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Introduction

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New Approaches to Nonlinear Waves

Part of the book series: Lecture Notes in Physics ((LNP,volume 908))

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Abstract

In the first chapter, we throw a brief glance at the topics presented in the following chapters and their place in the context of the general theory of nonlinear wave systems with dispersion. Starting with the concept of the wave resonance, we proceed through the formalism and presently known results in the theory of discrete and kinetic wave turbulence to the list of open questions and possible theoretical generalizations. At the end of the introductory chapter, we outline a few challenging problems in the area of matching theory and experiment, generally overlooked.

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Notes

  1. 1.

    There is no general theory answering the question whether indeed exact 4-wave resonances or approximate 3-wave resonances should be studied in this case. Even an unambiguous definition of approximate resonance is not yet available, [46]. This is the reason why we do not consider this topic, which undoubtedly has a great potential in applications, in the present volume. A brief overview of the problem is given in Sect. 1.2.3.

  2. 2.

    Modern terminology “discrete WTT” and “resonance clusters” has been first introduced almost 20 years later, in [18].

  3. 3.

    This definition has been used successfully in many physical applications for several decades. However, a rigorous mathematical proof of instability of the Stokes periodic wavetrain (within the Hamiltonian framework) was given by Bridges and Mielke almost 30 years later in [5].

References

  1. Annenkov, S.Y., Shrira, V.I.: Role of non-resonant interactions in the evolution of nonlinear random water wave fields. Fluid Mech. 561, 181–207 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Annenkov, S., Shrira, V.: Modelling the impact of squall on wind waves with the generalized kinetic equation. J. Phys. Ocean. 45(3), 807–812 (2015)

    Article  ADS  Google Scholar 

  3. Benjamin, T.B., Feir, J.E.: The disintegration of wavetrains in deep water, Part 1. Fluid Mech. 27, 417–31 (1967)

    Article  ADS  MATH  Google Scholar 

  4. Bridges, T.J.: A universal form for the emergence of the Korteweg – de Vries equation. Proc. R. Soc. Lond. A 469(2153), 20120707 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  5. Bridges, T.J., Mielke, A.: A proof of the Benjamin-Feir instability. Arch. Ration. Mech Anal. 133, 145–98 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Craik, A.D.: Wave Interactions and Fluid Flows. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  7. Craig, W., Sulem, C.: Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Gardner, C., Greene, J., Kruskal, M., Miura, R.: Method for Solving the Korteweg – de Vries Equation. Phys. Rev. Lett 19(19), 1095–1097 (1967)

    Article  ADS  Google Scholar 

  9. Grimshaw, R., Pelinovsky D., Pelinovsky E., Talipova T.: Wave group dynamics in weakly nonlinear long-wave models. Phys. D 159(1–2), 35–57 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hasselmann, K.: On the non-linear energy transfer in a gravity-wave spectrum, Part 1. General theory. Fluid Mech. 12(04), 481–500 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Hetmaniok, E., Nowak, I., Slota, D., Witula, R., Zielonka, A.: Solution of the inverse heat conduction problem with Neumann boundary condition by using the homotopy perturbation method. Therm. Sci. 17(3), 643–650 (2013)

    Article  Google Scholar 

  12. Izadian, J., Salahshour, S., Soheil Salahshour, S.: A numerical method for solving Volterra and Fredholm integral equations using homotopy analysis method. AWER Proc. Inf. Technol. Comput. Sci. 1 406–411 (2012)

    Google Scholar 

  13. Kartashova, E.: Model of laminated wave turbulence. JETP Lett. 83(7), 283–287 (2006)

    Article  ADS  Google Scholar 

  14. Kartashova, E.A.: Partitioning of ensembles of weakly interacting dispersing waves in resonators into disjoint classes. Phys. D 46(1), 43–56 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kartashova, E.A.: On properties of weakly nonlinear wave interactions in resonators. Phys. D 54, 125–34 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kartashova, E.A.: Weakly nonlinear theory of finite-size effects in resonators. Phys. Rev. Lett. 72, 2013–2016 (1994)

    Article  ADS  Google Scholar 

  17. Kartashova, E.: Exact and quasi-resonances in discrete water-wave turbulence. Phys. Rev. Lett. 98, 214502-1–214502-4 (2007)

    Google Scholar 

  18. Kartashova, E.A.: Discrete wave turbulence. Europhys. Lett. 87, 44001-1–44001-5 (2009)

    Google Scholar 

  19. Kartashova, E.: Nonlinear Resonance Analysis. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  20. Kartashova, E.: Energy transport in weakly nonlinear wave systems with narrow frequency band excitation. Phys. Rev. E 86, 041129 (2012)

    Article  ADS  Google Scholar 

  21. Kartashova, E.: Time scales and structures of wave interaction exemplified with water waves. Europhys. Lett. 102(4), 44005 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  22. Kartashova, E., L’vov, V.S.: A model of intra-seasonal oscillations in the Earth atmosphere. Phys. Rev. Lett. 98, 198501-1–198501-4 (2007)

    Google Scholar 

  23. Kartashova, E.A., Piterbarg, L.I., Reznik, G.M.: Weakly nonlinear interactions between Rossby waves on a sphere. Oceanology 29, 405–411 (1990)

    Google Scholar 

  24. Kashuri, A., Fundo, A., Kreku, M.: Mixture of a new integral transform and homotopy perturbation method for solving nonlinear partial differential equations. Adv. Pure Math. 3 317–323 (2013)

    Article  Google Scholar 

  25. Kharif, C., Pelinovsky E., Slunyaev, A.: Rogue Waves in the Ocean. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Berlin (2009)

    MATH  Google Scholar 

  26. Kuksin, S., Shirikyan, A.: Mathematics of Two-Dimensional Turbulence, p. 336. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  27. Levi-Civita, T.: Détermination rigoureuse des ondes permanents d’ampleur fini. Math. Ann. 93, 264 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lighthill M.J. (ed.): A Discussion on Nonlinear Theory of Wave Propagation in Dispersive Systems. Royal Society, London (1967)

    Google Scholar 

  29. Lvov, V.S., Pomyalov, A., Procaccia, I., Rudenko, O.: Finite-dimensional turbulence of planetary waves. Phys. Rev. E 80(6), 066319 (2009)

    Article  ADS  Google Scholar 

  30. Luke, J.C.: A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 375–397 (1967)

    Article  MathSciNet  ADS  Google Scholar 

  31. Mahmood, B., Manaa, S.A., Easif, F.H.: Homotopy analysis method for solving nonlinear diffusion equation with convection term. Int. J. Appl. Math. Res. 3(3), 244–250 (2014)

    Article  Google Scholar 

  32. Mastroberardino, A.: Homotopy analysis method applied to electrohydrodynamic flow. Commun. Nonlinear Sci. Numer. Simul. 16(7), 2730–2736 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Matinfar, M., Saeidy, M., Gharahsuflu, B.: Homotopy analysis method for systems of integro-differential equations. J. Intel. Fuzzy Syst. Appl. Eng. Technol. 26(3), 1095–1102 (2014)

    MathSciNet  Google Scholar 

  34. Nazarenko, S.: Wave Turbulence. Lecture Notes in Phyics, vol. 825. Springer, Berlin (2011)

    Google Scholar 

  35. Newell, A.C., Rumpf, B.: Wave Turbulence. Ann. Rev. Fluid Mech. 43(1), 59–78 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  36. Osborne, A.: Nonlinear Ocean Waves and the Inverse Scattering Transform. International Geophysics Series, vol. 97. Elsevier Academic, Amsterdam (2010)

    Google Scholar 

  37. Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1987)

    Book  MATH  Google Scholar 

  38. Phillips, O.M.: On the dynamics of unsteady gravity waves of finite amplitude, Part 1. The elementary interactions. Fluid Mech. 9, 193–217 (1960)

    Article  MATH  Google Scholar 

  39. Pushkarev, A.N.: On the Kolmogorov and frozen turbulence in numerical simulation of capillary waves. Eur. J. Mech. B/Fluids 18(3), 345–351 (1999)

    Article  ADS  MATH  Google Scholar 

  40. Pushkarev, A.N., Zakharov, V.E.: Turbulence of capillary waves - theory and numerical simulation. Phys. D 135(1–2), 98–116 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ruderman M., Talipova T., Pelinovsky E.: Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions. Plasma Phys. 74(5), 639–656 (2008)

    ADS  Google Scholar 

  42. Riemann, B.: Über die Fortpflanzung ebener Luftwellen von endlicher Schwindungsweite. Göttingen Abhandlungen VIII, 43 (1858)

    Google Scholar 

  43. Sajid, M.: Some Flow Problems in Differential Type Fluids: Series Solutions Using Homotopy Analysis Method (HAM). VDM (Verlag Dr. Müller, Germany) (2009)

    Google Scholar 

  44. Scott, A.: Encyclopedia of Nonlinear Science. Routledge, New-York (2004)

    Google Scholar 

  45. Stokes, G.G.: On the theories of internal friction of fluids in motion. Trans. Camb. Philos. Soc. 8, 287–305 (1845)

    Google Scholar 

  46. Tobisch, E.: What can go wrong when applying wave turbulence theory. E-print: arXiv:1406.3780 [physics.flu-dyn] (2014)

    Google Scholar 

  47. Whitham, G.B.: Linear and Nonlinear Waves. Pure and Applied Mathematics, Wiley, New York (1999)

    Google Scholar 

  48. Zabusky, N.J., Kruskal, M.D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)

    Article  ADS  MATH  Google Scholar 

  49. Zakharov, V.E., Filonenko, N.N.: Weak turbulence of capillary waves. J. Appl. Mech. Tech. Phys. 8(5), 37–40 (1967)

    Article  ADS  Google Scholar 

  50. Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194 (1968)

    Article  ADS  Google Scholar 

  51. Zakharov, V.E., Lvov, V.S., Falkovich, G.: Kolmogorov Spectra of Turbulence I. Wave Turbulence. Nonlinear Dynamics. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  52. Zakharov, V.E., Korotkevich, A.O., Pushkarev, A.N., Dyachenko, A.I.: Mesoscopic wave turbulence. JETP Lett. 82, 487–491 (2005)

    Article  Google Scholar 

  53. Zakharov, V.E., Ostrovsky L.A.: Modulation instability: the beginning. Phys. D 238, 540–548 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Johannes Kepler University, Linz, Austria

    Elena Tobisch

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  1. Elena Tobisch
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  1. Institute for Analysis, Johannes Kepler University, Linz, Austria

    Elena Tobisch

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Tobisch, E. (2016). Introduction. In: Tobisch, E. (eds) New Approaches to Nonlinear Waves. Lecture Notes in Physics, vol 908. Springer, Cham. https://doi.org/10.1007/978-3-319-20690-5_1

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