On the example of a nite dimensional approximation of the Kuramoto-Sivashinsky equation we show how topological methods may be successfully used in computer assisted proofs of the existence of heteroclinic connections in ordinary diierential equations.

A model consisting of a mixed Kuramoto-Sivashinsky-Kortewegde Vries equation, linearly coupled to an extra linear dissipative equation has been proposed in [1] in order to describe the surface waves on multi-layered liquid films and stability criteria are discussed using wave mode analysis. In this paper, we study the linear stability of solutions to the model from the viewpoint of energy estim...

For the 1-dimensional Kuramoto–Sivashinsky equation with random forcing term, existence and uniqueness of solutions is proved. Then, the Markovian semigroup is well defined; its properties are analyzed, in order to provide sufficient conditions for existence and uniqueness of invariant measures for this stochastic equation. Finally, regularity results are obtained by means of Girsanov theorem.

We analyse the nonlinear Kuramoto-Sivashinsky equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately models the dynamics and may be constructed systematically. The theory is applied after dividing the physical domain into small elements by introducing insulat...