I am reading a paper on the sine-gordon model. The solution for a two kink solution is given as: $$\phi=4\arctan\left(\frac{\sinh\frac{1}{2}(\theta_1-\theta_2)}{(a_{12})^\frac{1}{2}\cosh\frac{1}{2}(\theta_1+\theta_2)}\right)$$ where $\theta_1=m\gamma_1(x-v_1t)+\delta_1$, and similarly for $\theta_2$. Here $a_{12}, m, \gamma, \delta$ are all constants.

The author says that the limit as $t \to \pm \infty$, is

$$\lim \phi(x,t) \to 4\arctan(\exp [\theta_1+\eta_1^{\pm}])+4\arctan(\exp[\theta_2+\eta_2^{\pm}]) $$

where $\eta_1^{\pm}=-\eta_2^{\pm}=\pm\frac{1}{2}\ln(a_{12})$

I am having a hard time proving this. I did the expansion of the hyperbolic functions in terms of the exponents, and tried to limit the appropriate terms to 1, but I am not getting an answer.


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