Skip to main content
SpringerLink
Log in
Book cover

Encyclopedia of Complexity and Systems Science pp 80–92Cite as

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

  • Reference work entry

Definition of the Subject

Nonlinear phenomena play a significant role in many branches of applied sciences such as applied mathematics, physics, biology, chemistry, astronomy, plasma, and fluid dynamics. Nonlinear dispersive equations that govern these phenomena have the genuine soliton property. Solitons are pulses that propagate without any change of its identity, i. e., shape and speed, during their travel through a nonlinear dispersive medium [1,5,34]. Solitons resemble properties of a particle, hence the suffix on is used [19,20].

Solitons exist in many scientific branches, such as optical fiber photonics, fiber lasers, plasmas, molecular systems, laser pulses propagating in solids, liquid crystals, nonlinear optics, cosmology, and condensed‐matter physics. Based on its importance in many fields, a huge amount of research work has been conducted during the last four decades to make more progress in understanding the soliton phenomenon. A variety of very powerful algorithms has...

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   3,499.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   549.99
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

Abbreviations

Solitons :

Solitons appear as a result of a balance between a weakly nonlinear convection and a linear dispersion. The solitons are localized highly stable waves that retains its identity: shape and speed, upon interaction, and resemble particle like behavior. In the case of a collision, solitons undergo a phase shift.

Types of solitons:

Solitary waves , which are localized traveling waves, are asymptotically zero at large distances and appear in many structures such as solitons, kinks, peakons, cuspons, and compactons, among others. Solitons appear as a bell‐shaped sech profile. Kink waves rise or descend from one asymptotic state to another. Peakons are peaked solitary-wave solutions. Cuspons exhibit cusps at their crests. In the peakon structure, the traveling-wave solutions are smooth except for a peak at a corner of its crest. Peakons are the points at which spatial derivative changes sign so that peakons have a finite jump in the first derivative of the solution \( { u(x,t) } \). Unlike peakons, where the derivatives at the peak differ only by a sign, the derivatives at the jump of a cuspon diverge.

Compactons are solitons with compact spatial support such that each compacton is a soliton confined to a finite core or a soliton without exponential tails. Compactons are generated due to the delicate interaction between the effect of the genuine nonlinear convection and the genuinely nonlinear dispersion.

Adomian method:

The Adomian decomposition method approaches linear and nonlinear, and homogeneous and inhomogeneous differential and integral equations in a unified way. The method provides the solution in a rapidly convergent series with terms that are elegantly determined in a recursive manner. The method can be used to obtain closed-form solutions, if such solutions exist. A truncated number of the obtained series can be used for numerical purposes. The method was modified to accelerate the computational process. The noise terms phenomenon, which may appear for inhomogeneous cases, can give the exact solution in two iterations only.

Bibliography

PrimaryLiterature

  1. Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  2. Adomian G (1984) A new approach to nonlinear partial differential equations. J Math Anal Appl 102:420–434

    MathSciNet  MATH  Google Scholar 

  3. Boyd PJ (1997) Peakons and cashoidal waves: travelling wave solutions of the Camassa–Holm equation. Appl Math Comput 81(2–3):173–187

    MathSciNet  MATH  Google Scholar 

  4. Camassa R, Holm D (1993) An integrable shallow water equation with peaked solitons. Phys Rev Lett 71(11):1661–1664

    MathSciNet  ADS  MATH  Google Scholar 

  5. Drazin PG, Johnson RS (1996) Solitons: an introduction. Cambridge University Press, Cambridge

    Google Scholar 

  6. Helal MA, Mehanna MS (2006) A comparison between two different methods for solving KdV‐Burgers equation. Chaos Solitons Fractals 28:320–326

    MathSciNet  ADS  MATH  Google Scholar 

  7. Hereman W, Nuseir A (1997) Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math Comp Simul 43:13–27

    MathSciNet  MATH  Google Scholar 

  8. Hirota R (1971) Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons. Phys Rev Lett 27(18):1192–1194

    ADS  MATH  Google Scholar 

  9. Hirota R (1972) Exact solutions of the modified Korteweg–de Vries equation for multiple collisions of solitons. J Phys Soc Jpn 33(5):1456–1458

    ADS  Google Scholar 

  10. Korteweg DJ, de Vries G (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 5th Ser 36:422–443

    Google Scholar 

  11. Lenells J (2005) Travelling wave solutions of the Camassa–Holm equation. J Differ Equ 217:393–430

    MathSciNet  MATH  Google Scholar 

  12. Liu Z, Wang R, Jing Z (2004) Peaked wave solutions of Camassa–Holm equation. Chaos Solitons Fractals 19:77–92

    MathSciNet  ADS  MATH  Google Scholar 

  13. Malfliet W (1992) Solitary wave solutions of nonlinear wave equations. Am J Phys 60(7):650–654

    MathSciNet  ADS  MATH  Google Scholar 

  14. Malfliet W, Hereman W (1996) The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys Scr 54:563–568

    MathSciNet  ADS  MATH  Google Scholar 

  15. Malfliet W, Hereman W (1996) The tanh method: II. Perturbation technique for conservative systems. Phys Scr 54:569–575

    MathSciNet  ADS  MATH  Google Scholar 

  16. Rosenau P (1998) On a class of nonlinear dispersive‐dissipative interactions. Phys D 230(5/6):535–546

    Google Scholar 

  17. Rosenau P, Hyman J (1993) Compactons: solitons with finite wavelengths. Phys Rev Lett 70(5):564–567

    ADS  MATH  Google Scholar 

  18. Veksler A, Zarmi Y (2005) Wave interactions and the analysis of the perturbed Burgers equation. Phys D 211:57–73

    MathSciNet  MATH  Google Scholar 

  19. Wadati M (1972) The exact solution of the modified Korteweg–de Vries equation. J Phys Soc Jpn 32:1681–1687

    ADS  Google Scholar 

  20. Wadati M (2001) Introduction to solitons. Pramana J Phys 57(5/6):841–847

    ADS  Google Scholar 

  21. Wazwaz AM (1997) Necessary conditions for the appearance of noise terms in decomposition solution series. Appl Math Comput 81:265–274

    MATH  Google Scholar 

  22. Wazwaz AM (1998) A reliable modification of Adomian's decomposition method. Appl Math Comput 92:1–7

    MathSciNet  MATH  Google Scholar 

  23. Wazwaz AM (1999) A comparison between the Adomian decomposition method and the Taylor series method in the series solutions. Appl Math Comput 102:77–86

    MathSciNet  MATH  Google Scholar 

  24. Wazwaz AM (2000) A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl Math Comput 111(1):33–51

    MathSciNet  Google Scholar 

  25. Wazwaz AM (2001) Exact specific solutions with solitary patterns for the nonlinear dispersive K(m,n) equations. Chaos, Solitons and Fractals 13(1):161–170

    MathSciNet  ADS  Google Scholar 

  26. Wazwaz AM (2002) General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive K(n,n) equations in higher dimensional spaces. Appl Math Comput 133(2/3):229–244

    MathSciNet  MATH  Google Scholar 

  27. Wazwaz AM (2003) An analytic study of compactons structures in a class of nonlinear dispersive equations. Math Comput Simul 63(1):35–44

    MathSciNet  MATH  Google Scholar 

  28. Wazwaz AM (2003) Compactons in a class of nonlinear dispersive equations. Math Comput Model l37(3/4):333–341

    MathSciNet  Google Scholar 

  29. Wazwaz AM (2004) The tanh method for travelling wave solutions of nonlinear equations. Appl Math Comput 154(3):713–723

    MathSciNet  MATH  Google Scholar 

  30. Wazwaz AM (2005) Compact and noncompact structures for variants of the KdV equation. Int J Appl Math 18(2):213–221

    MathSciNet  MATH  Google Scholar 

  31. Wazwaz AM (2006) New kinds of solitons and periodic solutions to the generalized KdV equation. Numer Methods Partial Differ Equ 23(2):247–255

    MathSciNet  Google Scholar 

  32. Wazwaz AM (2006) Peakons, kinks, compactons and solitary patterns solutions for a family of Camassa–Holm equations by using new hyperbolic schemes. Appl Math Comput 182(1):412–424

    MathSciNet  MATH  Google Scholar 

  33. Wazwaz AM (2007) The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations. Appl Math Comput 184(2):1002–1014

    MathSciNet  MATH  Google Scholar 

  34. Whitham GB (1999) Linear and nonlinear waves. Wiley, New York

    MATH  Google Scholar 

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

    ADS  MATH  Google Scholar 

Books and Reviews

  1. Adomian G (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston

    MATH  Google Scholar 

  2. Burgers JM (1974) The nonlinear diffusion equation. Reidel, Dordrecht

    MATH  Google Scholar 

  3. Conte R, Magri F, Musette M, Satsuma J, Winternitz P (2003) Lecture Notes in Physics: Direct and Inverse methods in Nonlinear Evolution Equations. Springer, Berlin

    Google Scholar 

  4. Hirota R (2004) The Direct Method in Soliton Theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  5. Johnson RS (1997) A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  6. Kosmann-Schwarzbach Y, Grammaticos B, Tamizhmani KM (2004) Lecture Notes in Physics: Integrability of Nonlinear Systems. Springer, Berlin

    Google Scholar 

  7. Wazwaz AM (1997) A First Course in Integral Equations. World Scientific, Singapore

    MATH  Google Scholar 

  8. Wazwaz AM (2002) Partial Differential Equations: Methods and Applications. Balkema, The Netherlands

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Saint Xavier University, Chicago, USA

    Abdul-Majid Wazwaz

Authors
  1. Abdul-Majid Wazwaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

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

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

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag

About this entry

Cite this entry

Wazwaz, AM. (2009). Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation