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The tanh-Gordon equation is the following PDE:

$$ u_{xx}-u_{tt}= tanh(u)(1-tanh^2 (u))$$

There is a Langrangian $L$:

$$ L(u,u_x,u_t) = u_t^2 -u_x^2 - (tanh (u))^2$$

In first order, we have the Klein-Gordon equation. Moreover, it exists a soliton:

$$u(x,t)= argsinh (e^{ax+bt+c})$$

with $a^2 -b^2=4$.

Is the tanh-Gordon equation an integrable system like the sine-Gordon equation?

No (https://epubs.siam.org/doi/pdf/10.1137/0517058).

But $tanh$ is not a linear combination of exponentials, so that the question remains.

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