Asymptotic solving of a hyperbolic equation

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The solition and anti-solition nonlinear equation is given as:enter image description here

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

Resource: (solition) at page 38

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This is due to the fact that: $$\lim_{x\to-\infty}\sinh(x) \sim -e^{-x}$$ And that: $$\cosh x =\frac{1}{2}e^{-x} + \frac{1}{2}e^x$$