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

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.