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Rogue waves: Classification, measurement and data analysis, and hyperfast numerical modeling

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Abstract.

The last twenty years has seen the birth and subsequent evolution of a fundamental new idea in nonlinear wave research: Rogue waves, freak waves or extreme events in the wave field dynamics can often be classified as coherent structure solutions of the requisite nonlinear partial differential wave equations (PDEs). Since a large number of generic nonlinear PDEs occur across many branches of physics, the approach is widely applicable to many fields including the dynamics of ocean surface waves, internal waves, plasma waves, acoustic waves, nonlinear optics, solid state physics, geophysical fluid dynamics and turbulence (vortex dynamics and nonlinear waves), just to name a few. The first goal of this paper is to give a classification scheme for solutions of this type using the inverse scattering transform (IST) with periodic boundary conditions. In this context the methods of algebraic geometry give the solutions of particular PDEs in terms of Riemann theta functions. In the classification scheme the Riemann spectrum fully defines the coherent structure solutions and their mutual nonlinear interactions. I discuss three methods for determining the Riemann spectrum: (1) algebraic-geometric loop integrals, (2) Schottky uniformization and (3) the Nakamura-Boyd approach. I give an overview of several nonlinear wave equations and graph some of their coherent structure solutions using theta functions. The second goal is to discuss how theta functions can be used for developing data analysis (nonlinear Fourier) algorithms; nonlinear filtering techniques allow for the extraction of coherent structures from time series. The third goal is to address hyperfast numerical models of nonlinear wave equations (which are thousands of times faster than traditional spectral methods).

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References

  1. M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981)

  2. M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge, Cambridge, 1991)

  3. M.J. Ablowitz, B. Prinari, A.D. Trubatch, Discrete and Continuous Nonlinear Schroedinger Systems (Cambridge, Cambridge, 2004)

  4. N. Akhmediev, V.M. Eleonskii, N.E. Kulagin, Sov. Phys. JETP 62Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 894 (1985)

    Google Scholar 

  5. N. Akhmediev, I. Teoreticheskaya Fizika Matematicheskaya 69Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 189 (1986)

    MathSciNet  Google Scholar 

  6. N. Akhmediev, V.M. Elconskii, N.E. Kulagin, Theor. Math. Phys. 72Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 809 (1987)

    Article  MATH  Google Scholar 

  7. N. Akhmediev, D.R. Heatley, G.I. Stegeman, E.M. Wright, Phys. Rev. Lett. 65Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 1423 (1990)

    Article  ADS  Google Scholar 

  8. N. Akhmediev, N.V. Mitskevich, IEEEJ. Quantum Electron. 27Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 849 (1991)

    Article  ADS  Google Scholar 

  9. N. Akhmediev, A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, London, 1997)

  10. N. Akhmediev, Nature 413Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 267 (2001)

    Article  ADS  Google Scholar 

  11. N. Akhmediev, J.M. Soto-Crespo, Ph. Grelu, Phys. Lett. A 373Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 3124 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  12. N. Akhmediev, A. Ankiewicz, M. Taki, Phys. Lett. A 373Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 675 (2009)

    Article  ADS  Google Scholar 

  13. H.F. Baker, Abelian Functions: Abels Theorem and the Allied Theory of Theta Functions (Cambridge, Cambridge, 1897)

  14. E.D. Belokolos, A.I. Bobenko, V.Z. Enolskii, A.R. Its, V.B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations (Springer-Verlag, Berlin, 1994)

  15. T.B. Benjamin, J.F. Feir, J. Fluid Mech. 27Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 417 (1967)

    Article  MATH  ADS  Google Scholar 

  16. A.I. Bobenko, Dokl. Akad. Nauk SSSR 295Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 268 (1987)

    Google Scholar 

  17. A.I. Bobenko, D.A. Kubensky, Teor. Mat. Fiz. 72Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 352 (1987)

    MATH  Google Scholar 

  18. A.I. Bobenko, L.A. Bordag, Zap. LOMI 165Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 31 (1987)

    MATH  Google Scholar 

  19. A.I. Bobenko, L.A. Bordag, J. Phys. A, Math. Gen. 22Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 1259 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. J.P. Boyd, The double cnoidal wave of the Korteweg-deVries equation: An overview, J. Math. Phys. 25Discussion & Debate : Rogue Waves - Towards a Unifying Concept? 3390 (1984)

    Article  MATH  ADS  Google Scholar 

  21. J.P. Boyd, J. Math. Phys. 25Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 3402 (1984)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. J.P. Boyd, J. Math. Phys. 25Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 3415 (1984)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. J.P. Boyd, Adv. Appl. Mech 27Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 1 (1990)

    Article  MATH  Google Scholar 

  24. J.P. Boyd, S.E. Haupt, in Nonlinear Topics in Ocean Physics, edited by A.R. Osborne (Elsevier, Amsterdam, 1990)

  25. B. Deconinck, The Initial-Value Problem for Multiphase Solutions of the Kadomtsev-Petviashvili Equation, Ph. D. thesis, University of Colorado, Department of Applied Mathematics (1998)

  26. B. Deconinck, H. Segur, Physica D 123Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 123 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. B. Deconinck, M. van Hoeij, Physica D 152Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 28 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  28. B. Deconinck, M. Heil, A. Bobenko, M. van Hoeij, M. Schmies, Math. Comp. 73Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 1417 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. L.A. Dickey, Soliton Equations and Hamiltonian Systems (World Scientific, Singapore, 1991)

  30. P.G. Drazin, R.S. Johnson, Solitons: An Introduction (Cambridge University Press, Cambridge, 1989)

  31. B.A. Dubrovin, S.P. Novikov, Funct. Anal. Appl. 9Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 215 (1974a)

    Article  Google Scholar 

  32. B.A. Dubrovin, S.P. Novikov, Sov. Phys. JETP 40Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 1058 (1974b)

    MathSciNet  ADS  Google Scholar 

  33. B.A. Dubrovin, V.B. Matveev, S.P. Novikov, Russ. Math. Surveys 31Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 59 (1976)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. B.A. Dubrovin, Russian Math. Surveys 36Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 11 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  35. L.D. Faddeev, L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987)

  36. A.S. Fokas, P.M. Santini, Coherent structures in multidimensions, Phys. Rev. Lett. 63Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 1329 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  37. A.S. Fokas, P.M. Santini, Physica D 44Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 99 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  38. A.P. Fordy, Soliton Theory: A Survey of Results (Manchester University Press, Manchester, 1990)

  39. A.V. Gaponov-Grekhov, M.I. Rabinovich, Nonlinearities in Action (Springer-Verlag, Berlin, 1992)

  40. C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Phys. Rev. Lett. 19Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 1095 (1967)

    Article  MATH  ADS  Google Scholar 

  41. R.H.J. Grimshaw, Appl. Math. 73Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 1 (1985)

    MATH  MathSciNet  Google Scholar 

  42. R.H.J. Grimshaw, Internal solitary waves, in Advances in coastal and ocean engineering, Vol. IIIDiscussion & Debate : Rogue Waves - Towards a Unifying Concept? edited by P.L.F. Liu (World Scientific, Singapore, 1997), p. 1

  43. R.H.J. Grimshaw, L.A. Ostrovsky, V.I. Shrira, Y.A. Stepanyants, Long nonlinear surface, internal gravity waves in a rotating ocean, Surveys Geophys. 19Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 289 (1998)

    Article  ADS  Google Scholar 

  44. R.H.J. Grimshaw, Internal solitary waves, in Environmental Stratified Flows, edited by R. Grimshaw (Kluwer, Boston, 2001), p. 1

  45. R. Grimshaw (ed.), Korteweg-deVries equation, in Nonlinear waves in fluids: Recent advances and modern applications (Springer-Verlag, Berlin, 2005)

  46. R.H.J. Grimshaw (ed.), Solitary Waves in Fluids (WIT Press, Boston, 2007)

  47. J.L. Hammack, J. Fluid Mech. 60Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 769 (1973)

    Article  MATH  ADS  Google Scholar 

  48. J.L. Hammack, H. Segur, J. Fluid Mech. 65Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 289 (1974)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  49. J.L. Hammack, H. Segur, J. Fluid Mech. 84Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 337 (1978a)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  50. J.L. Hammack, H. Segur, J. Fluid Mech. 84Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 359 (1978b)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  51. E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge, Cambridge, 1990)

  52. A.R. Its, V.B. Matveev, Funct. Anal. Appl. 9Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 65 (1975a)

    Article  MATH  Google Scholar 

  53. A.R. Its, V.B. Matveev , Theor. Math. Phys. 23Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 343 (1975b)

    Article  Google Scholar 

  54. R.S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge University Press, Cambridge, 1997)

  55. C. Kharif, E. Pelinovsky, A. Slunyaev, Rogue Waves in the Ocean (Springer-Verlag, Berlin, 2009)

  56. C. Klein, Springer lecture notes in physics (Berlin, Germany: Springer, 2005)

  57. B.G. Konopelchenko, Inverse Problems 7Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 739 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  58. B.G. Konopelchenko, V.G. Dubrovsky, Phys. Lett. A 102Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 15 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  59. D.J. Korteweg, G. deVries, Philos. Mag. Ser. 5, 39Discussion & Debate : Rogue Waves - Towards a Unifying Concept? 422 (1895)

    Google Scholar 

  60. V.P. Kotljarov, A.R. Its, Dopovidi Akad. Nauk. UkRSR., Ser. A 11Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 965 (in Ukranian) (1976)

    Google Scholar 

  61. I.M. Krichever, Dokl. Akad. Nauk SSSR 298Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 802 (1988)

    MathSciNet  Google Scholar 

  62. H. Lamb, Hydrodynamics (Dover, New York, 1932)

  63. G.L. Lamb, Elements of Soliton Theory (John Wiley, New York, 1980)

  64. B.B. Matveev, Phil. Trans. R. Soc. A, 366Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 837 (2008)

    Article  MATH  ADS  Google Scholar 

  65. J.W. Miles, J. Fluid Mech. 79Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 171 (1977)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  66. A. Nakamura, J. Phys. Soc. Japan 48Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 653 (1980)

    Article  ADS  Google Scholar 

  67. A. Nakamura, Y. Matsuno, J. Phys. Soc. Japan 48Discussion & Debate : Rogue Waves - Towards a Unifying Concept? 1365 (1980)

    Article  ADS  Google Scholar 

  68. A.C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, 1985)

  69. A.C. Newell, J.V. Moloney, Nonlinear Optics (Addison-Wesely, 1992)

  70. S.P. Novikov, S.V. Manakov, L.P. Pitaevskii, V.E. Zakharov, Theory of solitons: The Inverse Scattering Method (Consultants Bureau, New York, 1984)

  71. A.R. Osborne (ed.), Nonlinear Topics in Ocean Physics (North Holland, Amsterdam, 1991), p. 996

  72. A.R. Osborne, M. Onorato, M. Serio, Phys. Lett. A 275Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 386 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  73. A.R. Osborne, Marine Structures 14Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 275 (2001)

    Article  Google Scholar 

  74. A.R. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform, in Scattering, edited by R. Pike and P. Sabatier (Academic Press, New York, 2002)

  75. A.R. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform (Academic Press, International Geophysics Series, Vol. 97, Boston, 2010), p. 944

  76. E. Pelinovsky, C. Kharif, Extreme Ocean Waves (Springer-Verlag, Berlin, 2008)

  77. M. Remoissenet, Waves Called Solitons (Springer, Berlin, 1999)

  78. H. Segur, J. Fluid Mech. 59Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 721 (1973)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  79. H. Segur, in Topics in Ocean Physics, edited by A.R. Osborne and P. Malanotte Rizzoli (North-Holland, Amsterdam, 1982)

  80. H. Segur, J.L. Hammack, J. Fluid Mech. 118Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 285

  81. H. Segur, D. Henderson, J. Carter, J. Hammack, C.-M. Li, D. Pheiff, K. Fluid Mech. 539Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 229 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  82. H. Segur, Integrable models of waves in shallow water, in Probability, Geometry and Integrable Systems, edited by M. Pinski and B. Birnir (Cambridge University Press, Cambridge, 2007)

  83. T. Soomere, Eur. Phys. J. Special Topics 185Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 81 (2010)

    Google Scholar 

  84. E.R. Tracy, Topics in nonlinear wave theory with applications, Ph.D. thesis, University of Maryland, Plasma Preprint UMLPF No. 85-006 (1984)

  85. E.R. Tracy, H.H. Chen, Phys. Rev. A 37Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 815 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  86. K. Trulsen, K. Dysthe, Wave Motion 24Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 281 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  87. V.E. Zakharov, J. App. Mech. Tech. Phys. (USSR) 2Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 190 (1968)

    ADS  Google Scholar 

  88. V.E. Zakharov, Sov. Phys. JETP 33Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 538 (1971)

    ADS  Google Scholar 

  89. V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP 34Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 62 (1972)

    MathSciNet  ADS  Google Scholar 

  90. V.E. Zakharov, A.M. Rubenchik, Prikl. Mat. Techn. Phys. 5Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 84 (1972)

    Google Scholar 

  91. V.E. Zakharov, E.A. Kuznetsov, Multi-scale expansions in the theory of systems integrable by the inverse scattering transform, in Solitons and Coherent Structures, edited by D.K. Campbell, A.C. Newell, R.J. Schrieffer and H. Segur (North-Holland, Amsterdam, 1986), p. 455

  92. V.E. Zakharov (ed.), Nonlinear Waves and Weak Turbulence (American Mathematical Society, Providence, 1998)

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  1. Dipartimento di Fisica Generale, Universita’ di Torino, Torino, Italy

    A.R. Osborne

  2. Nonlinear Waves Research Corporation, Arlington Virginia, USA

    A.R. Osborne

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Osborne, A. Rogue waves: Classification, measurement and data analysis, and hyperfast numerical modeling. Eur. Phys. J. Spec. Top. 185, 225–245 (2010). https://doi.org/10.1140/epjst/e2010-01251-x

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