Skip to main content
SpringerLink
Log in
Book cover

Rogue and Shock Waves in Nonlinear Dispersive Media pp 23–53Cite as

Integrability in Action: Solitons, Instability and Rogue Waves

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Physics ((LNP,volume 926))

Abstract

Integrable nonlinear equations modeling wave phenomena play an important role in understanding and predicting experimental observations. Indeed, even if approximate, they can capture important nonlinear effects because they can be derived, as amplitude modulation equations, by multiscale perturbation methods from various kind of wave equations, not necessarily integrable, under the assumption of weak dispersion and nonlinearity. Thanks to the mathematical property of being integrable, a number of powerful computational techniques is available to analytically construct special interesting solutions, describing coherent structures such as solitons and rogue waves, or to investigate patterns as those due to shock waves or behaviors caused by instability. This chapter illustrates a selection of these techniques, using first the ubiquitous Nonlinear Schrödinger (NLS) equation as a prototype integrable model, and moving then to the Vector Nonlinear Schrödinger (VNLS) equation as a natural extension to wave coupling.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Zabusky, N., Kruskal, M.: 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 

  2. Russell, J.S.: Report on waves. In: Report of the British Association for the Advancement of Science, vol. 14th Meeting (1845), pp. 311–390, plus plates 47–57. British Association for the Advancement of Science (1844)

    Google Scholar 

  3. Dauxois, T., Peyrard, M.: Physics of Solitons. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  4. Scott, A.: Nonlinear Science: Emergence and Dynamics of Coherent Structures. Oxford Texts in Applied and Engineering Mathematics. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  5. Remoissenet, M.: Waves Called Solitons: Concepts and Experiments. Springer, Berlin/Heidelberg (1994)

    Book  MATH  Google Scholar 

  6. Ablowitz, M., Clarkson, P.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  7. Toda, M.: Nonlinear Waves and Solitons. Japanese Series. Springer, Berlin (1989)

    MATH  Google Scholar 

  8. Drazin, P., Johnson, R.: Solitons: An Introduction. Cambridge Computer Science Texts. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  9. Newell, A.: Solitons in Mathematics and Physics. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1985)

    Book  MATH  Google Scholar 

  10. Novikov, S., Manakov, S., Pitaevskii, L., Zakharov, V.: Theory of Solitons: The Inverse Scattering Method. Contemporary Soviet Mathematics. Plenum, New York (1984)

    MATH  Google Scholar 

  11. Dodd, R., Eilbeck, J., Gibbon, J., Morris, H.: Solitons and Nonlinear Wave Equations. Academic, New York (1982)

    MATH  Google Scholar 

  12. Calogero, F., Degasperis, A.: Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations, vol. 1. North-Holland, Amsterdam (1982)

    MATH  Google Scholar 

  13. Ablowitz, M., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1981)

    Book  MATH  Google Scholar 

  14. Degasperis, A.: Resource letter sol-1: solitons. Am. J. Phys. 66 (6), 486–497 (1998)

    Article  ADS  Google Scholar 

  15. Babelon, O., Bernard, D., Talon, M.: Introduction to Classical Integrable Systems. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  16. Mikhailov, A. (ed.): Integrability. Lecture Notes in Physics. Springer, Berlin/Heidelberg (2008)

    MATH  Google Scholar 

  17. Zakharov, V. (ed.) What Is Integrability? Springer Series in Nonlinear Dynamics. Springer, Berlin/Heidelberg (1991)

    MATH  Google Scholar 

  18. Miwa, T., Jimbo, M., Date, E.: Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  19. Faddeev, L., Takhtajan, L.: Hamiltonian Methods in the Theory of Solitons. Classics in Mathematics. Springer, Berlin/Heidelberg (1987)

    Book  MATH  Google Scholar 

  20. Calogero, F.: Why are certain nonlinear PDEs both widely applicable and integrable? In: Zakharov, V.E. (ed.) What Is Integrability? Springer Series in Nonlinear Dynamics, pp. 1–62. Springer, Berlin/Heidelberg (1991)

    Google Scholar 

  21. Degasperis, A.: Multiscale expansion and integrability of dispersive wave equations. In: Mikhailov, A. (ed.) Integrability. Lecture Notes in Physics, vol. 767, pp. 215–244. Springer, Berlin/Heidelberg (2009)

    Google Scholar 

  22. Osborne, A.: Nonlinear Ocean Waves & the Inverse Scattering Transform. International Geophysics. Elsevier, Amsterdam (2010)

    MATH  Google Scholar 

  23. Infeld, E., Rowlands, G.: Nonlinear Waves, Solitons and Chaos. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  24. Shen, S.: A Course on Nonlinear Waves. Nonlinear Topics in the Mathematical Sciences. Springer, Dordrecht (1993)

    Book  MATH  Google Scholar 

  25. Whitham, G.: Linear and Nonlinear Waves. Wiley, New York (1974)

    MATH  Google Scholar 

  26. Fibich, G.: The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse. Applied Mathematical Sciences. Springer, Cham (2015)

    Book  MATH  Google Scholar 

  27. Ablowitz, M., Prinari, B., Trubatch, A.: Discrete and Continuous Nonlinear Schrödinger Systems. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  28. Sulem, C., Sulem, P.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Applied Mathematical Sciences. Springer, New York (1999)

    MATH  Google Scholar 

  29. Kelley, P.L.: Self-focusing of optical beams. Phys. Rev. Lett. 15, 1005–1008 (1965)

    Article  ADS  Google Scholar 

  30. Taniuti, T., Yajima, N.: Perturbation method for a nonlinear wave modulation. I. J. Math. Phys. 10 (8), 1369–1372 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  31. Taniuti, T., Yajima, N.: Special issue devoted to the Reductive Perturbation Method for Nonlinear Wave Propagation 55 (1974)

    Google Scholar 

  32. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media. Sov. J. Exp. Theor. Phys. 34, 62 (1972)

    ADS  MathSciNet  Google Scholar 

  33. Zakharov, V.E., Shabat, A.B.: Interaction between solitons in a stable medium. Sov. J. Exp. Theor. Phys. 37, 823 (1973)

    ADS  Google Scholar 

  34. Demontis, F., Prinari, B., van der Mee, C., Vitale, F.: The inverse scattering transform for the defocusing nonlinear Schrödinger equations with nonzero boundary conditions. Stud. Appl. Math. 131 (1), 1–40 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Biondini, G., Prinari, B.: On the spectrum of the Dirac operator and the existence of discrete eigenvalues for the defocusing nonlinear Schrödinger equation. Stud. Appl. Math. 132 (2), 138–159 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Biondini, G., Kovacic, G.: Inverse scattering transform for the focusing nonlinear Schrdinger equation with nonzero boundary conditions. J. Math. Phys. 55 (3), 031506-1–031506-22 (2014)

    Google Scholar 

  37. Cieslinski, J.L.: Algebraic construction of the Darboux matrix revisited. J. Phys. A Math. Theor. 42 (40), 404003 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Gu, C., Hu, A., Zhou, Z.: Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry. Mathematical Physics Studies. Springer, Dordrecht (2005)

    Book  MATH  Google Scholar 

  39. Rogers, C., Schief, W.: Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  40. Matveev, V., Salle, M.: Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  41. Doktorov, E., Leble, S.: A Dressing Method in Mathematical Physics. Mathematical Physics Studies. Springer, Dordrecht (2007)

    MATH  Google Scholar 

  42. Coley, A.: Bäcklund and Darboux Transformations: The Geometry of Solitons: AARMS-CRM Workshop, June 4-9, 1999, Halifax, N.S., Canada. CRM Proceedings and Lecture Notes. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  43. Degasperis, A., Lombardo, S.: Multicomponent integrable wave equations. Darboux-dressing transformation. J. Phys. A Math. Theor. 40 (5), 961–977 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Degasperis, A., Lombardo, S.: Multicomponent integrable wave equations. Soliton solutions. J. Phys. A Math. Theor. 42 (38), 385206 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Degasperis, A., Lombardo, S.: Rational solitons of wave resonant-interaction models. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 88 (5), 052914 (2013)

    Article  ADS  Google Scholar 

  46. Neugebauer, G., Meinel, R.: General N-soliton solution of the AKNS class on arbitrary background. Phys. Lett. A 100 (9), 467–470 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  47. Fan, E.: A unified and explicit construction of n-soliton solutions for the nonlinear Schrödinger equation. Commun. Theor. Phys. 36 (4), 401–404 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  48. Steudel, H., Meinel, R., Neugebauer, G.: Vandermonde-like determinants and N-fold Darboux/Bäcklund transformations. J. Math. Phys. 38 (9), 4692–4695 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Benjamin, T.B., Feir, J.E.: The disintegration of wave trains on deep water, Part 1. Theory. J. Fluid Mech. 27, 417–430 (1967)

    Article  ADS  MATH  Google Scholar 

  50. Agrawal, G.: Nonlinear Fiber Optics. Academic, New York (1995)

    MATH  Google Scholar 

  51. Zakharov, V., Ostrovsky, L.: Modulation instability: the beginning. Phys. D Nonlinear Phenom. 238, 540–548 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Kuznetsov, E.: Solitons in a parametrically unstable plasma. Sov. Phys. Dokl. (Engl. Transl.); (United States) 22, 507–508 (1977)

    Google Scholar 

  53. Ma, Y.: The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud. Appl. Math. 60 (1), 43–58 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Akhmediev, N., Eleonskii, V., Kulagin, N.: Exact first-order solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 72 (2), 809–818 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  55. Peregrine, D.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B Appl. Math. 25, 16–43 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  56. Hopkin, M.: Sea snapshots will map frequency of freak waves. Nature 430, 492 (2004)

    Article  ADS  Google Scholar 

  57. Müller, P., Garrett, C., Osborne, A.: Rogue waves – the fourteenth ‘Aha Huliko’a Hawaiian winter workshop. Oceanography 18, 66–75 (2005)

    Article  Google Scholar 

  58. Perkins, P.: Dashing rogues: freak ocean waves pose threat to ships, deep-sea oil platforms. Sci. News 170 (21), 328–329 (2006)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  60. Akhmediev, N., Pelinovsky, E.: Discussion & debate: rogue waves – towards a unifying concept? Eur. Phys. J. Spec. Top. 185, 1–4 (2010)

    Article  Google Scholar 

  61. Pelinovsky, E., Kharif, C. (eds.): Extreme Ocean Waves. Springer, Cham (2008)

    Google Scholar 

  62. Erkintalo, M., Genty, G., Dudley, J.M.: Rogue-wave-like characteristics in femtosecond supercontinuum generation. Opt. Lett. 34 (16), 2468–2470 (2009)

    Article  ADS  Google Scholar 

  63. Bonatto, C., Feyereisen, M., Barland, S., Giudici, M., Masoller, C., Leite, J., Tredicce, J.: Deterministic optical rogue waves. Phys. Rev. Lett. 107, 053901 (2011)

    Article  ADS  Google Scholar 

  64. Stenflo, L., Shukla, P.K.: Nonlinear acoustic-gravity waves. J. Plasma Phys. 75, 841–847 (2009)

    Article  ADS  Google Scholar 

  65. Bailung, H., Sharma, S.K., Nakamura, Y.: Observation of Peregrine solitons in a multicomponent plasma with negative ions. Phys. Rev. Lett. 107, 255005 (2011)

    Article  ADS  Google Scholar 

  66. Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Matter rogue waves. Phys. Rev. A 80, 033610 (2009)

    Article  ADS  Google Scholar 

  67. Ankiewicz, A., Kedziora, D.J., Akhmediev, N.: Rogue wave triplets. Phys. Lett. A 375 (28–29), 2782–2785 (2011)

    Article  ADS  MATH  Google Scholar 

  68. Baronio, F., Degasperis, A., Conforti, M., Wabnitz (2012) Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves. Phys. Rev. Lett. 109, 044102 (2012)

    Google Scholar 

  69. Ohta, Y., Yang, J.: General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 468 (2142), 1716–1740 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  70. He, J., Zhang, H., Wang, L., Porsezian, K., Fokas, A.: Generating mechanism for higher-order rogue waves. Phys. Rev. E 87, 052914 (2013)

    Article  ADS  Google Scholar 

  71. Baronio, F., Conforti, M., Degasperis, A., Lombardo, S.: Rogue waves emerging from the resonant interaction of three waves. Phys. Rev. Lett. 111, 114101 (2013)

    Article  ADS  Google Scholar 

  72. Chabchoub, A., Akhmediev, N.: Observation of rogue wave triplets in water waves. Phys. Lett. A 377 (38), 2590–2593 (2013)

    Article  ADS  MATH  Google Scholar 

  73. Chen, S., Grelu, P., Soto-Crespo, J.M.: Dark- and bright-rogue-wave solutions for media with long-wave-short-wave resonance. Phys. Rev. E 89, 011201 (2014)

    Article  ADS  Google Scholar 

  74. Chen, S., Soto-Crespo, J.M., Grelu, P.: Dark three-sister rogue waves in normally dispersive optical fibers with random birefringence. Opt. Express 22 (22), 27632–27642 (2014)

    Article  ADS  Google Scholar 

  75. Chen, S.: Dark and composite rogue waves in the coupled Hirota equations. Phys. Lett. A 378 (38–39), 2851–2856 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  76. Ling, L., Guo, B., Zhao, L.: High-order rogue waves in vector nonlinear Schrödinger equations. Phys. Rev. E 89, 041201 (2014)

    Article  ADS  Google Scholar 

  77. Chen, S., Mihalache, D.: Vector rogue waves in the Manakov system: diversity and compossibility. J. Phys. A Math. Theor. 48 (21), 215202 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  78. Chen, S., Soto-Crespo, J.M., Grelu, P.: Watch-hand-like optical rogue waves in three-wave interactions. Opt. Express 23 (1), 349–359 (2015)

    Article  ADS  Google Scholar 

  79. Manakov, S.V.: On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov. J. Exp. Theor. Phys. 38, 248 (1974)

    ADS  Google Scholar 

  80. Onorato, M., Proment, D., Toffoli, A.: Freak waves in crossing seas. Eur. Phys. J. Spec. Top. 185 (1), 45–55 (2010)

    Article  Google Scholar 

  81. Grönlund, A., Eliasson, B., Marklund, M.: Evolution of rogue waves in interacting wave systems. EPL Lett. J. Explor. Front. Phys. 86 (2), 24001 (2009)

    Google Scholar 

  82. Onorato, M., Osborne, A.R., Serio, M.: Modulational instability in crossing sea states: a possible mechanism for the formation of freak waves. Phys. Rev. Lett. 96, 014503 (2006)

    Article  ADS  Google Scholar 

  83. Kivshar, Y., Agrawal, G.: Optical Solitons: From Fibers to Photonic Crystals. Elsevier, Amsterdam (2003)

    Google Scholar 

  84. Mumtaz, S., Essiambre, R., Agrawal, G.P.: Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations. J. Lightwave Technol. 31 (3), 398–406 (2013)

    Article  ADS  Google Scholar 

  85. Yaman, F., Li, G.: Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation. IEEE Photon. J. 1 (2), 144–152 (2009)

    Article  Google Scholar 

  86. Winter, M., Bunge, C., Setti, D., Petermann, K.: A statistical treatment of cross-polarization modulation in DWDM systems. J. Lightwave Technol. 27 (17), 3739–3751 (2009)

    Article  ADS  Google Scholar 

  87. Evangelides, J., S. G., Mollenauer, L.F., Gordon, J.P., Bergano, N.S.: Polarization multiplexing with solitons. J. Lightwave Technol. 10 (1), 28–35 (1992)

    Google Scholar 

  88. Kevrekidis, P., Frantzeskakis, D., Carretero-González, R.: Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment. Springer Series on Atomic, Optical, and Plasma Physics. Springer, Berlin/Heidelberg (2007)

    Google Scholar 

  89. Wang, D., Zhang, D., Yang, J.: Integrable properties of the general coupled nonlinear Schrödinger equations. J. Math. Phys. 51 (2), 023510 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  90. Prinari, B., Ablowitz, M.J., Biondini, G.: Inverse scattering transform for the vector nonlinear Schrdinger equation with nonvanishing boundary conditions. J. Math. Phys. 47, 063508 (2006). doi:http://dx.doi.org/10.1063/1.2209169

    Google Scholar 

  91. Forest, M.G., McLaughlin, D.W., Muraki, D.J., Wright, O.C.: Nonfocusing instabilities in coupled, integrable nonlinear Schrödinger PDEs. J. Nonlinear Sci. 10 (3), 291–331 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  92. Frisquet, B., Kibler, B., Fatome, J., Morin, P., Baronio, F., Conforti, M., Millot, G., Wabnitz, S.: Polarization modulation instability in a Manakov fiber system. Phys. Rev. A 92, 053854 (2015)

    Article  ADS  Google Scholar 

  93. Baronio, F., Conforti, M., Degasperis, A., Lombardo, S., Onorato, M., Wabnitz, S.: Vector rogue waves and baseband modulation instability in the defocusing regime. Phys. Rev. Lett. 113, 034101 (2014)

    Article  ADS  Google Scholar 

  94. Baronio, F., Chen, S., Grelu, P., Wabnitz, S., Conforti, M.: Baseband modulation instability as the origin of rogue waves. Phys. Rev. A 91, 033804 (2015)

    Article  ADS  Google Scholar 

  95. Mikhailov, A., Shabat, A., Sokolov, V.: The symmetry approach to classification of integrable equations. In: Zakharov, V. (ed.) What Is Integrability? Springer Series in Nonlinear Dynamics, pp. 115–184. Springer, Berlin/Heidelberg (1991)

    Google Scholar 

  96. Mikhailov, A., Novikov, V.: Perturbative symmetry approach. J. Phys. A Math. Gen. 35 (22), 4775 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  97. Lombardo, S., Mikhailov, A.: Reductions of integrable equations: dihedral group. J. Phys. A Math. Gen. 37 (31), 7727 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  98. Ablowitz, M., Kaup, D., Newell, A., Segur, H.: Method for solving the sine-Gordon equation. Phys. Rev. Lett. 30, 1262–1264 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  99. Ablowitz, M., Kaup, D., Newell, A., Segur, H.: The inverse scattering transform- Fourier analysis for nonlinear problems. Stud. Appl. Math. 53 (4), 249–315 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  100. Rajaraman, R.: Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory. North-Holland Personal Library. North-Holland, Amsterdam (1982)

    MATH  Google Scholar 

  101. Mikhailov, A.: Integrability of the two-dimensional Thirring model. JETP Lett. (Pis’ma Zh. Eksp. Teor. Fiz. 23, 356–358) 23, 320–323 (1976)

    Google Scholar 

  102. Kuznetsov, E., Mikhailov, A.: On the complete integrability of the two-dimensional classical Thirring model. Theor. Math. Phys. 30 (3), 193–200 (1977)

    Article  MathSciNet  Google Scholar 

  103. Kaup, D.: The three-wave interaction – a nondispersive phenomenon. Stud. Appl. Math. 55 (1), 9–44 (1976)

    Article  MathSciNet  Google Scholar 

  104. Degasperis, A., Conforti, M., Baronio, F., Wabnitz, S., Lombardo, S.: The three-wave resonant interaction equations: spectral and numerical methods. Lett. Math. Phys. 96 (1–3), 367–403 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  105. Gerdjikov, V., Ivanov, R., Kyuldjiev, A.: On the N-wave equations and soliton interactions in two and three dimensions. Wave Motion 48 (8), 791–804 (2011). Nonlinear waves in action: propagation and interaction

    Google Scholar 

  106. Gerdjikov, V., Grahovski, G., Kostov, N.: On N-wave type systems and their gauge equivalent. Eur. Phys. J. B Condens. Matter Complex Syst. 29 (2), 243–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  107. Newell, A.: Long waves–short waves; a solvable model. SIAM J. Appl. Math. 35 (4), 650–664 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  108. Benney, D.: A general theory for interactions between short and long waves. Stud. Appl. Math. 35 (56), 81–94 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  109. Zhu, J., Kuang, Y.: CUSP solitons to the long-short waves equation and the \(\overline{\partial }\)-dressing method. Rep. Math. Phys. 75 (2), 199–211 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  110. Chowdhury, R., Chanda, P.K.: To the complete integrability of long wave–short wave interaction equations. J. Math. Phys. 27 (3), 707–709 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  111. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539 (1970)

    ADS  MATH  Google Scholar 

  112. Davey, A., Stewartson, K.: On three-dimensional packets of surface waves. Proc. R. Soc. Lond. Ser. A 338, 101–110 (1974)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  113. Guo, B., Ling, L., Liu, Q.P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)

    Article  ADS  Google Scholar 

  114. He, J., Wang, L., Li, L., Porsezian, K., Erdélyi, R.: Few-cycle optical rogue waves: complex modified Korteweg de Vries equation. Phys. Rev. E 89, 062917 (2014)

    Article  ADS  Google Scholar 

  115. Guo, B., Ling, L., Liu, Q.P.: High-order solutions and generalized Darboux transformations of derivative nonlinear Schrödinger equations. Stud. Appl. Math. 130 (4), 317–344 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  116. Degasperis, A., Wabnitz, S., Aceves, A.: Bragg grating rogue wave. Phys. Lett. A 379 (14–15), 1067–1070 (2015)

    Article  MATH  Google Scholar 

  117. Chow, K., Chan, H., Kedziora, D., Grimshaw, R.: Rogue wave modes for the long wave-short wave resonance model. J. Phys. Soc. Jpn. 82 (7), 074001 (2013)

    Article  ADS  Google Scholar 

  118. Li, C., He, J., Porsezian, K.: Rogue waves of the Hirota and the Maxwell-Bloch equations. Phys. Rev. E 87, 012913 (2013)

    Article  ADS  Google Scholar 

  119. Akhmediev, N., Soto-Crespo, J., Devine, N., Hoffmann, N.: Rogue wave spectra of the Sasa–Satsuma equation. Phys. D Nonlinear Phenom. 294, 37–42 (2015)

    Article  MathSciNet  Google Scholar 

  120. Soto-Crespo, J.M., Devine, N., Hoffmann, N.P., Akhmediev, N.: Rogue waves of the Sasa-Satsuma equation in a chaotic wave field. Phys. Rev. E 90, 032902 (2014)

    Article  ADS  Google Scholar 

  121. Chen, S.: Twisted rogue-wave pairs in the Sasa-Satsuma equation. Phys. Rev. E 88, 023202 (2013)

    Article  ADS  Google Scholar 

  122. Dubard, P., Matveev, V.: Multi-rogue waves solutions: from the NLS to the KP-I equation. Nonlinearity 26 (12), R93 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  123. Dubard, P., Gaillard, P., Klein, C.P., Matveev, V.: On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation. Eur. Phys. J. Spec. Top. 185, 247–258 (2010)

    Article  Google Scholar 

  124. Ohta, Y., Yang, J.: Dynamics of rogue waves in the Davey-Stewartson II equation. J. Phys. A Math. Theor. 46 (10), 105202 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica, “Sapienza” Università degli Studi di Roma, Roma, Italy

    Antonio Degasperis

  2. Department of Mathematics and Information Sciences, Northumbria University, Newcastle upon Tyne, UK

    Sara Lombardo

Authors
  1. Antonio Degasperis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sara Lombardo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sara Lombardo .

Editor information

Editors and Affiliations

  1. Dipartimento di Fisica, Università di Torino, Torino, Italy

    Miguel Onorato

  2. Institut Non Linéaire de Nice, Centre National de la Recherche Scientifique, CNRS, Université de Nice-Sophia Antipolis, Valbonne, France

    Stefania Resitori

  3. Dipartimento di Ingegneria dell' Informazione, Università di Brescia, Brescia, Italy

    Fabio Baronio

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Degasperis, A., Lombardo, S. (2016). Integrability in Action: Solitons, Instability and Rogue Waves. In: Onorato, M., Resitori, S., Baronio, F. (eds) Rogue and Shock Waves in Nonlinear Dispersive Media. Lecture Notes in Physics, vol 926. Springer, Cham. https://doi.org/10.1007/978-3-319-39214-1_2

Download citation

Publish with us

Policies and ethics

Search

Navigation