics (White Plains, New York: IBM Data Processing Division, 1965) p. 43
  29. Kruskal M D Nonlinear Wave Motion Vol. 15 (Lectures in Appl. Math., 1974); Ed. A C Newell) Nonlinear Wave Motion (Providence, RI: AMS) p. 61
  30. Zabusky N S J. Comput. Phys. 43 195 (1981)
  31. Drazin P G, Johnson R S Solitons: An Introduction (Cambridge: Cambridge University Press, 1990)
  32. Dodd R K et al Solitons And Nonlinear Equations (New York: Academic Press, 1982)
  33. Freeman N C, Davey A Proc. R. Soc. London Ser. A 344 427 (1975)
  34. Kadomtsev B B, Petviashvili V I Dokl. Akad. Nauk SSSR 192 753 (1970); Kadomtsev B B, Petviashvili V I Sov. Phys. Dokl. 15 539 (1970)
  35. Zakharov V E, Shabat A B Zh. Eksp. Teor. Fiz. 61 118 (1971); Zakharov V E, Shabat A B Sov. Phys. JETP 34 62 (1972)
  36. Hasimoto H, Ono H J. Phys. Soc. Jpn. 33 805 (1972)
  37. Zakharov V E J. Appl. Mech. Tech. Phys 2 190 (1968)
  38. Benney D J, Roskes G J. Stud. Appl. Math. 48 377 (1969)
  39. Davey A, Stewartson K Proc. R. Soc. London Ser. A 338 101 (1974)
  40. Djordjevic V D, Redekopp L G J. Fluid Mech. 79 703 (1977)
  41. Ablowitz M, Segur H J. Fluid Mech. 92 691 (1979)
  42. Craik A D D Wave Interactions And Fluid Flows (Cambridge: Cambridge University Press, 1985)
  43. Mei C C The Applied Dynamics Of Ocean Surface Waves (New York: Wiley, 1983)
  44. Infeld E, Rowlands G Nonlinear Waves, Solitons And Chaos (Cambridge: Cambridge University Press, 1992)
  45. Zeytounian R Kh J. Eng. Math. 28 261 (1994)
  46. Peregrine O H J. Fluid Mech. 27 815 (1967); Peregrine O H J. Fluid Mech. 25 311 (1966)
  47. Ono H J. Phys. Soc. Jpn. 32 332 (1972)
  48. Johnson R S Proc. Cambridge Philos. Soc. 73 183 (1973)
  49. Gardner C S, Greene J M, Kruskal M D, Miura R M Phys. Rev. Lett. 19 1095 (1967)
  50. Gardner C S et al Commun. Pure Appl. Math. 27 97 (1974)
  51. Masden O S, Mei C C J. Fluid Mech. 39 781 (1969)
  52. Knickerbocker C J, Newell A C J. Fluid Mech. 98 803 (1980)
  53. Freeman N C Adv. Appl. Mech. 20 1 (1980)
  54. Rosales R R, Papnicolaou G C Stud. Appl. Math. 68 89 (1983)
  55. Benilov E S Stud. Appl. Math. 87 1 (1992)
  56. Xue-Nong Chen Phys. Fluids A1 ((12)) 2058 (1989)
  57. Sugimoto N, Kakutani T J. Phys. Soc. Jpn. 43 1469 (1977)
  58. Jeffrey A, Kawahara T Asymptotic Methods In Nonlinear Wave Theory (London: Pitman, 1982)
  59. Shinbrot M Lectures On Fluid Mechanics (New York: Gordon and Breach, 1973)
  60. Friedrichs K O, Hyers D H Commun. Pure Appl. Math. 7 517 (1954)
  61. Gerber R J. Math. (Paris) 34 185 (1955)
  62. Lighthill J Waves In Fluids (Cambridge: Cambridge University Press, 1979)
  63. Whitham G B Linear And Nonlinear Waves (New York: Wiley, 1974)
  64. Debnath L ""Bifurcation and nonlinear instability in applied mathematics"" Nonlinear Analysis (Ed. Th M Rassias) (Singapore: World Scientific, 1987) p. 161
  65. Zeytounian R Kh "Nonlinear Long Surface Waves in Shallow Water" Preprint (Universite de Lille I, LML, 1993) p. 224; Zeytounian R Kh Notes Sur Les Ecoulements Rotationnels De Fluides Parfaits Lecture Notes In Physics, Vol. 27) (Berlin: Springer, 1974)
  66. Luke J C J. Fluid Mech. 27 395 (1967)
  67. Brekhovskikh L M, Goncharova V V Vvedenie V Mekhaniku Sploshnykh Sred (v Prilozhenii K Teorii Voln, Introduction To Mechanics Of Continuous Media: Applications To The Theory Of Waves) (Moscow: Nauka, 1985)
  68. Satsuma J J. Phys. Soc. Jpn. 40 286 (1976)
  69. Segur H, Finkel A Stud. Appl. Math. 73 183 (1985)
  70. Miles J W J. Fluid Mech. 79 157 (1977)
  71. Chu V C, Mei Ch C J. Fluid Mech. 41 873 (1970)
  72. Whitham G B J. Fluid Mech. 27 399 (1967)
  73. Davey A J. Fluid Mech. 53 769 (1972)
  74. Benjamin T D, Feir J E J. Fluid Mech. 27 417 (1967)
  75. Johnson R S Proc. R. Soc. London Ser. A 357 131 (1977)
  76. Anker D, Freeman N C Proc. R. Soc. London Ser. A 360 529 (1978)
  77. Zakharov V E, Shabat A B Funct. Anal. Appl. 8 226 (1974)
  78. Johnson R S Proc. R. Soc. London Ser. A 347 537 (1976)
  79. Johnson R S Nonlinear Waves (Ed. L Debnath) (Cambridge: Cambridge University Press, 1983) p. 25
  80. Liu P L-F, Yoon S B, Kirby J T J. Fluid Mech. 153 185 (1985)
  81. Kakutani T J. Phys. Soc. Jpn. 30 272 (1971)
  82. Johnson R S J. Fluid Mech. 60 813 b) ((1973)
  83. Kaup D J, Newell A C Proc. R. Soc. London Ser. A 361 413 (1978)
  84. Russell J S ""Report on waves"" Report Of The Fourteenth Meeting Of The British Association For The Advancement Of Science, York, 1844 (London: John Murray, 1844) p. 311-390, + 57 plates
  85. Lax P D Commun. Pure Appl. Math. 21 467 (1968)
  86. Zakharov V E, Faddeev L D Funct. Anal. Appl. 5 280 (1972)
  87. Jeffrey A, Kakutani T SIAM Review 14 582 (1972)
  88. Scott A C, Chu F Y F, McLaughlin D W ""The soliton: a new concept in applied science"" Proc. IEEE 61 1443 (1973)
  89. Miura R M SIAM Review 18 412 (1976); Miura R M Backlund Transformation, The Inverse Scattering Method, Solitons, And Their Applications Lecture Notes In Mathematics Vol. 515) (Berlin: Springer, 1976)
  90. Calogero F Nonlinear Evolution Equations Solvable By The Spectral Transform Method (London: Pitman, 1978)
  91. Bhatnagar P L Nonlinear Waves In One-Dimensional Dispersive Systems (Oxford: Oxford University Press, 1979)
  92. Kadomtsev B B Phenomenes Collectifs Dans Les Plasmas (Moscow: Mir, 1979)
  93. Miles J W J. Fluid Mech. 106 131 (1981)
  94. Lamb C L Elements Of Soliton Theory (New York: Wiley, 1980)
  95. Bullough R K, Caudry P W Solitons (Berlin: Springer, 1980)
  96. Zakharov V E, Manakov S V, Novikov S P, Pitaevskii L P Teoriya Solitonov. Metod Obratnoi Zadachi (Moscow: Nauka, 1980); Novikov S P, Manakov S V, Pitaevskii L P, Zakharov V E Theory Of Solitons, The Inverse Scattering Method (New York: Plenum, 1984)
  97. Ablowitz M J, Segur H Solitons And The Inverse Spectral Transform (Philadelphia: SIAM, 1981)
  98. Taniuti T, Nishihara K Nonlinear Waves (London: Pitman, 1983)
  99. Debnath L Nonlinear Waves (Cambridge: Cambridge University Press, 1983)
  100. Debnath L Advances In Nonlinear Waves Vol. Vols 1, 11 (London: Pitman, 1984)
  101. Atiyah M, Gibbon J D, Wilson G Philos. Trans. R. Soc. London Ser. A 315 333 (1985)
  102. Takhtadjian L A, Faddeev L D Gamil’tonov Podkhod V Teorii Solitonov (Hamiltonian Approach To Soliton Theory) (Moscow: Nauka, 1980)
  103. Ablowitz M J, Fuchssteiner B, Kruskal M Topics In Soliton Theory And Exactly Solvable Nonlinear Equations (Singapore: World Scientific, 1987)
  104. Drazin P G Solitons (Cambridge: Cambridge University Press, 1983)
  105. Gel’fand I M, Levitan B M Am. Math. Soc. Translation 1 253 (1955)
  106. Wahlquist H D, Estabrook F B Phys. Rev. Lett. 31 1386 (1973)
  107. Eisenhart L P A Treatise On The Differential Geometry Of Curves And Surfaces (Ginn, 1909), reprinted New York: Dover, 1960)
  108. Hirota R Phys. Rev. Lett. 27 1192 (1971)
  109. Hopf E Commun. Pure Appl. Math. 3 201 (1950)
  110. Cole J D Q. Appl. Math. 9 225 (1951)
  111. Krasovskii Yu P Zh. Vychisl. Mat-Mat. Fiz. 1 5 (1961)
  112. Lavrentiev M A Sib. Mat. Zh. 3 ((5)) 715 (1962)
  113. Lavrentiev M A J. Anal. Math. XIX 217 (1967)
  114. Ter-Krikorov A M Prikl. Mat. Mekh. 24 622 (1960)
  115. Ter-Krikorov A M Zh. Vychisl. Mat. Mat. Fiz. 1 (1961)
  116. Ter-Krikorov A M J. Mec. 11 ((3)) 351 (1963)
  117. Ovsiannikov L V Actes, Congres Intern. Maths. 3 137 (1970)
  118. Moiseev N N Mekhanika V SSSR Za 50 Let (Mechanics In The USSR In 50 Years) Vol. Vol. 2 (Moscow: Nauka, 1970) p. 55
  119. Showalter R E J. Math. Pure Appl. 56 ((2)) 123 (1977)
  120. Keady G, Norbury J Math. Proc. Cambridge Philos. Soc. 83 137 (1978)
  121. Ovsyannikov L V Trudy-75 Let Akademiku Petrovskomu (Contributions To Celebrate 75th Birthday Of Academician Petrovshii) (Moscow: Izd. Mosk. Univ., 1978)
  122. Kano T, Nishida T J. Math. Kyoto Univ. 19 335 (1979)
  123. Berger N SIA M J. Appl. Math. 31 438 (1976)
  124. Shen M C Nonlinear Waves (Ed. L Debnath) (Cambridge: Cambridge University Press, 1983) p. 69
  125. Kato T Lecture Notes In Mathematics 448 25 (1975)
  126. Kato T Research Notes In Mathematics 53 293 (1980)
  127. Amick C J, McLeod J B SAA CM 1 ((2)) 127 (1991)
  128. Kirchgassner K Adv. Appl. Mech. 26 135 (1988)
  129. Kano T, Nishida T Osaka J. Math. 23 389 (1986)
  130. Craig W Commun. Part. Differ. Eq. 10 787 (1985)
  131. Kevorkian J, Cole J D Perturbation Methods In Applied Mathematics (Berlin: Springer, 1981)
  132. Craig W, Sulem C, Sulem P L Nonlinearity 5 497 (1992)
  133. Nalimov V I Dinamika Spl. Sredy 18 104 (1974)
  134. Yosihara H Publ. RIM S Kyoto Univ. 18 49 (1982)
  135. Coifman R R, Meyer Y Proc. Symp. Pure Math. Soc. 43 71 (1985)
  136. Keulegan G H, Patterson G W J. Res. Natl. Bur. Stand. 24 47 (1940)
  137. Benjamin T B J. Fluid Mech. 25 241 (1966)
  138. Benjamin T B ""Nonlinear wave motions"" Lectures In Applied Mathematics Vol. Vol. 15 (Ed. A C Newell) (Providence: AMS, 1974) p. 3
  139. Benjamin T B, Bona J G, Mahoney J J Philos. Trans. R. Soc. London Ser. A 272 47 (1972)
  140. Whitham G B J. Fluid Mech. 22 273 (1965)
  141. Whitham G B J. Fluid Mech. 44 373 (1970)
  142. Jimenez J, Whitham G B Proc. R. Soc. London Ser. A 349 277 (1976)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions