Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

(3+1)-Dimensional Nonlinear Equations and Couplings of Fifth-Order Equations in the Solitary Waves Theory: Multiple Soliton Solutions

  • Abdul-Majid WazwazEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_5-7
  • 608 Downloads

Glossary

Solitons

Solitons appear as a result of a balance between a weakly nonlinear convection and a linear dispersion. The soliton is a localized highly stable wave that retains its identity – shape and speed – upon interaction and resembles particle-like behavior. It is a localized solitary wave so that it decays or approaches a constant at infinity. In the case of a collision, solitons undergo a phase shift. The stability of solitons stems from the delicate equilibrium between the two effects of nonlinearity and dispersion.

Types of Traveling Waves

Traveling waves appear in many scientific and engineering applications in solitary wave theory. Solitons, kinks, peakons, cuspons, compactons, negatons, positons, complexitons, and others are examples of solitary waves. Solitons are localized wave packets which are asymptotically zero at large distances with exponential wings or tails. Kink waves are solitons that rise or descend from one asymptotic state to another, and hence another...

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

Bibliography

Primary Literature

  1. Blaszak M, Szablikowski B, Silindir B (2012) Construction and separability of nonlinear soliton integrable couplings. Appl Math Comput 219(2012):1866–1873zbMATHMathSciNetCrossRefGoogle Scholar
  2. Geng X (2003) Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations. J Phys A Math Gen 36:2289–2303zbMATHCrossRefADSGoogle Scholar
  3. Hereman W, Nuseir A (1997) Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math Comput Simul 43:13–27zbMATHMathSciNetCrossRefGoogle Scholar
  4. Hietarinta J (1987) A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations. J Math Phys 28(8):1732–1742zbMATHMathSciNetCrossRefADSGoogle Scholar
  5. Hirota R (1971) Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett 27(18):1192–1194zbMATHCrossRefADSGoogle Scholar
  6. Hirota R (1972) Exact solutions of the modified Korteweg-de Vries equation for multiple collisions of solitons. J Phys Soc Jpn 33(5):1456–1458CrossRefADSGoogle Scholar
  7. Hirota R, Ito M (1983) Resonance of solitons in one dimension. J Phys Soc Jpn 52(3):744–748MathSciNetCrossRefADSGoogle Scholar
  8. Kadomtsev BB, Petviashvili VI (1970) On the stability of solitary waves in weakly dispersive media. Sov Phys Dokl 15:539–541zbMATHADSGoogle Scholar
  9. Lenells J (2005) Travelling wave solutions of the Camassa-Holm equation. J Differ Equ 217:393–430zbMATHMathSciNetCrossRefADSGoogle Scholar
  10. Liu Z, Wang R, Jing Z (2004) Peaked wave solutions of Camassa–Holm equation. Chaos Solitons Fractals 19:77–92zbMATHMathSciNetCrossRefADSGoogle Scholar
  11. Ma WX, Fuchssteiner B (1996) Integrable theory of the perturbation equations. Chaos Solitons Fractals 7:1227–1250zbMATHMathSciNetCrossRefADSGoogle Scholar
  12. Ma WX, Abdeljabbar A, Asaad MG (2011) Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation. Appl Math Comput 217:10016–10023zbMATHMathSciNetCrossRefGoogle Scholar
  13. Malfliet W (1992) Solitary wave solutions of nonlinear wave equations. Am J Phys 60(7):650–654zbMATHMathSciNetCrossRefADSGoogle Scholar
  14. Malfliet W, Hereman W (1996a) The tanh method:I. Exact solutions of nonlinear evolution and wave equations. Phys Scr 54:563–568zbMATHMathSciNetCrossRefADSGoogle Scholar
  15. Malfliet W, Hereman W (1996b) The tanh method:II. Perturbation technique for conservative systems. Phys Scr 54:569–575zbMATHMathSciNetCrossRefADSGoogle Scholar
  16. Shen H-F, Tu M-H (2011) On the constrained B-type Kadomtsev-Petviashvili equation: Hirota bilinear equations and Virasoro symmetry. J Math Phys 52:032704MathSciNetCrossRefADSGoogle Scholar
  17. Veksler A, Zarmi Y (2005) Wave interactions and the analysis of the perturbed Burgers equation. Phys D 211:57–73zbMATHMathSciNetCrossRefGoogle Scholar
  18. Wadati M (1972) The exact solution of the modified Korteweg-de Vries equation. J Phys Soc Jpn 32:1681–1687CrossRefADSGoogle Scholar
  19. Wadati M (2001) Introduction to solitons. Pramana J Phys 57(5/6):841–847CrossRefADSGoogle Scholar
  20. Wazwaz AM (2007a) Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh-coth method. Appl Math Comput 190:633–640zbMATHMathSciNetCrossRefGoogle Scholar
  21. Wazwaz AM (2007b) Multiple-front solutions for the Burgers equation and the coupled Burgers equations. Appl Math Comput 190:1198–1206zbMATHMathSciNetCrossRefGoogle Scholar
  22. Wazwaz AM (2007c) New solitons and kink solutions for the Gardner equation. Commun Nonlinear Sci Numer Simul 12(8):1395–1404zbMATHMathSciNetCrossRefADSGoogle Scholar
  23. Wazwaz AM (2007d) Multiple-soliton solutions for the Boussinesq equation. Appl Math Comput 192:479–486zbMATHMathSciNetCrossRefGoogle Scholar
  24. Wazwaz AM (2008a) The Hirota’s direct method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Ito seventh-order equation. Appl Math Comput 199(1):133–138zbMATHMathSciNetCrossRefGoogle Scholar
  25. Wazwaz AM (2008b) Multiple-front solutions for the Burgers-Kadomtsev-Petvisahvili equation. Appl Math Comput 200:437–443zbMATHMathSciNetCrossRefGoogle Scholar
  26. Wazwaz AM (2008c) Multiple-soliton solutions for the Lax-Kadomtsev-Petvisahvili (Lax- KP) equation. Appl Math Comput 201(1/2):168–174zbMATHMathSciNetCrossRefGoogle Scholar
  27. Wazwaz AM (2008d) The Hirota’s direct method for multiple-soliton solutions for three model equations of shallow water waves. Appl Math Comput 201(1/2):489–503zbMATHMathSciNetCrossRefGoogle Scholar
  28. Wazwaz AM (2008e) Multiple-soliton solutions of two extended model equations for shallow water waves. Appl Math Comput 201(1/2):790–799zbMATHMathSciNetCrossRefGoogle Scholar
  29. Wazwaz AM (2008f) Single and multiple-soliton solutions for the (2 + 1)-dimensional KdV equation. Appl Math Comput 204:20–26zbMATHMathSciNetCrossRefGoogle Scholar
  30. Wazwaz AM (2008g) Solitons and singular solitons for the Gardner-KP equation. Appl Math Comput 204:162–169zbMATHMathSciNetCrossRefGoogle Scholar
  31. Wazwaz AM (2008h) Regular soliton solutions and singular soliton solutions for the modified Kadomtsev-Petviashvili equations. Appl Math Comput 204:817–823zbMATHMathSciNetCrossRefGoogle Scholar
  32. Wazwaz AM (2009a) Adomian decomposition method applied to nonlinear evolution equations in solitons theory. In: Meyers RA (ed) Encyclopedia of complexity and systems science. Springer, HeibelbergGoogle Scholar
  33. Wazwaz AM (2011a) Multi-front waves for extended form of modified Kadomtsev–Petviashvili equation. Appl Math Mech 32(7):875–880Google Scholar
  34. Wazwaz AM (2011b) Distinct kinds of multiple soliton solutions for a (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation. Phys Scr 84:055006Google Scholar
  35. Wazwaz AM (2011c) A new fifth order nonlinear integrable equation: multiple soliton solutions. Physica Scripta 83:015012Google Scholar
  36. Wazwaz AM (2011d) A new generalized fifth-order nonlinear integrable equation. Phys Scr 83:035003Google Scholar
  37. Wazwaz AM (2012) Multiple soliton solutions for some (3 + 1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy. Appl Math Lett 25(1):1936–1940zbMATHMathSciNetCrossRefGoogle Scholar
  38. Zabusky NJ, Kruskal MD (1965) Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243zbMATHCrossRefADSGoogle Scholar
  39. Zhang Y, Tam H (2010) Three kinds of coupling integrable couplings of the Kortewegde Vries hierarchy of evolution equations. J Math Phys 51:043510MathSciNetCrossRefADSGoogle Scholar
  40. Zhaqilao (2012) A generalized AKNS hierarchy, bi-Hamiltonian structure, and Darboux transformation. Commun Nonlinear Sci Numer Simul 17:2319–2332zbMATHMathSciNetCrossRefADSGoogle Scholar

Books and Reviews

  1. Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  2. Drazin PG, Johnson RS (1996) Solitons: an introduction. Cambridge University Press, CambridgeGoogle Scholar
  3. Hirota R (2004) The direct method in soliton theory. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  4. Wazwaz AM (2002) Partial differential equations: methods and applications. Balkema Publishers, LisseGoogle Scholar
  5. Wazwaz AM (2009b) Partial differential equations: methods and solitary waves theory. Springer/HEP, Berlin/BeijingGoogle Scholar
  6. Wazwaz AM (1997) A first course in integral equations. World Scientific, SingaporezbMATHCrossRefGoogle Scholar
  7. Wazwaz AM (2011e) Linear and nonlinear integral equations. Springer/HEP, Berlin/BeijingGoogle Scholar
  8. Whitham GB (1999) Linear and nonlinear waves. Wiley–Interscience Series, New YorkzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsSaint Xavier UniversityChicagoUSA