International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 70, Pages 3839-3848

Multiplicity and structures for traveling wave solutions of the Kuramoto-Sivashinsky equation

Bao-Feng Feng

Department of Mathematics, The University of Texas – Pan American, Edinburg 78541-2999, TX, USA

Received 6 May 2004

Copyright © 2004 Bao-Feng Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Kuramoto-Sivashinsky (KS) equation is known as a popular prototype to represent a system in which the transport of energy through nonlinear mode coupling produces a balance between long wavelength instability and short wavelength dissipation. Existing numerical results indicate that the KS equation admits three classes (namely, regular shock, oscillatory shock, and solitary wave) of nonperiodic traveling wave solutions and families of multiple solutions in each class. However, the details of multiple solutions are still unclear because of numerical accuracy. In this work, a rational spectral approach is used to compute these multiple traveling wave solutions. Owing to the high accuracy of the employed method, the new families of regular shock waves are found and the fine structure of each family is recognized.