Skip to main content
SpringerLink
Log in

Numerical Study of the Generalized Korteweg–de Vries Equations with Oscillating Nonlinearities and Boundary Conditions

  • Original Article
  • Published:
Water Waves Aims and scope Submit manuscript

Abstract

The focus here is upon the generalized Korteweg–de Vries equation,

$$\begin{aligned} u_t + u_x + \frac{1}{p} \left( u^p \right) _x +u_{xxx} \, = \, 0, \end{aligned}$$

where \(p = 2, 3, \ldots \). When \(p \ge 5\), it is thought that the equation is not globally well posed in time for \(L_2\)-based Sobolev class data. Various numerical simulations carried out by multiple research groups indicate that solutions can blowup in finite time for large, smooth initial data. This is known to be the case in the critical case \(p = 5\), but remains a conjecture for supercritical values of p. Studied here are methods for controlling this potential blow up. Several candidates are put forward; the addition of dissipation or of higher order dispersion are two obvious candidates. However, these apparently can only work for a limited range of nonlinearities. However, the introduction of high frequency temporal oscillations appear to be more effective. Both temporal oscillation of the nonlinearity and of the boundary condition in an initial-boundary-value configuration are considered. The bulk of the discussion will turn around this prospect in fact.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Abdullaev, F.Kh., Caputo, J.G., Kraenkel, R.A., Malomed, B.A.: Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length. Phys. Rev. A 67, 013605 (2003)

  2. Abe, K., Inoue, O.: Fourier expansion solution of the Korteweg–de Vries equation. J. Comput. Phys. 34(2), 202–210 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Albert, J.P., Bona, J.L., Felland, M.: A criterion for the formation of singularities for the generalized Korteweg–de Vries equation. Mat. Aplic. Comp. 5(1), 3–11 (1988)

    MathSciNet  MATH  Google Scholar 

  4. Alexander, M.E., Morris, J.L.: Galerkin methods applied to some model equations for non-linear dispersive waves. J. Comput. Phys. 30(3), 428–451 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baker, G.A., Dougalis, V.A., Karakashian, O.A.: Convergence of Galerkin approximations for the Korteweg–de Vries equation. Math. Comput. 40(162), 419–433 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A 272(1220), 47–78 (1972)

  7. Boczar-Karakiewicz, B., Bona, J.L., Romanczyk, W., Thornton, E.G.: Modeling the dynamics of the bar system at Duck, N.C., USA. In: Proc. 26th International Conference on Coastal Engr. (Copenhagen) American Soc. Civil Engrs., New York, pp. 2877–2887 (1998)

  8. Bona, J.L.: Convergence of periodic wave trains in the limit of large wavelength. Appl. Sci. Res. 37, 21–33 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bona, J.L., Chen, H., Karakashian, O.A., Xing, Y.: Conservative, discontinuous Galerkin-methods for the generalized Korteweg–de Vries equation. Math. Comput. 82(283), 1401–1432 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bona, J.L., Dougalis, V.A., Karakashian, O.A.: Fully discrete Galerkin methods for the Korteweg–de Vries equation. Comput. Math. Appl. Ser. A 12(7), 859–884 (1986)

    MathSciNet  MATH  Google Scholar 

  11. Bona, J.L., Dougalis, V.A., Karakashian, O.A., McKinney, W.R.: Computations of blow-up and decay for periodic solutions of the generalized Korteweg–de Vries–Burgers equation. Appl. Numer. Math. 10(3–4), 335–355 (1992). A Festschrift to honor Professor Garrett Birkhoff on his eightieth birthday

  12. Bona, J.L., Dougalis, V.A., Karakashian, O.A., McKinney, W.R.: Conservative, high-order numerical schemes for the generalized Korteweg–de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A 351(1695), 107–164 (1995)

  13. Bona, J.L., Dougalis, V.A., Karakashian, O.A., McKinney, W.R.: The effect of dissipation on solutions of the generalized Korteweg–de Vries equation. J. Comput. Appl. Math. 74, 127–154 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bona, J.L., Chen, H., Sun, S.-M., Zhang, B.-Y.: Comparison of quarter-plane and two-point boundary value problems: the KdV equation. Discrete Contin. Dyn. Syst. Ser. B 7(3), 465–495 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Bona, J.L., Chen, H., Hong, Y.: Blowup in the face of higher-order dissipation for the generalized Korteweg–de Vries equation (in preparation)

  16. Bona, J.L., Lenells, J.: The KdV equation on the half line: time periodicity and mass transport. SIAM J. Math. Anal. 52(2), 1009–1039 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bona, J.L., Luo, L.: A generalized Korteweg–de Vries equation in a quarter plane. Contemp. Math. 221, 59–125 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bona, J.L., Pritchard, W.G., Scott, L.R.: An evaluation of a water wave model. Philos. Trans. R. Soc. Lond. Ser. A 302, 457–510 (1981)

  19. Bona, J.L., Smith, R.: The initial-value problem for the Korteweg–de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A 278, 555–601 (1975)

  20. Bona, J.L., Souganidis, P.E., Strauss, W.A.: Stability and instability of solitary waves of Korteweg–de Vries type. Proc. R. Soc. Lond. Ser. A 411(1841), 395–412 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bona, J.L., Sun, S.-M., Zhang, B.-Y.: Forced ocsillations of a damped Korteweg–de Vries equation in a quarter plane. Commun. Contemp. Math. 5(3), 369–400 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bona, J.L., Sun, S.-M., Zhang, B.-Y.: A nonhomogeneous boundary-value problem for the Korteweg–de Vries equation posed on a finite domain. Commun. Partial Differ. Equ. 28(7–8), 1391–1436 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bona, J.L., Sun, S.-M., Zhang, B.-Y.: A non-homogeneous boundary-value problem for the Korteweg–de Vries equation posed on a finite domain. II. J. Differ. Equ. 247(9), 2558–2596 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bona, J.L., Winther, R.: The Korteweg–de Vries equation, posed in a quarter plane. SIAM J. Math. Anal. 14(6), 1056–1106 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bona, J.L., Winther, R.: The Korteweg–de Vries Equation, posed in a quarter plane, continuous dependence results. Differ. Integr. Equ. 2(2), 228–250 (1989)

    MATH  Google Scholar 

  26. Bona, J.L., Wu, J.: Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane. Discrete Contin. Dyn. Syst. 23(4), 1141–1168 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Boussinesq, J.: Essai sur la theorie des eaux courantes, pp. 1–680. Memoires presentes par divers savants l’Acad. des Sci. Inst. Nat. France, XXIII (1877)

  28. Canuto, Cl., Yousuff Hussaini, M., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer, New York (1988)

  29. Carvajal, X., Panthee, M., Scialom, M.: On the critical KDV equation with time-oscillating nonlinearity. Differ. Integr. Equ. 24(5–6), 541–567 (2011)

    MathSciNet  MATH  Google Scholar 

  30. Cazenave, T., Scialom, M.: A Schrödinger equation with time-oscillating nonlinearity. Revista Matemática Complutense 23, 321–339 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Chan, T.F., Kerkhoven, T.: Fourier methods with extended stability intervals for the Korteweg–de Vries equation. SIAM J. Numer. Anal. 22(3), 441–454 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  32. Chen, H.: Long period limit of nonlinear, dispersive waves. Differ. Integr. Equ. 19(4), 463–480 (2006)

    MathSciNet  MATH  Google Scholar 

  33. Colliander, J.E., Kenig, C.E.: The generalized Korteweg–de Vries equation on the half line. Commun. Partial Differ. Equ. 27(11–12), 2187–2266 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Damergi, I., Goubet, O.: Blow-up solutions to the nonlinear Schrödinger equation with oscillating nonlinearities. J. Math. Anal. Appl. 352(1), 336–344 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Dauxois, T.: Fermi, Pasta, Ulam, and a mysterious lady. Physics Today, pp. 56–57 (2008)

  36. Di Menza, L., Goubet, O.: Stabilizing blow up solutions to nonlinear Schrödinger equations. Commun. Pure Appl. Anal. 16(3), 1059–1081 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  37. Dougalis, V.A., Karakashian, O.A.: On some high-order accurate fully discrete Galerkin methods for the Korteweg–de Vries equation. Math. Comput. 45(172), 329–345 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  38. Dubrovin, B., Grava, T., Klein, C.: Numerical study of breakup in generalized Korteweg–de Vries and Kawahara equations. SIAM J. Appl. Math. 71(4), 983–1008 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Erdogan, M.B., Tzirakis, N.: Lecture Notes II: on local and global theory for the KdV equation. https://faculty.math.illinois.edu/~tzirakis/KdV.pdf

  40. Faminskii, A.V.: An initial boundary-value problem in a half-strip for the Korteweg–de Vries equation in fractional-order Sobolev spaces. Commun. Partial Differ. Equ. 29, 1653–1695 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  41. Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems Document LA-1940. Los Alamos National Laboratory (1955)

  42. Gabitov, I.R., Lushnikov, P.M.: Nonlinearity management in a dispersion-managed system. Opt. Lett. 27(2), 113–115 (2002)

    Article  Google Scholar 

  43. Gardner, C.S., Morikawa, G.K.: Similarity in the asymptotic behavior of collision-free hydromagnetic waves and water waves. Report MF-2, NYO-9080, Courant Institute of Mathematical Sciences, New York University (1960)

  44. Guo, B.-Y.: Spectral Methods and Their Applications. World Scientific Publishing Co., River Edge (1998)

    Book  MATH  Google Scholar 

  45. Hammack, J.L.: A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech. 60(4), 769–799 (1973)

    Article  MATH  Google Scholar 

  46. Hammack, J.L., Segur, H.: The Korteweg–de Vries equation and water waves. Part 2. Comparison with experiments. J. Fluid Mech. 65(2), 289–314 (1974)

  47. Heinrichs, W.: Spectral approximation of third-order problems. J. Sci. Comput. 14(3), 275–289 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kato, T.: On the Korteweg–de Vries equation. Manuscr. Math. 28, 89–99 (1979)

    Article  MATH  Google Scholar 

  49. Kato, T.: On the Cauchy problem for the (generalized) Korteweg–de Vries equation. In: Studies in Applied Mathematics, Adv. Math. Suppl. Stud., vol. 8, pp. 93–128. Academic Press, New York (1983)

  50. Kenig, C.E., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46(4), 527–620 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  51. Klein, C., Peter, R.: Numerical study of blow-up and dispersive shocks in solutions to generalized Korteweg–de Vries equations. Phys. D 304(305), 52–78 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  52. Klein, C., Saut, J.-C.: Numerical study of blow up and stability of solutions of generalized Kadomtsev–Petviashvili equations. J. Nonlinear Sci. 22(5), 763–811 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. Konotop, V.V., Pacciani, P.: Collapse of solutions of the nonlinear Schrödinger equation with a time-dependent nonlinearity: application to Bose–Einstein condensates. Phys. Rev. Lett. 94, 240405 (2005)

    Article  Google Scholar 

  54. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39(240), 422–443 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  55. Ma, H., Sun, W.: A Legendre–Petrov–Galerkin and Chebyshev collocation method for third-order differential equations. SIAM J. Numer. Anal. 38(5), 1425–1438 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  56. Maday, Y., Quarteroni, A.: Error analysis for spectral approximation of the Korteweg–de Vries equation. RAIRO Modél. Math. Anal. Numér. 22(3), 499–529 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  57. Merle, F.: Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. Am. Math. Soc. 14(3), 555–578 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  58. Miura, R.M.: The Korteweg–de Vries equation: a survey of results. SIAM Rev. 18(3), 412–459 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  59. Levy, D., Shu, C.-W., Yan, J.: Local discontinuous Galerkin methods for nonlinear dispersive equations. J. Comput. Phys. 196(2), 751–772 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  60. Nunes, W.V.L.: Global well-posedness for the transitional Korteweg–de Vries equation. Appl. Math. Lett. 11(5), 15–20 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  61. Panthee, M., Scialom, M.: On the supercritical KdV equation with time-oscillating nonlinearity. NoDEA Nonlinear Differ. Equ. Appl. 20(3), 1191–1212 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  62. Pavoni, D.: Single and multidomain Chebyshev collocation methods for the Korteweg–de Vries equation. Calcolo 25(4), 311–346 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  63. Pelinovsky, E., Polukhina, O., Slunyaev, A., Talipova, T.: Internal solitary waves, Ch. 4. In: WIT Transactions on State of the Art in Science and Engineering, vol. 9. WIT Press, Southampton (2007)

  64. Scott, A.C., Chu, F.Y.F., McLaughlin, D.W.: The soliton: a new concept in applied science. Proc. IEEE 61, 1443–1483 (1973)

    Article  MathSciNet  Google Scholar 

  65. Shen, J.: A new dual-Petrov–Galerkin method for third and higher odd-order differential equations: application to the KdV equation. SIAM J. Numer. Anal. 41(5), 1595–1619 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  66. Shen, J., Tang, T.: Spectral and high-order methods with applications. Mathematics Monograph Series, vol. 3. Science Press, Beijing (2006)

  67. Shen, J., Tang, T., Wang, L.-L.: Spectral methods, Springer Series in Computational Mathematics, vol. 41. Springer, Heidelberg (2011). Algorithms, analysis and applications

  68. Skogestad, J.O., Kalisch, H.: A boundary value problem for the KdV equation: comparison of finite-difference and Chebyshev methods. Math. Comput. Simul. 80(1), 151–163 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  69. Steinerberger, S.: Dispersion dynamics for the defocusing generalized Korteweg–de Vries equation. Proc. Am. Math. Soc. 143(2), 789–800 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  70. Tao, T.: Nonlinear dispersive equations: local and global analysis. CBMS Regional Conference Series in Mathematics, vol. 106. American Math. Soc., Providence (2006)

  71. Tao, T.: Two remarks on the generalised Korteweg–de Vries equation. Discrete Contin. Dyn. Syst. 18(1), 1–14 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  72. Thiab, T.R., Ablowitz, M.J.: IST numerical schemes for nonlinear evolution equations of physical interest. In: Numerical Approximation of Partial Differential Equations (Madrid, 1985), North-Holland Math. Stud, vol. 133, pp. 425–433. North-Holland, Amsterdam (1987)

  73. Trefethen, L.N.: Spectral Methods in MATLAB, Software, Environments, and Tools, vol. 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000)

    Book  Google Scholar 

  74. Yan, J., Shu, C.-W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40(2), 769–791 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  75. Yi, N., Huang, Y., Liu, H.: A direct discontinuous Galerkin method for the generalized Korteweg–de Vries equation: energy conservation and boundary effect. J. Comput. Phys. 242, 351–366 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  76. Zabusky, N.J., Galvin, C.: Shallow-water waves, the Korteweg–de Vries equation and solitons. J. Fluid Mech. 47(4), 811–824 (1971)

    Article  Google Scholar 

  77. Zharnitsky, V., Grenier, E., Jones, C.K.R.T., Turitsyn, S.K.: Stabilizing effects of dispersion management. Phys. D 152(153), 794–817 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by a research grant from the University of Illinois at Chicago. The work of Y. Hong was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (NRF-2021R1A2C1093579).

Author information

Author notes

  1. Jerry Bona and Youngjoon Hong contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, USA

    Jerry Bona

  2. Department of Mathematics, Sungkyunkwan University, Suwon, South Korea

    Youngjoon Hong

Authors
  1. Jerry Bona
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Youngjoon Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jerry Bona.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bona, J., Hong, Y. Numerical Study of the Generalized Korteweg–de Vries Equations with Oscillating Nonlinearities and Boundary Conditions. Water Waves 4, 109–137 (2022). https://doi.org/10.1007/s42286-022-00057-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42286-022-00057-5

Keywords

Search

Navigation