On the Variational Approach for Analyzing the Stability of Solutions of Evolution Equations

The eigenvalue problems arise in the analysis of stability of traveling waves or rest state solutions are currently dealt with, using the Evans function method. In the literature, it had been shown that, use of this method is not straightforward even in very simple examples. Here an extended variational method to solve the eigenvalue problem for the higher order differential equations is suggested. The extended method is matched to the well known variational iteration method. The criteria for validity of the eigen-functions and eigenvalues obtained is presented. Attention is focused to find eigenvalue and eigenfunction solutions of the Kuramoto-Slivashinsky and (K[p,q]) equation.