Hi all. I have some questions about the dispersion relation in the study of waves. First of all, why do we always assume a plane wave solution when we want to obtain a dispersion relation? Second, is "assuming a plane wave solution" a general way to obtian all dispersion relations? for both linear and nonlinear wave equations? So, what is the dispersion relation for the KdV equation? I can hardly see anyone deriving the dispersio relation for nonlinear equations like KdV and NLS etc, why?