# Is the tanh-Gordon equation an integrable system?

+ 2 like - 1 dislike
862 views

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?

asked Oct 21, 2019
edited Oct 22, 2019

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

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\varnothing$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). Please complete the anti-spam verification