(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am currently trying to show a the sine-Gordon equation is satisfied by a 'soliton-antisoliton' solution. Basically, I need to differentiate twice w.r.t. t and x separately and plug in everything as usual but my expression is getting extremely complicated and I don't know how to deal with the sin(4arctan(...)) term (this will become clear in the relevant equations).

2. Relevant equations

sine-Gordon equation: (∂^2 θ)/(∂t^2) - (∂^2 θ)/(∂x^2) + sin(θ)=0

where the solution is θ(x,t)=4arctan[ (sinh{(ut)/SQRT(1-u^2)}) / (u cosh{x/SQRT(1-u^2)}) ]

3. The attempt at a solution

My attempt is rather messy so it might be a better idea if I don't write my working. However, what I have tried is differentiating directly but the expression didn't simplify and I didn't know what to do with sin(4arctan(...)). I was then told to rewrite sinh and cosh as exp functions but that still leaves a very complicated expression. I even tried doing it on Matlab (with cosh and sinh, not exp) but the LHS of the sine-Gordon equation doesn't simplify to 0 on Matlab, possibly due to the sine and arctan combining.. causing complication? Could anyone give me some tips?

Many thanks. I hope I have posted this in the correct thread!

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# Homework Help: Showing the sine-Gordon equation is satisfied.

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