# Asymptotic solving of a hyperbolic equation

The solition and anti-solition nonlinear equation is given as:

My problem is that, how do we get the next equation after considering asyptotic behaviour?

Resource: (solition) at page 38

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What is $\Delta$? Is $u$ related to $\bar{t}$ in any way? – Antonio Vargas Aug 14 '13 at 21:06
Can you please check this book (users.ugent.be/~ddudal/sol&mon.pdf) at page 38? – Complex Guy Aug 15 '13 at 19:14
I'm confused why you ask what $\Delta$ is. You yourself linked me to the pdf which says $$\Delta = ((1-u^2)/u)\ln u.$$ – Antonio Vargas Aug 27 '13 at 20:00
Ohh my bad, then we just need to transform the hyperbolic term into exponential? – Complex Guy Aug 27 '13 at 20:02
It seems plausible, but I haven't carried out the calculations. – Antonio Vargas Aug 27 '13 at 20:03

This is due to the fact that: $$\lim_{x\to-\infty}\sinh(x) \sim -e^{-x}$$ And that: $$\cosh x =\frac{1}{2}e^{-x} + \frac{1}{2}e^x$$
Shouldn't that be $e^{-x}/2$? – marty cohen Aug 31 '13 at 6:33
What will happen when you impose the $- \infty$? – Complex Guy Aug 31 '13 at 10:58