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Korteweg–de Vries Equation (KdV) and Modified Korteweg–de Vries Equations (mKdV), Semi-analytical Methods for Solving the

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

In this study, some semi‐analytical/numerical methods are applied to solve the Korteweg–de Vries (KdV) equation and the modifiedKorteweg–de Vries (mKdV) equation, which are characterized by the solitary wave solutions of the classical nonlinear equations that lead tosolitons. Here, the classical nonlinearequations of interest usually admit for the existence of a special type of the traveling wave solutions which are either solitary waves orsolitons. These approaches are based on the choice of a suitable differential operator which may be ordinary or partial, linear or nonlinear,deterministic or stochastic. It does not require discretization, and consequently massive computation.

In this scheme the solution is performed in the form of a convergent power series with easily computable components. This section isparticularly concerned with the Adomian decomposition method (ADM ) and the results obtained are compared to those obtained by the...

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Abbreviations

Korteweg–de Vries equation :

The classical nonlinear equations of interest usually admit for the existence of a special type of the traveling wave solutions , which are either solitary waves or solitons.

Modified Korteweg–de Vries:

This equation is a modified form of the classical KdV equation in the nonlinear term.

Soliton :

This concept can be regarded as solutions of nonlinear partial differential equations.

Exact solution :

A solution to a problem that contains the entire physics and mathematics of a problem, as opposed to one that is approximate, perturbative, closed, etc.

Adomian decomposition method , Homotopy analysis method, Homotopy perturbation method and Variational perturbation method :

These are some of the semi‐analytic/numerical methods for solving ODE or PDE in literature.

Bibliography

Primary Literature

  1. Abbasbandy S (2006) The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 360:109–13

    MathSciNet  ADS  MATH  Google Scholar 

  2. Abdou MA, Soliman AA (2005) New applications of variational iteration method. Phys D 211:1–8

    MathSciNet  MATH  Google Scholar 

  3. Abdou MA, Soliman AA (2005) Variational iteration method for solving Burger's and coupled Burger's equations. J Comput Appl Math 181:245–251

    MathSciNet  ADS  MATH  Google Scholar 

  4. Adomian G (1988) A Review of the Decomposition Method in Applied Mathematics. J Math Anal Appl 135:501–544

    MathSciNet  MATH  Google Scholar 

  5. Adomian G, Rach R (1992) Noise Terms in Decomposition Solution Series. Comput Math Appl 24:61–64

    MathSciNet  MATH  Google Scholar 

  6. Adomian G (1994) Solving Frontier Problems of Physics: The decomposition method. Kluwer, Boston

    MATH  Google Scholar 

  7. Chowdhury MSH, Hashim I, Abdulaziz O (2009) Comparison of homotopy analysis method and homotopy‐perturbation method for purely nonlinear fin-type problems. Commun Nonlinear Sci Numer Simul 14:371–378

    MathSciNet  ADS  MATH  Google Scholar 

  8. Debtnath L (1997) Nonlinear Partial Differential Equations for Scientist and Engineers. Birkhauser, Boston

    Google Scholar 

  9. Debtnath L (2007) A brief historical introduction to solitons and the inverse scattering transform – a vision of Scott Russell. Int J Math Edu Sci Techn 38:1003–1028

    Google Scholar 

  10. Domairry G, Nadim N (2008) Assessment of homotopy analysis method and homotopy perturbation method in non‐linear heat transfer equation. Intern Commun Heat Mass Trans 35:93–102

    Google Scholar 

  11. Drazin PG, Johnson RS (1989) Solutions: An Introduction. Cambrige University Press, Cambridge

    Google Scholar 

  12. Edmundson DE, Enns RH (1992) Bistable light bullets. Opt Lett 17:586

    ADS  Google Scholar 

  13. Fermi E, Pasta J, Ulam S (1974) Studies of nonlinear problems. American Math Soci, Providence, pp 143–156

    Google Scholar 

  14. Gardner CS, Greene JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg–de Vries equation. Phys Rev Lett 19:1095–1097

    ADS  MATH  Google Scholar 

  15. Gardner CS, Greene JM, Kruskal MD, Miura RM (1974) Korteweg–de Vries equation and generalizations, VI, Methods for exact solution. Commun Pure Appl Math 27:97–133

    MathSciNet  MATH  Google Scholar 

  16. Geyikli T, Kaya D (2005) An application for a Modified KdV equation by the decomposition method and finite element method. Appl Math Comp 169:971–981

    MathSciNet  MATH  Google Scholar 

  17. Geyikli T, Kaya D (2005) Comparison of the solutions obtained by B‑spline FEM and ADM of KdV equation. Appl Math Comp 169:146–156

    MathSciNet  MATH  Google Scholar 

  18. Hayat T, Abbas Z, Sajid M (2006) Series solution for the upper‐convected Maxwell fluid over a porous stretching plate. Phys Lett A 358:396–403

    ADS  MATH  Google Scholar 

  19. Hayat T, Sajid M (2007) On analytic solution of thin film flow of a fourth grade fluid down a vertical cylinder. Phys Lett A 361:316–322

    ADS  MATH  Google Scholar 

  20. He JH (1997) A new approach to nonlinear partial differential equations. Commun Nonlinear Sci Numer Simul 2:203–205

    Google Scholar 

  21. He JH (1999) Homotopy perturbation technique. Comput Math Appl Mech Eng 178:257–262

    MATH  Google Scholar 

  22. He JH (1999) Variation iteration method – a kind of non‐linear analytical technique: some examples. Int J Nonlinear Mech 34:699–708

    MATH  Google Scholar 

  23. He JH (2000) A coupling method of homotopy technique and a perturbation technique for nonlinear problems. Int J Nonlinear Mech 35:37–43

    MATH  Google Scholar 

  24. He JH (2003) Homotopy perturbation method a new nonlinear analytical technique. Appl Math Comput 135:73–79

    MathSciNet  MATH  Google Scholar 

  25. He JH (2006) Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 20:1141–1199

    ADS  MATH  Google Scholar 

  26. He JH, Wu XH (2006) Construction of solitary solution and compacton‐like solution by variational iteration method. Chaos Solitons Fractals 29:108–113

    MathSciNet  ADS  MATH  Google Scholar 

  27. Helal MA, Mehanna MS (2007) A comparative study between two different methods for solving the general Korteweg–de Vries equation. Chaos Solitons Fractals 33:725–739

    MathSciNet  ADS  MATH  Google Scholar 

  28. Hirota R (1971) Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys Rev Lett 27:1192–1194

    ADS  MATH  Google Scholar 

  29. Hirota R (1973) Exact N‑solutions of the wave equation of long waves in shallow water and in nonlinear lattices. J Math Phys 14:810–814

    MathSciNet  ADS  MATH  Google Scholar 

  30. Hirota R (1976) Direct methods of finding exact solutions of nonlinear evolution equations. In: Miura RM (ed) Bäcklund Transformations. Lecture Notes in Mathematics, vol 515. Springer, Berlin, pp 40–86

    Google Scholar 

  31. Inan IE, Kaya D (2006) Some Exact Solutions to the Potential Kadomtsev–Petviashvili Equation. Phys Lett A 355:314–318

    MathSciNet  ADS  MATH  Google Scholar 

  32. Inan IE, Kaya D (2006) Some Exact Solutions to the Potential Kadomtsev–Petviashvili Equation. Phys Lett A 355:314–318

    MathSciNet  ADS  MATH  Google Scholar 

  33. Inan IE, Kaya D (2007) Exact Solutions of the Some Nonlinear Partial Differential Equations. Phys A 381:104–115

    MathSciNet  Google Scholar 

  34. Kaya D (2003) A Numerical Solution of the Sine–Gordon Equation Using the Modified Decomposition Method. Appl Math Comp 143:309–317

    MathSciNet  MATH  Google Scholar 

  35. Kaya D (2006) The Exact and Numerical Solitary‐wave Solutions for Generalized Modified Boussinesq Equation. Phys Lett A 348:244–250

    MathSciNet  ADS  MATH  Google Scholar 

  36. Kaya D Some Methods for the Exact and Numerical Solutions of Nonlinear Evolution Equations. Paper presented at the 2nd International Conference on Mathematics: Trends and Developments (ICMTD), Cairo, Egypt, Dec 2007, pp 27–30 (in press)

    Google Scholar 

  37. Kaya D, Al‐Khaled K (2007) A Numerical Comparison of a Kawahara Equation. Phys Lett A 363:433–439

    Google Scholar 

  38. Kaya D, El-Sayed SM (2003) An Application of the Decomposition Method for the Generalized KdV and RLW Equations. Chaos Solitons Fractals 17:869–877

    MathSciNet  ADS  MATH  Google Scholar 

  39. Kaya D, El-Sayed SM (2003) Numerical Soliton‐like Solutions of the Potential Kadomstev–Petviashvili Equation by the Decomposition Method. Phys Lett A 320:192–199

    MathSciNet  ADS  MATH  Google Scholar 

  40. Kaya D, El-Sayed SM (2003) On a Generalized Fifth Order KdV Equations. Phys Lett A 310:44–51

    MathSciNet  ADS  MATH  Google Scholar 

  41. Kaya D, Inan IE (2005) A Convergence Analysis of the ADM and an Application. Appl Math Comp 161:1015–1025

    MathSciNet  MATH  Google Scholar 

  42. Khater AH, El‐Kalaawy OH, Helal MA (1997) Two new classes of exact solutions for the KdV equation via Bäcklund transformations. Chaos Solitons Fractals 8:1901–1909

    Google Scholar 

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

    MATH  Google Scholar 

  44. Liao SJ (1992) The proposed homotopy analysis technique for the solution of nonlinear problems. Ph D Thesis, Shanghai Jiao Tong University

    Google Scholar 

  45. Liao SJ (1995) An approximate solution technique not depending on small parameters: A special example. Int J Nonlinear Mech 30:371

    MATH  Google Scholar 

  46. Liao SJ (2003) Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton

    Google Scholar 

  47. Liao SJ (2004) On the homotopy analysis method for nonlinear problems. Appl Math Comp 147:499

    MATH  Google Scholar 

  48. Liao SJ (2004) On the homotopy analysis method for nonlinear problems. Appl Math Comput 147:499–513

    MathSciNet  MATH  Google Scholar 

  49. Liao SJ (2005) Comparison between the homotopy analysis method and homotopy perturbation method. Appl Math Comp 169:1186–1194

    MATH  Google Scholar 

  50. Light Bullet Home Page, http://www.sfu.ca/%7Erenns/lbullets.html

  51. Miura RM (1968) Korteweg–de Vries equations and generalizations, I; A remarkable explicit nonlinear transformations. J Math Phys 9:1202–1204

    MathSciNet  ADS  MATH  Google Scholar 

  52. Miura RM, Gardner CS, Kruskal MD (1968) Korteweg–de Vries equations and generalizations, II; Existence of conservation laws and constants of motion. J Math Phys 9:1204–1209

    MathSciNet  ADS  MATH  Google Scholar 

  53. Momani S, Abuasad S (2006) Application of He's variational iteration method to Helmholtz equation. Chaos Solitons Fractals 27:1119–1123

    MathSciNet  ADS  MATH  Google Scholar 

  54. Polat N, Kaya D, Tutalar HI (2006) A Analytic and Numerical Solution to a Modified Kawahara Equation and a Convergence Analysis of the Method. Appl Math Comp 179:466–472

    MathSciNet  MATH  Google Scholar 

  55. Ramos JI (2008) On the variational iteration method and other iterative techniques for nonlinear differential equations. Appl Math Comput 199:39–69

    MathSciNet  MATH  Google Scholar 

  56. Rayleigh L (1876) On Waves. Lond Edinb Dublin Philos Mag 5:257

    Google Scholar 

  57. Russell JS (1844) Report on Waves. In: Proc 14th meeting of the British Association for the Advancement of Science, BAAS, London

    Google Scholar 

  58. Sajid M, Hayat T (2008) Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations, Nonlinear Analysis. Real World Appl 9:2296–2301

    MathSciNet  MATH  Google Scholar 

  59. Sajid M, Hayat T (2008) The application of homotopy analysis method for thin film flow of a third order fluid. Chaos Solitons Fractal 38:506–515

    MathSciNet  ADS  MATH  Google Scholar 

  60. Sajid M, Hayat T, Asghar S (2007) Comparison between HAM and HPM solutions of thin film flows of non‐Newtonian fluids on a moving belt. Nonlinear Dyn 50:27–35

    MathSciNet  MATH  Google Scholar 

  61. Shawagfeh N, Kaya D (2004) Series Solution to the Pochhammer–Chree Equation and Comparison with Exact Solutions. Comp Math Appl 47:1915–1920

    MathSciNet  MATH  Google Scholar 

  62. Ugurlu Y, Kaya D Exact and Numerical Solutions for the Cahn–Hilliard Equation. Comput Math Appl

    Google Scholar 

  63. Wazwaz AM (1997) Necessary Conditions for the Appearance of Noise Terms in Decomposition Solution Series J Math Anal Appl 5:265–274

    Google Scholar 

  64. Wazwaz AM (2002) Partial Differential Equations: Methods and Applications. Balkema, Rotterdam

    MATH  Google Scholar 

  65. Wazwaz AM (2007) Analytic study for fifth-order KdV-type equations with arbitrary power nonlinearities. Comm Nonlinear Sci Num Sim 12:904–909

    MathSciNet  MATH  Google Scholar 

  66. Wazwaz AM (2007) A variable separated ODE method for solving the triple sine‐Gordon and the triple sinh‐Gordon equations. Chaos Solitons Fractals 33:703–710

    MathSciNet  ADS  MATH  Google Scholar 

  67. Wazwaz AM (2007) The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. App Math Comp 187:1131–1142

    MathSciNet  MATH  Google Scholar 

  68. Wazwaz AM (2007) The variational iteration method for solving linear and nonlinear systems of PDEs. Comput Math Appl 54:895–902

    MathSciNet  MATH  Google Scholar 

  69. Wazwaz AM, Helal MA (2004) Variants of the generalized fifth-order KdV equation with compact and noncompact structures. Chaos Solitons Fractals 21:579–589

    MathSciNet  ADS  MATH  Google Scholar 

  70. Whitham GB (1974) Linear and Nonlinear Waves. Wiley, New York

    MATH  Google Scholar 

  71. Zabusky NJ, Kruskal MD (1965) Interactions of solitons in collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243

    ADS  MATH  Google Scholar 

Books and Reviews

  1. The following, referenced by the end of the paper, is intended to give some useful for further reading.

    Google Scholar 

  2. For another obtaining of the KdV equation for water waves, see Kevorkian and Cole (1981); one can see the work of the Johnson (1972) for a different water-wave application with variable depth, for waves on arbitrary shears in the work of Freeman and Johnson (1970) and Johnson (1980) for a review of one and two‐dimensional KdV equations. In addition to these; one can see the book of Drazin and Johnson (1989) for some numerical solutions of nonlinear evolution equations. In the work of the Zabusky, Kruskal and Deam (F1965) and Eilbeck (F1981), one can see the motion pictures of soliton interactions. See a comparison of the KdV equation with water wave experiments in Hammack and Segur (1974).

    Google Scholar 

  3. For further reading of the classical exact solutions of the nonlinear equations can be seen in the works: the Lax approach is described in Lax (1968); Calogero and Degasperis (1982, A.20), the Hirota's bilinear approach is developed in Matsuno (1984), the Bäckland transformations are described in Rogers and Shadwick (1982); Lamb (1980, Chap. 8), the Painleve properties is discussed by Ablowitz and Segur (1981, Sect. 3.8), In the book of Dodd, Eilbeck, Gibbon and Morris (1982, Chap. 10) can found review of the many numerical methods to solve nonlinear evolution equations and shown many of their solutions.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Firat University, Elazig, Turkey

    Doğan Kaya

  2. Dean of Engineering Faculty, Ardahan University, Ardahan, Turkey

    Doğan Kaya

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  1. Doğan Kaya
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Kaya, D. (2009). Korteweg–de Vries Equation (KdV) and Modified Korteweg–de Vries Equations (mKdV), Semi-analytical Methods for Solving the. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_305

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