Skip to main content
SpringerLink
Log in
Book cover

Nonlinear Systems, Vol. 1 pp 339–368Cite as

On the Numerical Approximation to Generalized Ostrovsky Equations: I

A Numerical Method and Computation of Solitary-Wave Solutions

  • Chapter
  • First Online:

  • 1191 Accesses

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

In the present chapter, two numerical procedures to simulate the dynamics of generalized versions of the Ostrovsky equation are presented. First, a numerical method to approximate the corresponding periodic initial-value problem is introduced. The scheme consists of a spatial discretization based on Fourier collocation methods, which is justified by the presence of nonlocal terms. Due to the stiff character of the semidiscretization in space, the time integration is performed with a fourth-order, diagonally implicit Runge-Kutta method, which provides additional theoretical and computational properties. The second point treated in this chapter concerns the solitary-wave solutions of the equations. Their numerical generation is carried out by using a Petviashvili-type method, along with acceleration techniques. The resulting procedure is able to compute both classical and generalized solitary waves in an efficient way. The speed-amplitude relation and the asymptotic behaviour of the waves are studied from the computed profiles.

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   89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
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

Learn about institutional subscriptions

References

  1. Ablowitz, M.J., Musslimani, Z.H.: Spectral renormalization method for computing self-localized solutions to nonlinear systems. Opt. Lett. 30, 2140–2142 (2005)

    Article  ADS  Google Scholar 

  2. Álvarez, J., Durán, A.: Petviashvili type methods for traveling wave computations: I. Analysis of convergence. J. Comput. Appl. Math. 266, 39–51 (2014)

    Article  MathSciNet  Google Scholar 

  3. Álvarez, J., Durán, A.: Petviashvili type methods for traveling wave computations: II. Acceleration techniques. Math. Comput. Simul. 123, 19–36 (2016)

    Google Scholar 

  4. Apel, J.R., Ostrovsky, L.A., Stepanyants, Y.A., Lynch, J.F.: Internal solitons in the ocean. WHOI Tech. Rep. (2006)

    Google Scholar 

  5. Benilov, E.S.: On the surface waves in a shallow channel with an uneven bottom. Stud. Appl. Math. 87, 1–14 (1992)

    Article  MathSciNet  Google Scholar 

  6. Bona, J.L., Dougalis, V.A., Mitsotakis, D.E.: Numerical solution of KdV-KdV systems of Boussinesq equations I. The numerical scheme and generalized solitary waves. Math. Comput. Simul. 74, 214–228 (2007)

    Article  MathSciNet  Google Scholar 

  7. Boyd, J.P.: Weakly nonlocal solitary waves and beyond-all-orders asymptotics: generalized solitons and hyperasymptotic perturbation theory. In: Mathematics and Its Applications, vol. 442. Kluwer, Amsterdam (1998)

    Chapter  Google Scholar 

  8. Boyd, J.P., Chen, G.Y.: Five regimes of the quasi-cnoidal, steadily translating waves of the rotation-modified Korteweg-de Vries (“Ostrovsky”) equation. Wave Motion 35, 141–155 (2002)

    Article  MathSciNet  Google Scholar 

  9. Brezinski, C.: Convergence acceleration during the 20th century. J. Comput. Appl. Math. 122, 1–21 (1975)

    Article  MathSciNet  Google Scholar 

  10. Cano, B.: Conserved quantities of some Hamiltonian wave equations after full discretizations. Numer. Math. 103, 197–223 (2006)

    Article  MathSciNet  Google Scholar 

  11. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York, Heidelberg, Berlin (1988)

    Book  Google Scholar 

  12. Chen, G.Y., Boyd, J.P.: Analytical and numerical studies of weakly nonlocal solitary waves of the rotation-modified Korteweg-de Vries equation. Physica D 155, 201–222 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  13. Choudhury, S.R.: Solitary-wave families of the Ostrovsky equation: an approach via reversible systems theory and normal forms. Chaos, Solitons Fract. 33, 1468–1479 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  14. Choudhury, S.R., Ivanov, R.I., Liu, Y.: Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation. Chaos, Solitons Fract. 34, 544–550 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  15. Costanzino, N., Manukian, V., Jones, C.K.R.T.: Solitary waves of the regularized short pulse and Ostrovsky equations. SIAM J. Math. Anal. 41(5), 2088–2106 (2009)

    Article  MathSciNet  Google Scholar 

  16. Dougalis, V.A., Durán, A., López-Marcos, M.A., Mitsotakis, D.E.: A numerical study of the stability of solitary waves of the Bona-Smith family of Boussinesq systems. J. Nonlinear Sci. 17, 569–607 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  17. Dougalis, V.A., Durán, A., Mitsotakis, D.E.: Numerical solution of the Benjamin equation. Wave Motion 52, 194–215 (2015)

    Article  MathSciNet  Google Scholar 

  18. Durán, A., Sanz-Serna, J.M.: The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation. IMA J. Numer. Anal. 20, 235–261 (2000)

    Article  MathSciNet  Google Scholar 

  19. de Frutos, J., Sanz-Serna, J.M.: An easily implementable fourth-order method for the time integration of wave problems. J. Comput. Phys. 103, 160–168 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  20. Galkin, V.N., Stepanyants, Y.A.: On the existence of stationary solitary waves in a rotating fluid. J. Appl. Math. Mech. 55, 939–943 (1991)

    Article  MathSciNet  Google Scholar 

  21. Gilman, O.A., Grimshaw, R., Stepanyants, Y.A.: Approximate analytical and numerical solutions of the stationary Ostrovsky equation. Stud. Appl. Math. 95, 115–126 (1995)

    Article  MathSciNet  Google Scholar 

  22. Gilman, O.A., Grimshaw, R., Stepanyants, Y.A.: Dynamics of internal solitary waves in a rotating fluid. Dyn. Atm. Ocean 23(1), 403–411 (1995)

    ADS  Google Scholar 

  23. Grimshaw, R.H.: Evolution equations for weakly nonlinear, long internal waves in a rotating fluid. Stud. Appl. Math. 73, 1–33 (1985)

    Article  MathSciNet  Google Scholar 

  24. Grimshaw, R.H.: Internal solitary waves. In: Liu (ed.) Advances in Coastal and Ocean Engineering, pp. 1–30. World Scientific, Singapore (1997)

    Google Scholar 

  25. Grimshaw, R.H., He, J.M., Ostrovsky, L.A.: Terminal damping of a solitary wave due to radiation in rotational systems. Stud. Appl. Math. 10, 197–210 (1998)

    Article  MathSciNet  Google Scholar 

  26. Grimshaw, R.H., Helfrich, K.R., Johnson, E.R.: Experimental study of the effect of rotation on nonlinear internal waves. Phys. Fluids 25, 0566,021–05660,223 (2013)

    Article  ADS  Google Scholar 

  27. Grimshaw, R.H., Ostrovsky, L.A., Shira, V.I., Stepanyants, Y.A.: Long nonlinear surface and internal gravity waves in a rotating ocean. Surv. Gheophys. 19, 289–338 (1998)

    Article  ADS  Google Scholar 

  28. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. In: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, New York, Heidelberg, Berlin (2004)

    Google Scholar 

  29. Helfrich, K.R.: Decay and return of internal solitary waves with rotation. Phys. Fluids 19, O26,601 (2007)

    Article  ADS  Google Scholar 

  30. Helfrich, K.R., Melville, W.K.: Long nonlinear internal waves. Ann. Rev. Fluid Mech. 38, 395–425 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  31. Hunter, J.K.: Numerical solutions of some nonlinear dispersive wave equations. In: E.L. Allgower (ed.), K.G.E. Computational Solutions of Nonlinear Systems of Equations. Lectures in Applied Mathematics, vol. 26, pp. 301–316. AMS, Providence (1990)

    Google Scholar 

  32. Iooss, G., Adelmeyer, M.: Topics in Bifurcation Theory and Applications. World Scientific, Singapore (1998)

    MATH  Google Scholar 

  33. Isaza, P., Mejía, J.: Global Cauchy problem for the Ostrovsky equation. Nonl. Anal. 67, 1482–1503 (2007)

    Article  MathSciNet  Google Scholar 

  34. Jbilous, K., Sadok, H.: Vector extrapolation methods. applications and numerical comparisons. J. Comput. Appl. Math. 122, 149–165 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  35. Lakoba, T., Yang, Y.: A generalized Petviashvili method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity. J. Comput. Phys. 226, 1668–1692 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  36. Lakoba, T., Yang, Y.: A mode elimination technique to improve convergence of iteration methods for finding solitary waves. J. Comput. Phys. 226, 1693–1709 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  37. Leonov, A.I.: The effect of earth rotation on the propagation of weak nonlinear surface and internal long oceanic waves. Annal. New York Acad. Sci. 373, 150–159 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  38. Levandosky, S.: On the stability of solitary waves of a generalized Ostrovsky equation. Technical Report. (2006)

    Google Scholar 

  39. Levandosky, S., Liu, Y.: Stability of solitary waves of a generalized Ostrovsky equation. SIAM J. Math. Anal. 38, 985–1011 (2006)

    Article  MathSciNet  Google Scholar 

  40. Levandosky, S., Liu, Y.: Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Disc. Cont. Dyn. Syst. Ser. B 7, 793–806 (2007)

    Article  MathSciNet  Google Scholar 

  41. Linares, F., Milanés, A.: Local and global well-posedness for the Ostrovsky equation. J. Diff. Eq. 222, 325–340 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  42. Liu, Y., Varlamov, V.: Stability of solitary waves and weak rotation limit for the Ostrovsky equation. J. Diff. Eq. 203, 159–183 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  43. Lombardi, E.: Topics in Bifurcation Theory and ApplicationsOscillatory Integral and Phenomena Beyond all Algebraic Orders. Springer, Berlin (2000)

    Google Scholar 

  44. Obregon, M.A., Stepanyants, Y.A.: On numerical solution of the Gardner-Ostrovsky equation. Math. Model Nat. Phenom. 7(2), 113–130 (2012)

    Article  MathSciNet  Google Scholar 

  45. Ostrovsky, L.A.: Nonlinear internal waves in a rotating ocean. Okeanologia 18, 181–191 (1978)

    Google Scholar 

  46. Ostrovsky, L.A., Stepanyants, Y.A.: Nonlinear surface and internal waves in rotating fluids. In: Nonlinear Waves 3, pp. 106–128. Springer, New York (1990)

    Chapter  Google Scholar 

  47. Pelinovsky, D.E., Stepanyants, Y.A.: Convergence of Petviashvili’s iteration method for numerical approximation of stationary solutions of nonlinear wave equations. SIAM J. Numer. Anal. 42, 1110–1127 (2004)

    Article  MathSciNet  Google Scholar 

  48. Pelloni, B., Dougalis, V.A.: Numerical solution of some nonlocal nonlinear dispersive wave equations. J. Nonlinear Sci. 10, 1–22 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  49. Pelloni, B., Dougalis, V.A.: Error estimates for a fully discrete spectral scheme for a class of nonlinear, nonlocal dispersive wave equations. Appl. Numer. Math. 37, 95–107 (2001)

    Article  MathSciNet  Google Scholar 

  50. Petviashvili, V.I.: Equation of an extraordinary soliton. Soviet J. Plasma Phys. 2, 257–258 (1976)

    ADS  Google Scholar 

  51. Sanz-Serna, M., J., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)

    Book  Google Scholar 

  52. Shira, V.: Propagation of long nonlinear waves in a layer of a rotating fluid. Iza. Akad. Nauk SSSR, Fiz Atmosfery i Okeana 17, 76–81 (1981)

    Google Scholar 

  53. Shira, V.: On long essentially nonlinear waves in a rotating ocean. Iza. Akad. Nauk SSSR, Fiz Atmosfery i Okeana 22, 395–405 (1986)

    Google Scholar 

  54. Sidi, A.: Convergence and stability of minimal polynomial and reduced rank extrapolation algorothms. SIAM J. Numer. Anal. 23, 197–209 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  55. Sidi, A., Ford, W.F., Smith, D.A.: Acceleration of convergence of vector sequences. SIAM J. Numer. Anal. 23, 178–196 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  56. Smith, D.A., Ford, W.F., Sidi, A.: Extrapolation methods for vector sequences. SIAM Rev. 29, 199–233 (1987)

    Article  MathSciNet  Google Scholar 

  57. Thomée, V., Vasudeva Murthy, A.S.: A numerical method for the Benjamin-Ono equation. BIT 38, 597–611 (1998)

    Article  MathSciNet  Google Scholar 

  58. Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)

    Book  Google Scholar 

  59. Tsuwaga, K.: Well-posedness and weak rotation limit for the Ostrovsky equation. J. Diff. Eq. 247, 3163–3180 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  60. Varlamov, V., Liu, Y.: Cauchy problem for the Ostrovsky equation. Disc. Dyn. Syst. 10, 731–751 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by Spanish Ministerio de Economía y Competitividad under the Research Grant MTM2014-54710-P. The author would like to thank Professors V. Dougalis, D. Dutykh and D. Mitsotakis for fruitful discussions and so important suggestions.

Author information

Authors and Affiliations

  1. Applied Mathematics Department, University of Valladolid, 47011, Valladolid, Spain

    Ángel Durán

Authors
  1. Ángel Durán
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ángel Durán .

Editor information

Editors and Affiliations

  1. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Victoriano Carmona

  2. Departamento de Física Aplicada I, Universidad de Sevilla, Sevilla, Spain

    Jesús Cuevas-Maraver

  3. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Fernando Fernández-Sánchez

  4. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Elisabeth García- Medina

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, Á. (2018). On the Numerical Approximation to Generalized Ostrovsky Equations: I. In: Carmona, V., Cuevas-Maraver, J., Fernández-Sánchez, F., García- Medina, E. (eds) Nonlinear Systems, Vol. 1. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-66766-9_12

Download citation

Publish with us

Policies and ethics

Search

Navigation