Observations on the tanh–coth expansion method for finding solutions to nonlinear evolution equations

The ‘tanh–coth expansion method’ for finding solitary travelling-wave solutions to nonlinear evolution equations has been used extensively in the literature. It is a natural extension to the basic tanh-function expansion method which was developed in the 1990s. It usually delivers three types of solution, namely a tanh-function expansion, a coth-function expansion, and a tanh–coth expansion. It is known that, for every tanh-function expansion solution, there is a corresponding coth-function expansion solution. It is shown that there is a tanh–coth expansion solution that is merely a disguised version of the coth solution. In many papers, such tanh–coth solutions are erroneously claimed to be ‘new’. However, other tanh–coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh-function method. Similar remarks apply to tan, cot and tan–cot expansion solutions.