Travelling wave solutions to the Kuramoto–Sivashinsky equation

Combining the approaches given by Baldwin [Baldwin D et al. Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs. J Symbol Comput 2004;37:669–705], Peng [Peng YZ. A polynomial expansion method and new general solitary wave solutions to KS equation. Comm Theor Phys 2003;39:641–2] and by Schürmann [Schürmann HW, Serov VS. Weierstrass’ solutions to certain nonlinear wave and evolution equations. Proc progress electromagnetics research symposium, 28–31 March 2004, Pisa. p. 651–4; Schürmann HW. Traveling-wave solutions to the cubic–quintic nonlinear Schrödinger equation. Phys Rev E 1996;54:4312–20] leads to a method for finding exact travelling wave solutions of nonlinear wave and evolution equations (NLWEE). The first idea is to generalize ansätze given by Baldwin and Peng to find elliptic solutions of NLWEEs. Secondly, conditions used by Schürmann to find physical (real and bounded) solutions and to discriminate between periodic and solitary wave solutions are used. The method is shown in detail by evaluating new solutions of the Kuramoto–Sivashinsky equation.