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



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...

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsSaint Xavier UniversityChicagoUSA