Skip to main content
SpringerLink
Log in

Peregrine’s System Revisited

  • Chapter
  • First Online:
Nonlinear Waves and Pattern Dynamics

Abstract

In 1967, D. H. Peregrine proposed a Boussinesq-type model for long waves in shallow waters of varying depth Peregrine (J Fluid Mech 27:815–827, 1967, [70]). This prominent paper turned a new leaf in coastal hydrodynamics along with contributions by Serre (La Houille Blanche 8:374–388, 1953, [72]) and Green and Naghdi (J Fluid Mech 78:237–246, 1976, [47]) and many others since then. Several modern Boussinesq-type systems stem from these pioneering works. In the present work, we revise the long wave model traditionally referred to as the Peregrine system. Namely, we propose a modification of the governing equations, which is asymptotically similar to the initial model for weakly nonlinear waves, while preserving an additional symmetry of the complete water wave problem. This modification procedure is called the invariantization. We show that the improved system has well-conditioned dispersive terms in the swash zone, hence allowing for efficient and stable run-up computations.

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

Access this chapter

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 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Notes

  1. 1.

    The steady version of the celebrated Serre–Green–Naghdi equations can be traced back up to Lord Rayleigh [59].

  2. 2.

    Notice, please, that this scaling is different from \(\mathscr {X}_{3}\) given in Sect. 2.1.

  3. 3.

    The asymptotic argument holds here since this term is \(\mathscr {O}(\mu ^{2})\).

  4. 4.

    We do not take here the conservative variables \((H,\, Q)\) since the reconstruction procedure is more accurate and robust in physical variables \((H,\,u)\).

References

  1. M. Antuono, V. Y. Liapidevskii, and M. Brocchini. Dispersive Nonlinear Shallow-Water Equations. Studies in Applied Mathematics, 122(1):1–28, 2009.

    Article  MathSciNet  Google Scholar 

  2. E. Audusse. Modélisation hyperbolique et analyse numérique pour les écoulements en eaux peu profondes. PhD thesis, Université Paris {VI}, 2004.

    Google Scholar 

  3. T. J. Barth. Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations. Lecture series - van Karman Institute for Fluid Dynamics, 5:1–140, 1994.

    Google Scholar 

  4. T. J. Barth and M. Ohlberger. Finite Volume Methods: Foundation and Analysis. In E. Stein, R. de Borst, and T. J. R. Hughes, editors, Encyclopedia of Computational Mechanics. John Wiley & Sons, Ltd, Chichester, UK, nov 2004.

    Google Scholar 

  5. G. K. Batchelor. An introduction to fluid dynamics, volume 61 of Cambridge mathematical library. Cambridge University Press, 2000.

    Google Scholar 

  6. S. A. Beisel, L. B. Chubarov, D. Dutykh, G. S. Khakimzyanov, and N. Y. Shokina. Simulation of surface waves generated by an underwater landslide in a bounded reservoir. Russ. J. Numer. Anal. Math. Modelling, 27(6):539–558, 2012.

    MathSciNet  MATH  Google Scholar 

  7. S. Bellec and M. Colin. On the existence of solitary waves for Boussinesq type equations and Cauchy problem for a new conservative model. Adv. Differential Equations, 21(9/10):945–976, 2016.

    MathSciNet  MATH  Google Scholar 

  8. G. Bellotti and M. Brocchini. On the shoreline boundary conditions for Boussinesq-type models. Int. J. Num. Meth. in Fluids, 37(4):479–500, 2001.

    Article  ADS  Google Scholar 

  9. G. Bellotti and M. Brocchini. On using Boussinesq-type equations near the soreline: a note of caution. Ocean Engineering, 29:1569–1575, 2002.

    Article  Google Scholar 

  10. T. B. Benjamin and P. J. Olver. Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech, 125:137–185, 1982.

    Article  ADS  MathSciNet  Google Scholar 

  11. F. Benkhaldoun and M. Seaïd. New finite-volume relaxation methods for the third-order differential equations. Commun. Comput. Phys., 4:820–837, 2008.

    MathSciNet  MATH  Google Scholar 

  12. E. N. Bernard and V. V. Titov. Improving tsunami forecast skill using deep ocean observations. Mar. Technol. Soc. J., 40(4):23–26, 2007.

    Google Scholar 

  13. G. W. Bluman, A. F. Cheviakov, and S. C. Anco. Applications of Symmetry Methods to Partial Differential Equations. Springer, New York, 2010.

    Book  Google Scholar 

  14. P. Bogacki and L. F. Shampine. A 3(2) pair of Runge-Kutta formulas. Appl. Math. Lett., 2(4):321–325, 1989.

    Article  MathSciNet  Google Scholar 

  15. J. V. Boussinesq. Théorie de l’intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. C.R. Acad. Sci. Paris Sér. A-B, 72:755–759, 1871.

    Google Scholar 

  16. J. P. Boyd. Solitons from sine waves: analytical and numerical methods for non-integrable solitary and cnoidal waves. Physica D, 21:227–246, 1986.

    Article  ADS  Google Scholar 

  17. J. P. Boyd. Chebyshev and Fourier Spectral Methods. New York, 2nd edition, 2000.

    Google Scholar 

  18. J. P. Boyd. A comparison of numerical algorithms for Fourier extension of the first, second and third kinds. J. Comput. Phys., 178:118–160, 2002.

    Article  ADS  MathSciNet  Google Scholar 

  19. J. P. Boyd. Deleted residuals, the QR-factored Newton iteration, and other methods for formally overdetermined determinate discretizations of nonlinear eigenproblems for solitary, cnoidal, and shock waves. J. Comput. Phys., 179:216–237, 2002.

    Article  ADS  MathSciNet  Google Scholar 

  20. M. Brocchini. A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics. Proc. R. Soc. A, 469(2160):20130496, oct 2013.

    Article  ADS  MathSciNet  Google Scholar 

  21. J. Chambarel, C. Kharif, and J. Touboul. Head-on collision of two solitary waves and residual falling jet formation. Nonlin. Processes Geophys., 16:111–122, 2009.

    Article  ADS  Google Scholar 

  22. A. F. Cheviakov. Computation of fluxes of conservation laws. J. Eng. Math., 66(1-3):153–173, mar 2010.

    Article  ADS  MathSciNet  Google Scholar 

  23. C. I. Christov. An energy-consistent dispersive shallow-water model. Wave Motion, 34:161–174, 2001.

    Article  MathSciNet  Google Scholar 

  24. L. B. Chubarov, G. S. Khakimzyanov, and N. Y. Shokina. Numerical modelling of surface water waves arising due to movement of underwater landslide on irregular bottom slope. In Notes on Numerical Fluid Mechanics and Multidisciplinary Design: Computational Science and High Performance Computing IV, pages 75–91. Springer-Verlag, Berlin, Heidelberg, vol. 115 edition, 2011.

    Google Scholar 

  25. D. Clamond and D. Dutykh. Practical use of variational principles for modeling water waves. Phys. D, 241(1):25–36, 2012.

    Article  MathSciNet  Google Scholar 

  26. D. Clamond, D. Dutykh, and D. Mitsotakis. Conservative modified Serre–Green–Naghdi equations with improved dispersion characteristics. Comm. Nonlin. Sci. Num. Sim., 45:245–257, 2017.

    Article  ADS  MathSciNet  Google Scholar 

  27. A. I. Delis, M. Kazolea, and N. A. Kampanis. A robust high-resolution finite volume scheme for the simulation of long waves over complex domains. Int. J. Numer. Meth. Fluids, 56:419–452, 2008.

    Article  MathSciNet  Google Scholar 

  28. J. W. Demmel. Applied Numerical Linear Algebra. SIAM, Philadelphia, 1997.

    Google Scholar 

  29. M. Di Risio, G. Bellotti, A. Panizzo, and P. De Girolamo. Three-dimensional experiments on landslide generated waves at a sloping coast. Coastal Engineering, 56(5-6):659–671, 2009.

    Article  Google Scholar 

  30. V. A. Dougalis, A. Durán, M. A. Lopez-Marcos, and D. E. Mitsotakis. A numerical study of the stability of solitary waves of Bona-Smith family of Boussinesq systems. J. Nonlinear Sci., 17:595–607, 2007.

    Article  MathSciNet  Google Scholar 

  31. A. Duran, D. Dutykh, and D. Mitsotakis. On the Galilean Invariance of Some Nonlinear Dispersive Wave Equations. Stud. Appl. Math., 131(4):359–388, nov 2013.

    Article  MathSciNet  Google Scholar 

  32. D. Dutykh, D. Clamond, P. Milewski, and D. Mitsotakis. Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations. Eur. J. Appl. Math., 24(05):761–787, 2013.

    Article  MathSciNet  Google Scholar 

  33. D. Dutykh and F. Dias. Water waves generated by a moving bottom. In A. Kundu, editor, Tsunami and Nonlinear waves, pages 65–96. Springer Verlag (Geo Sc.), 2007.

    Google Scholar 

  34. D. Dutykh and F. Dias. Tsunami generation by dynamic displacement of sea bed due to dip-slip faulting. Mathematics and Computers in Simulation, 80(4):837–848, 2009.

    Article  MathSciNet  Google Scholar 

  35. D. Dutykh and F. Dias. Influence of sedimentary layering on tsunami generation. Computer Methods in Applied Mechanics and Engineering, 199(21-22):1268–1275, 2010.

    Article  ADS  MathSciNet  Google Scholar 

  36. D. Dutykh and H. Kalisch. Boussinesq modeling of surface waves due to underwater landslides. Nonlin. Processes Geophys., 20(3):267–285, may 2013.

    Article  ADS  Google Scholar 

  37. D. Dutykh, T. Katsaounis, and D. Mitsotakis. Finite volume schemes for dispersive wave propagation and runup. J. Comput. Phys., 230(8):3035–3061, apr 2011.

    Article  ADS  MathSciNet  Google Scholar 

  38. D. Dutykh and D. Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete and Continuous Dynamical Systems - Series B, 13(4):799–818, 2010.

    Article  MathSciNet  Google Scholar 

  39. D. Dutykh, D. Mitsotakis, S. A. Beisel, and N. Y. Shokina. Dispersive waves generated by an underwater landslide. In E. Vazquez-Cendon, A. Hidalgo, P. Garcia-Navarro, and L. Cea, editors, Numerical Methods for Hyperbolic Equations: Theory and Applications, pages 245–250. CRC Press, Boca Raton, London, New York, Leiden, 2013.

    Chapter  Google Scholar 

  40. D. Dutykh, R. Poncet, and F. Dias. The VOLNA code for the numerical modeling of tsunami waves: Generation, propagation and inundation. Eur. J. Mech. B/Fluids, 30(6):598–615, 2011.

    Article  ADS  MathSciNet  Google Scholar 

  41. K. S. Erduran, S. Ilic, and V. Kutija. Hybrid finite-volume finite-difference scheme for the solution of Boussinesq equations. Int. J. Numer. Meth. Fluids, 49:1213–1232, 2005.

    Article  MathSciNet  Google Scholar 

  42. J. Fenton. A ninth-order solution for the solitary wave. J. Fluid Mech, 53(2):257–271, 1972.

    Article  ADS  Google Scholar 

  43. E. D. Fernandez-Nieto, F. Bouchut, D. Bresch, M. J. Castro-Diaz, and A. Mangeney. A new Savage-Hutter type models for submarine avalanches and generated tsunami. J. Comput. Phys., 227(16):7720–7754, 2008.

    Article  ADS  MathSciNet  Google Scholar 

  44. A. G. Filippini, S. Bellec, M. Colin, and M. Ricchiuto. On the nonlinear behaviour of Boussinesq type models: Amplitude-velocity vs amplitude-flux forms. Coastal Engineering, 99:109–123, 2015.

    Article  Google Scholar 

  45. J.-M. Ghidaglia, A. Kumbaro, and G. Le Coq. Une méthode volumes-finis à flux caractéristiques pour la résolution numérique des systèmes hyperboliques de lois de conservation. C. R. Acad. Sci. I, 322:981–988, 1996.

    MATH  Google Scholar 

  46. G. Golub and C. Van Loan. Matrix Computations. J. Hopkins University Press, 3rd ed. edition, 1996.

    Google Scholar 

  47. A. E. Green and P. M. Naghdi. A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech., 78:237–246, 1976.

    Article  ADS  Google Scholar 

  48. S. T. Grilli and P. Watts. Modeling of waves generated by a moving submerged body. Applications to underwater landslides. Engineering Analysis with boundary elements, 23:645–656, 1999.

    Article  Google Scholar 

  49. C. B. Harbitz, F. Lovholt, G. Pedersen, S. Glimsdal, and D. G. Masson. Mechanisms of tsunami generation by submarine landslides - a short review. Norwegian Journal of Geology, 86(3):255–264, 2006.

    Google Scholar 

  50. A. Harten. ENO schemes with subcell resolution. J. Comput. Phys, 83:148–184, 1989.

    Article  ADS  MathSciNet  Google Scholar 

  51. A. Harten and S. Osher. Uniformly high-order accurate nonscillatory schemes. I. SIAM J. Numer. Anal., 24:279–309, 1987.

    Article  ADS  Google Scholar 

  52. N. J. Higham. Accuracy and Stability of Numerical Algorithms. SIAM Philadelphia, 2nd ed. edition, 2002.

    Google Scholar 

  53. H. Kalisch. Stability of solitary waves for a nonlinearly dispersive equation. Discrete and Continuous Dynamical Systems, 10:709–717, 2004.

    Article  MathSciNet  Google Scholar 

  54. G. S. Khakimzyanov, D. Dutykh, and Z. I. Fedotova. Dispersive shallow water wave modelling. Part III: Model derivation on a globally spherical geometry. Commun. Comput. Phys., 23(2):315–360, 2018.

    Google Scholar 

  55. G. S. Khakimzyanov, D. Dutykh, Z. I. Fedotova, and D. E. Mitsotakis. Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat space. Commun. Comput. Phys., 23(1):1–29, 2018.

    Google Scholar 

  56. N. E. Kolgan. Finite-difference schemes for computation of three dimensional solutions of gas dynamics and calculation of a flow over a body under an angle of attack. Uchenye Zapiski TsaGI [Sci. Notes Central Inst. Aerodyn], 6(2):1–6, 1975.

    Google Scholar 

  57. P. L.-F. Liu, P. Lynett, and C. E. Synolakis. Analytical solutions for forced long waves on a sloping beach. J. Fluid Mech., 478:101–109, 2003.

    Article  ADS  MathSciNet  Google Scholar 

  58. M. S. Longuet-Higgins and J. Fenton. On the Mass, Momentum, Energy and Circulation of a Solitary Wave. II. Proc. R. Soc. A, 340(1623):471–493, 1974.

    Article  ADS  MathSciNet  Google Scholar 

  59. J. W. S. Lord Rayleigh. On Waves. Phil. Mag., 1:257–279, 1876.

    Article  Google Scholar 

  60. P. A. Madsen, H. B. Bingham, and H. A. Schaffer. Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis. Proc. R. Soc. Lond. A, 459:1075–1104, 2003.

    Article  ADS  MathSciNet  Google Scholar 

  61. P. A. Madsen and H. A. Schaffer. A review of Boussinesq-type equations for surface gravity waves. Adv. Coastal Ocean Engng, 5:1–94, 1999.

    Article  Google Scholar 

  62. P. A. Madsen, H. A. Sorensen, and H. A. Schaffer. Surf zone dynamics simulated by a Boussinesq-type model. Part I. Model description and cross-shore motion of regular waves. Coastal Engineering, 32:255–287, 1997.

    Article  Google Scholar 

  63. O. Nwogu. Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterway, Port, Coastal and Ocean Engineering, 119:618–638, 1993.

    Article  Google Scholar 

  64. E. A. Okal. Seismic Parameters Controlling Far-field Tsunami Amplitudes: A Review. Natural Hazards, 1:67–96, 1988.

    Article  Google Scholar 

  65. E. A. Okal and C. E. Synolakis. A theoretical comparison of tsunamis from dislocations and landslides. Pure and Applied Geophysics, 160:2177–2188, 2003.

    Article  ADS  Google Scholar 

  66. E. A. Okal and C. E. Synolakis. Source discriminants for near-field tsunamis. Geophys. J. Int., 158:899–912, 2004.

    Article  ADS  Google Scholar 

  67. P. J. Olver. Applications of Lie groups to differential equations, volume 107 (2nd e of Graduate Texts in Mathematics. Springer-Verlag, 1993.

    Book  Google Scholar 

  68. F. Pascal. Sur des méthodes d’approximation effectives et d’analyse numérique pour les équations de la mécanique de fluides. Habilitation à diriger des recherches, Université de Paris-Sud, 2002.

    Google Scholar 

  69. E. Pelinovsky and A. Poplavsky. Simplified model of tsunami generation by submarine landslides. Physics and Chemistry of the Earth, 21(12):13–17, 1996.

    Article  ADS  Google Scholar 

  70. D. H. Peregrine. Long waves on a beach. J. Fluid Mech., 27:815–827, 1967.

    Article  ADS  Google Scholar 

  71. J. Sandee and K. Hutter. On the development of the theory of the solitary wave. A historical essay. Acta Mechanica, 86:111–152, 1991.

    Article  MathSciNet  Google Scholar 

  72. F. Serre. Contribution à l’étude des écoulements permanents et variables dans les canaux. La Houille blanche, 8:374–388, 1953.

    Article  Google Scholar 

  73. L. F. Shampine and M. W. Reichelt. The MATLAB ODE Suite. SIAM J. Sci. Comput., 18:1–22, 1997.

    Article  MathSciNet  Google Scholar 

  74. G. Söderlind. Digital filters in adaptive time-stepping. ACM Trans. Math. Software, 29:1–26, 2003.

    Article  MathSciNet  Google Scholar 

  75. G. Söderlind and L. Wang. Adaptive time-stepping and computational stability. J. Comp. Appl. Math., 185(2):225–243, 2006.

    Article  ADS  MathSciNet  Google Scholar 

  76. C. E. Synolakis. The runup of solitary waves. J. Fluid Mech., 185:523–545, 1987.

    Article  ADS  MathSciNet  Google Scholar 

  77. C. E. Synolakis and E. N. Bernard. Tsunami science before and beyond Boxing Day 2004. Phil. Trans. R. Soc. A, 364:2231–2265, 2006.

    Article  ADS  MathSciNet  Google Scholar 

  78. S. Tinti, E. Bortolucci, and C. Chiavettieri. Tsunami Excitation by Submarine Slides in Shallow-water Approximation. Pure appl. geophys., 158:759–797, 2001.

    Article  ADS  Google Scholar 

  79. V. V. Titov, F. I. Gonzalez, E. N. Bernard, M. C. Eble, H. O. Mofjeld, J. C. Newman, and A. J. Venturato. Real-Time Tsunami Forecasting: Challenges and Solutions. Natural Hazards, 35:41–58, 2005.

    Article  Google Scholar 

  80. M. I. Todorovska, A. Hayir, and M. D. Trifunac. A note on tsunami amplitudes above submarine slides and slumps. Soil Dynamics and Earthquake Engineering, 22:129–141, 2002.

    Article  Google Scholar 

  81. E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin, Heidelberg, 2009.

    Book  Google Scholar 

  82. F. Ursell. The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc., 49:685–694, 1953.

    Article  ADS  MathSciNet  Google Scholar 

  83. B. van Leer. Towards the ultimate conservative difference scheme V: a second order sequel to Godunov’ method. J. Comput. Phys., 32:101–136, 1979.

    Article  ADS  Google Scholar 

  84. B. van Leer. Upwind and High-Resolution Methods for Compressible Flow: From Donor Cell to Residual-Distribution Schemes. Commun. Comput. Phys., 1:192–206, 2006.

    MATH  Google Scholar 

  85. P. Watts, F. Imamura, and S. T. Grilli. Comparing model simulations of three benchmark tsunami generation cases. Science of Tsunami Hazards, 18(2):107–123, 2000.

    Google Scholar 

  86. T. Y. Wu. Long Waves in Ocean and Coastal Waters. Journal of Engineering Mechanics, 107:501–522, 1981.

    Google Scholar 

  87. T. Y. T. Wu. Generation of upstream advancing solitons by moving disturbances. J. Fluid Mech., 184:75–99, 1987.

    Article  ADS  MathSciNet  Google Scholar 

  88. Y. Xing and C.-W. Shu. High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys., 208:206–227, 2005.

    Article  ADS  MathSciNet  Google Scholar 

  89. J. Yang. Nonlinear Waves in Integrable and Nonintegrable Systems. Society for Industrial and Applied Mathematics, Philadelphia, jan 2010.

    Google Scholar 

  90. J. A. Zelt. The run-up of nonbreaking and breaking solitary waves. Coastal Engineering, 15:205–246, 1991.

    Article  Google Scholar 

  91. J. G. Zhou, D. M. Causon, D. M. Ingram, and C. G. Mingham. Numerical solutions of the shallow water equations with discontinuous bed topography. Int. J. Numer. Meth. Fluids, 38:769–788, 2002.

    Article  Google Scholar 

Download references

Acknowledgements

D. Dutykh and A. Durán acknowledge the support from project MTM2014-54710-P entitled ‘Numerical Analysis of Nonlinear Nonlocal Evolution Problems’ (NANNEP). D. Mitsotakis was supported by the Marsden Fund administered by the Royal Society of New Zealand.

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, E.T.S.I. Telecomunicación, Universidad de Valladolid, Campus Miguel Delibes, Paseo de Belen 15, 47011, Valladolid, Spain

    Angel Durán

  2. Université Savoie Mont Blanc, LAMA, UMR 5127 CNRS, Campus Scientifique, 73376, Le Bourget-du-Lac Cedex, France

    Denys Dutykh

  3. Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS, LAMA, 73000, Chambéry, France

    Denys Dutykh

  4. School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington, 6140, New Zealand

    Dimitrios Mitsotakis

Authors
  1. Angel Durán
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Denys Dutykh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dimitrios Mitsotakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Denys Dutykh .

Editor information

Editors and Affiliations

  1. Normandie Université, UNICAEN, UNIROUEN, CNRS, UMR 6143, M2C, Caen, France

    Nizar Abcha

  2. Department of Nonlinear Geophysical Processes, Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia

    Efim Pelinovsky

  3. Normandie Université, Unihavre, CNRS, UMR 6294, LOMC, Le Havre CX, France

    Innocent Mutabazi

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Durán, A., Dutykh, D., Mitsotakis, D. (2018). Peregrine’s System Revisited. In: Abcha, N., Pelinovsky, E., Mutabazi, I. (eds) Nonlinear Waves and Pattern Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-78193-8_1

Download citation

Publish with us

Policies and ethics

Search

Navigation