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

Parkes, E.J. (2010) Observations on the tanh-coth expansion method for finding solutions to nonlinear evolution equations. Applied Mathematics and Computation, 217. pp. 1749-1754. ISSN 0096-3003

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Abstract

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.