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Using the Painlevé test, it is shown that the only integrable nonlinear Klein–Gordon equations uₓₜ = f(u) with f a linear combination of exponentials are the Liouville, sine-Gordon (or sink-Gordon) and Mikhailov equations. In particular, the double sine-Gordon equation is not integrable.
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Clarkson et al. (1986) studied this question.
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